Publication#
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Submitted or under preparation#
36. chen.foondun.ea:23:global#
[CFHS23] Le Chen, Mohammud Foondun, Jingyu Huang & Michael Salins (2023) ‘Global solution for superlinear stochastic heat equation on \(\mathbb{R}^d\) under Osgood-type conditions’, preprint arXiv:2310.02153
Details
Global solution for superlinear stochastic heat equation on \(\mathbb{R}^d\) under Osgood-type conditions
Le Chen, Mohammud Foondun, Jingyu Huang, and Michael Salins
Abstract: We study the stochastic heat equation (SHE) on \(\mathbb{R}^d\) subject to a centered Gaussian noise that is white in time and colored in space.The drift term is assumed to satisfy an Osgood-type condition and the diffusion coefficient may have certain related growth. We show that there exists a random field solution that does not explode in finite time. This complements and improves upon recent results on blow-up of solutions to stochastic partial differential equations.
Keywords. Global solution; Stochastic heat equation; Reaction-diffusion; Dalang’s condition; superlinear growth; Osgood-type conditions.
[CFHS23] Le Chen, Mohammud Foondun, Jingyu Huang & Michael Salins (2023) ‘Global solution for superlinear stochastic heat equation on \(\mathbb{R}^d\) under Osgood-type conditions’, preprint arXiv:2310.02153
@article{chen.foondun.ea:23:global,
title = {Global solution for superlinear stochastic heat equation on $\mathbb{R}^d$ under Osgood-type conditions},
author = {Le Chen and Mohammud Foondun and Jingyu Huang and Michael Salins},
year = {2023},
month = {October},
journal = {preprint arXiv:2310.02153},
url = {http://arXiv.org/abs/2310.02153}
}
References: Athreya et al. [AJM21]; Cerrai and Röckner [CR04]; Chen and Huang [CH19a]; Chen and Huang [CH23b]; Da Prato and Zabczyk [DPZ14]; Dalang et al. [DKM+09]; Dalang [Dal99]; Dalang et al. [DKZ19]; Fernández Bonder and Groisman [FBG09]; Foondun and Nualart [FN21]; Foondun and Nualart [FN22]; Fujita [Fuj66]; Geiß and Manthey [GM94]; Khoshnevisan [Kho14]; Millet and Sanz-Solé [MSS21]; Mueller [Mue91b]; Olver et al. [OLBC10]; Ondreját [Ond04]; Osgood [Osg98]; Quittner and Souplet [QS19]; Salins [Sal22a]; Salins [Sal22b]; Salins [Sal22c]; Seidler [Sei10]; Shang and Zhang [SZ21]; Shang and Zhang [SZ22]; Walsh [Wal86]; Zhang [Zha10];
35. chen.xia:23:asymptotic#
[CX23] Le Chen & Panqiu Xia (2023) ‘Asymptotic properties of stochastic partial differential equations in the sublinear regime’, preprint arXiv:2306.06761
Details
Asymptotic properties of stochastic partial differential equations in the sublinear regime
Le Chen and Panqiu Xia
Abstract: In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds and almost sure spatial asymptotic properties for the solutions. These results shed light on the smoothing intermittency effect under weak diffusion (i.e., sublinear growth) previously observed by Zeldovich et al. [ZMRS87]. The sample-path spatial asymptotics obtained in this paper partially bridge a gap in earlier works of Conus et al. [CJK13, CJKS13b], which focused on two extreme scenarios: a linear diffusion coefficient and a bounded diffusion coefficient. Our approach is highly robust and applicable to a variety of stochastic partial differential equations, including the one-dimensional stochastic wave equation and the stochastic fractional diffusion equations.
MSC 2010 subject classifications: Primary 60H15; Secondary 35R60.
Keywords: stochastic partial differential equations, sublinear growth, asymptotic concavity, moment bounds, intermittency, spatial asymptotics.
[CX23] Le Chen & Panqiu Xia (2023) ‘Asymptotic properties of stochastic partial differential equations in the sublinear regime’, preprint arXiv:2306.06761
@article{chen.xia:23:asymptotic,
title = {Asymptotic properties of stochastic partial differential equations in the sublinear regime},
author = {Le Chen and Panqiu Xia},
year = {2023},
month = {June},
journal = {preprint arXiv:2306.06761},
url = {http://arXiv.org/abs/2306.06761}
}
References: Amir et al. [ACQ11]; Balan and Conus [BC16]; Bauinov and Simeonov [BS92]; Bihari [Bih56]; Carmona and Molchanov [CM94]; Chen and Dalang [CD14a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen and Eisenberg [CE22a]; Chen and Eisenberg [CE22b]; Chen et al. [CGS22]; Chen et al. [CHN19]; Chen and Huang [CH19a]; Chen and Huang [CH23b]; Chen and Kim [CK19]; Chen and Kim [CK20]; Chen et al. [CFHS23]; Chen [Che15]; Conus et al. [CJKS13b]; Conus and Khoshnevisan [CK12]; Conus et al. [CJK13]; Conus et al. [CJKS14]; Corwin [Cor12]; Dalang et al. [DKM+09]; Dalang [Dal99]; Dalang et al. [DKZ19]; Dalang and Mueller [DM09]; Dalang and Sanz-Solé [DSS05]; Dawson et al. [DIP89]; Dawson [Daw93]; Etheridge [Eth00]; Fang and Zhang [FZ05]; Foondun and Khoshnevisan [FK09]; Grafakos [Gra14]; Guo et al. [GSS23]; Guttorp and Gneiting [GG06]; Hu and Wang [HW21]; Hu et al. [HWXZ23]; Huang et al. [HLN17a]; Joseph et al. [JKM17]; Kallenberg [Kal02]; Khoshnevisan [Kho14]; Khoshnevisan et al. [KKX17]; König [Kon16]; Konno and Shiga [KS88]; LaSalle [LaS49]; Mueller [Mue91a]; Mueller et al. [MMP14]; Mueller et al. [MMQ11]; Mueller et al. [MMR21]; Mueller and Perkins [MP92]; Mytnik [Myt98]; Mytnik and Perkins [MP11]; Mytnik et al. [MPS06]; Niculescu and Persson [NP18]; Olver et al. [OLBC10]; Perkins [Per02]; Reimers [Rei89]; Salins [Sal22b]; Sanz-Solé and Sarrà [SSS02]; Shiga [Shi94]; Walsh [Wal86]; Xiong [Xio13]; Zeldovich et al. [ZRS90]; Zeldovich et al. [ZMRS87];
Published or to appear#
34. chen.ouyang.ea:23:parabolic#
[COV23] Le Chen, Cheng Ouyang & William Vickery (2023) ‘Parabolic Anderson model with colored noise on torus’, preprint arXiv:2308.10802
Details
Parabolic Anderson model with colored noise on torus
Le Chen, Cheng Ouyang, William Vickery
To appear in Bernoulli
Abstract: We construct an intrinsic family of Gaussian noises on the \(d\)-dimensional flat torus \(\mathbb{T}^d\). It is the analogue of the colored noise on \(\mathbb{R}^d\) and allows us to study stochastic PDEs on the torus in the Itô sense in high dimensions. With this noise, we consider the parabolic Anderson model (PAM) with measure-valued initial conditions and establish some basic properties of the solution, including a sharp upper and lower bound for the moments and Hölder continuity in space and time. The study of the toy model of \(\mathbb{T}^d\) in the present paper is a first step in our effort to understand how geometry and topology play a role in the behavior of stochastic PDEs on general (compact) manifolds.
MSC 2010 subject classifications: Primary 60H15, Secondary: 60G60, 37H15.
Keywords: stochastic heat equation on torus; Dalang’s condition; measure-valued initial condition; Brownian bridge; moment asymptotics; intermittency; moment Lyapunov exponent; theta function.
[COV23] Le Chen, Cheng Ouyang & William Vickery (2023) ‘Parabolic Anderson model with colored noise on torus’, preprint arXiv:2308.10802
@article{chen.ouyang.ea:23:parabolic,
title = {Parabolic Anderson model with colored noise on torus},
author = {Le Chen and Cheng Ouyang and William Vickery},
year = {2023},
month = {August},
journal = {preprint arXiv:2308.10802},
url = {http://arXiv.org/abs/2308.10802}
}
References: Balan and Chen [BC18]; Baudoin et al. [BOTW22]; Baxter and Brosamler [BB76]; Brosamler [Bro83]; Candil et al. [CCL23]; Chen and Dalang [CD15b]; Chen and Huang [CH19a]; Chen and Kim [CK19]; Cosco et al. [CSZ21]; Dalang [Dal99]; Foondun and Khoshnevisan [FK13]; Gelbaum [Gel14]; Hu et al. [HNS11]; Mayorcas and Singh [MS23]; Olver et al. [OLBC10]; Walsh [Wal86];
33. chen.kuzgun.ea:23:on#
[CKMX23] Le Chen, Sefika Kuzgun, Carl Mueller & Panqiu Xia (2023) ‘On the radius of self-repellent fractional Brownian motion’, preprint arXiv:2308.10889
Details
On the radius of self-repellent fractional Brownian motion
Le Chen, Sefika Kuzgun, Carl Mueller, and Panqiu Xia
To appear in Journal of Statistical Physics
Abstract: We study the radius of gyration \(R_T\) of a self-repellent fractional Brownian motion \(\left\{B^H_t\right\}_{0\le t\le T}\) taking values in \(\mathbb{R}^d\). Our sharpest result is for \(d=1\), where we find that with high probability,
For \(d>1\), we provide upper and lower bounds for the exponent \(\nu\), but these bounds do not match.
[CKMX23] Le Chen, Sefika Kuzgun, Carl Mueller & Panqiu Xia (2023) ‘On the radius of self-repellent fractional Brownian motion’, preprint arXiv:2308.10889
MSC 2010 subject classifications: Primary 60H15. Secondary 35R60.
Keywords: fractional Brownian motion, self-avoiding, self-repellent, Girsanov theorem.
@article{chen.kuzgun.ea:23:on,
title = {On the radius of self-repellent fractional Brownian motion},
author = {Le Chen and Sefika Kuzgun and Carl Mueller and Panqiu Xia},
year = {2024},
month = {January},
journal = {Journal of Statistical Physics},
url = {https://link.springer.com/article/10.1007/s10955-023-03227-y}
}
References: Adler and Taylor [AT07]; Bauerschmidt et al. [BDCGS12]; Biagini et al. [BHZ08]; Biswas and Cherayil [BC95b]; Bock et al. [BBC+15]; Bolthausen [Bol90]; Bornales et al. [BOS13]; Brydges and Spencer [BS85]; Domb and Joyce [DJ72]; Edwards [Edw65]; Fixman [Fix62]; Greven and den Hollander [GdH93]; Grothaus et al. [GOdSS11]; Hara and Slade [HS91]; Hara and Slade [HS92]; Hu and Nualart [HN05]; Madras [Mad14]; Madras and Slade [MS93]; Mueller and Neuman [MN23]; Norros et al. [NVV99]; Rosen [Ros87]; Slade [Sla19];
32. chen.eisenberg:22:invariant#
[CE22b] Le Chen & Nicholas Eisenberg (2022) ‘Invariant measures for the nonlinear stochastic heat equation with no drift term’, J. Theoret. Probab. (pending revision, preprint arXiv:2209.04771)
Details
Invariant measures for the nonlinear stochastic heat equation with no drift term
Le Chen and Nicholas Eisenberg
To appeare in Journal of Theoretical Proabibility
Abstract: This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation
where \(b\) is assumed to be a globally Lipschitz continuous function and the noise \(\dot{W}\) is a centered and spatially homogeneous Gaussian noise that is white in time. We identify a set of nearly optimal conditions on the initial data, the correlation measure of the noise, and the weight function \(\rho\), which together guarantee the existence of an invariant measure in the weighted space \(L^2_\rho(\mathbb{R}^d)\). In particular, our result covers the textit{parabolic Anderson model} (i.e., the case when \(b(u) = \lambda u\)) starting from the Dirac delta measure.
[CE22b] Le Chen & Nicholas Eisenberg (2022) ‘Invariant measures for the nonlinear stochastic heat equation with no drift term’, J. Theoret. Probab. (pending revision, preprint arXiv:2209.04771)
MSC 2010 subject classifications: Primary 60H15. Secondary 60H07, 60F05.
keyworkds: stochastic heat equation, parabolic Anderson model, invariant measure, Dirac delta initial condition, weighted \(L^2\) space, Matérn class of correlation functions, Bessel kernel
@article{chen.eisenberg:22:invariant,
title = {Invariant measures for the nonlinear stochastic heat equation with no drift term},
author = {Le Chen and Nicholas Eisenberg},
year = {2022},
month = {September},
journal = {J. Theoret. Probab. Feb. 2024},
url = {http://arXiv.org/abs/2209.04771}
}
References: Amir et al. [ACQ11]; Assing and Manthey [AM03]; Billingsley [Bil99]; Brzezniak and G\c atarek [BGcatarek99]; Carmona and Molchanov [CM94]; Cerrai [Cer01]; Cerrai [Cer03]; Chen and Dalang [CD15c]; Chen and Huang [CH19a]; Chen et al. [CHKK19]; Chen and Kim [CK19]; Da Prato et al. [DPKwapienZ87]; Da Prato and Zabczyk [DPZ96]; Da Prato and Zabczyk [DPZ14]; Dalang et al. [DKM+09]; Dalang [Dal99]; Dalang and Quer-Sardanyons [DQS11]; Dunlap et al. [DGRZ21]; Eckmann and Hairer [EH01]; Foondun and Khoshnevisan [FK09]; Grafakos [Gra14]; Gu and Li [GL20]; Loh et al. [LSW21]; Misiats et al. [MSY16]; Misiats et al. [MSY20]; Olver et al. [OLBC10]; Peszat and Zabczyk [PZ97]; Sanz-Solé and Sarrà [SSS02]; Stein [Ste99]; Tessitore and Zabczyk [TZ98a]; Walsh [Wal86];
31. chen.guo.ea:22:moments#
[CGS22] Le Chen, Yuhui Guo & Jian Song (2022) ‘Moments and asymptotics for a class of SPDEs with space-time white noise’, preprint arXiv:2206.10069, to appear in Trans. Amer. Math. Soc.
Details
Moments and asymptotics for a class of SPDEs with space-time white noise
Le Chen, Yuhui Guo, and Jian Song
To appeare in Transactions of American Mathematical Scociety
Abstract: In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition:
with constants \(\lambda\ne 0\) and \(\nu>0\), where \(\partial^{\beta}_t\) is the Caputo fractional derivative of order \(\beta\in(0,2]\), \(I_{t}^{\gamma}\) refers to the Riemann-Liouville integral of order \(\gamma \ge 0\), and \(\left(-\Delta\right)^{\alpha/2}\) is the standard fractional/power of Laplacian with \(\alpha>0\). We concentrate on the scenario where the noise \(\dot{W}\) is the space-time white noise. The existence and uniqueness of solution in the Itô-Skorohod sense is obtained under Dalang’s condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the \(p\)-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the \(p\)-th moment Lyapunov exponents. In particular, by letting \(\beta = 2\), \(\alpha = 2\), \(\gamma = 0\), and \(d = 1\), we confirm the following standing conjecture for the stochastic wave equation:
The method for the lower bounds is inspired by a recent work of Hu and Wang [HW22], where the authors focus on the space-time colored Gaussian noise case.
MSC 2010 subject classifications: Primary 60H15; Secondary 60G60, 26A33, 37H15, 60H07.
Keywords: stochastic partial differential equation, stochastic heat/wave equation, space-time white noise, Dalang’s condition, moment asymptotics, intermittency, moment Lyapunov exponent.
[CGS22] Le Chen, Yuhui Guo & Jian Song (2022) ‘Moments and asymptotics for a class of SPDEs with space-time white noise’, preprint arXiv:2206.10069, to appear in Trans. Amer. Math. Soc.
@article{chen.guo.ea:22:moments,
title = {Moments and asymptotics for a class of SPDEs with space-time white noise},
author = {Le Chen and Yuhui Guo and Jian Song},
year = {2022},
month = {June},
journal = {preprint arXiv:2206.10069, to appear in Trans. Amer. Math. Soc.},
url = {https://www.arxiv.org/abs/2206.10069}
}
References: Ablowitz and Fokas [AF03a]; Balan et al. [BCC22]; Balan and Song [BS19]; Carlen and Krée [CK91]; Carmona and Molchanov [CM94]; Chen [Che17a]; Chen and Hu [CH23a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15c]; Chen and Dalang [CD15b]; Chen and Eisenberg [CE23]; Chen and Hu [CH22]; Chen et al. [CHHH17]; Chen et al. [CHN19]; Chen et al. [CHN21]; Chen and Huang [CH19a]; Chen and Kim [CK19]; Chen and Kim [CK20]; Chen [Che15]; Chen et al. [CHSS18]; Chen et al. [CHSX15]; Conus et al. [CJKS13a]; Conus and Khoshnevisan [CK12]; Cranston and Molchanov [CM07]; Dalang et al. [DKM+09]; Dalang [Dal99]; Dalang and Mueller [DM09]; Dalang et al. [DMT08]; Debbi [Deb06]; Debbi and Dozzi [DD05]; Diethelm [Die10]; Eidelman et al. [EIK04]; Erdélyi et al. [EMOT54]; Foondun and Khoshnevisan [FK09]; Fox [Fox61]; Hochberg [Hoc78]; Hu [Hu17]; Hu and Wang [HW22]; Khoshnevisan [Kho14]; Kilbas and Saigo [KS04]; Kilbas et al. [KST06]; Krylov [Kry60]; Mathai et al. [MSH10]; Mijena and Nane [MN15]; Olver et al. [OLBC10]; Podlubny [Pod99]; Prudnikov et al. [PBM90]; Samko et al. [SKM93]; Song et al. [SSX20]; Walsh [Wal86]; Zeldovich et al. [ZMRS88];
30. candil.chen.ea:23:parabolic#
[CCL23] David Candil, Le Chen & Cheuk Yin Lee (2023) ‘Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds’, preprint arXiv:2301.06435, to appear in Stoch. Partial Differ. Equ. Anal. Comput.
Details
Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds
David Candil, Le Chen, Cheuk Yin Lee
To appeare in stochastic Partial Differential Equations: Analysis and Computation
Abstract: We consider nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish existence, uniqueness and moment bounds of the random field solution under measure-valued initial data \(\nu\). We also study the two-point correlation function of the solution and obtain explicit upper and lower bounds. For \(C^{1, \alpha}\)-domains with Dirichlet condition, the initial data \(\nu\) is not required to be a finite measure and the moment bounds can be improved under the weaker condition that the leading eigenfunction of the differential operator is integrable with respect to \(|\nu|\). As an application, we show that the solution is fully intermittent for sufficiently high level \(\lambda\) of noise under the Dirichlet condition, and for all \(\lambda > 0\) under the Neumann condition.
MSC 2010 subject classifications: Primary 60H15; Secondary 35R60.
Keywords: Parabolic Anderson model, stochastic heat equation, Dirichlet/Neumann boundary conditions, Lipschitz domain, intermittency, two-point correlation, rough initial conditions.
[CCL23] David Candil, Le Chen & Cheuk Yin Lee (2023) ‘Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds’, preprint arXiv:2301.06435, to appear in Stoch. Partial Differ. Equ. Anal. Comput.
@article{candil.chen.ea:23:parabolic,
title = {Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds},
author = {David Candil and Le Chen and Cheuk Yin Lee},
year = {2023},
month = {January},
journal = {preprint arXiv:2301.06435, to appear in Stoch. Partial Differ. Equ. Anal. Comput.},
url = {http://arXiv.org/abs/2301.06435}
}
References: Amir et al. [ACQ11]; Balan and Conus [BC16]; Balan et al. [BJQS17]; Candil [Can22]; Carmona and Molchanov [CM94]; Cerrai [Cer01]; Chen and Dalang [CD15c]; Chen and Dalang [CD15b]; Chen et al. [CHN17]; Chen et al. [CHN19]; Chen et al. [CHN21]; Chen and Huang [CH19a]; Chen and Kim [CK17]; Chen and Kim [CK19]; Conus et al. [CJKS14]; Corwin [Cor12]; Da Prato and Zabczyk [DPZ14]; Dalang [Dal99]; Dalang and Quer-Sardanyons [DQS11]; Davies [Dav90]; Folland [Fol95]; Foondun and Joseph [FJ14]; Foondun and Khoshnevisan [FK09]; Foondun and Nualart [FN15]; Grisvard [Gri85]; Guerngar and Nane [GN20]; Henrot and Pierre [HP05]; Khoshnevisan [Kho14]; Khoshnevisan and Kim [KK15]; Liu et al. [LTF17]; Mueller [Mue91b]; Mueller and Nualart [MN08]; Nualart [Nua18]; Olver et al. [OLBC10]; Ouhabaz and Wang [OW07]; Polyanin and Nazaikinskii [PN16]; Prévôt and Röckner [PR07]; Riahi [Ria13]; Saloff-Coste [SC92]; Saloff-Coste [SC10]; Shiga [Shi94]; Trèves [Treves75]; Walsh [Wal86];
29. chen.eisenberg:23:interpolating#
[CE23] Chen, Le & Eisenberg, Nicholas (2023) ‘Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics’, Stoch. Partial Differ. Equ. Anal. Comput. 11, 1203–1253. <https://doi.org/10.1007/s40072-022-00258-6>
Details
Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics
Le Chen and Nicholas Eisenberg
Abstract: In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global \(L^p(\Omega)\)-solution exists for all \(p\ge 2\). In this case, we derive exact moment asymptotics following the same strategy as that in a recent work by Balan et al [BCC22]. In the case when there exists only a local solution, we determine the precise deterministic time, \(T_2\), before which a unique \(L^2(\Omega)\)-solution exists, but after which the series corresponding to the \(L^2(\Omega)\) moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.
MSC 2010 subject classifications: Primary 60H15; Secondary 60H07, 37H15.
Keywords: stochastic partial differential equations, Caputo derivatives, Riemann-Liouville fractional integral, fractional Laplacian, Malliavin calculus, Skorohod integral, exact moment asymptotics, time-independent Gaussian noise, white noise, global and local solutions.
[CE23] Chen, Le & Eisenberg, Nicholas (2023) ‘Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics’, Stoch. Partial Differ. Equ. Anal. Comput. 11, 1203–1253. <https://doi.org/10.1007/s40072-022-00258-6>
@article{chen.eisenberg:23:interpolating,
author = {Chen, Le and Eisenberg, Nicholas},
title = {Interpolating the stochastic heat and wave equations with time-independent noise: solvability and exact asymptotics},
journal = {Stoch. Partial Differ. Equ. Anal. Comput.},
fjournal = {Stochastic Partial Differential Equations. Analysis and Computations},
volume = {11},
year = {2023},
number = {3},
pages = {1203--1253},
issn = {2194-0401},
mrclass = {60H15 (35R11 37H15 60H07)},
mrnumber = {4624137},
doi = {10.1007/s40072-022-00258-6},
url = {https://doi.org/10.1007/s40072-022-00258-6}
}
References: Balan et al. [BCC22]; Balan and Song [BS17]; Bass et al. [BCR09]; Chen [Che17a]; Chen et al. [CHN19]; Chen et al. [CHHH17]; Chen [Che12]; Chen [Che17b]; Chen [Che19]; Chen and Li [CL04]; Chen et al. [CHSX15]; Chen et al. [CHSS18]; Chen et al. [CDOT21]; Dalang [Dal09]; Hairer and Labbé [HL15]; Hairer and Labbé [HL18]; Hu [Hu01]; Hu [Hu02]; Lê [Le16]; Lieb and Loss [LL01]; Mijena and Nane [MN15]; Nualart and Nualart [NN18];
28. chen.huang:23:superlinear#
[CH23b] Chen, Le & Huang, Jingyu (2023) ‘Superlinear stochastic heat equation on \(\mathbb{R}^d\)’, Proc. Amer. Math. Soc. 151, 4063–4078. <https://doi.org/10.1090/proc/16436>
Details
Superlinear stochastic heat equation on \(\mathbb{R}^d\)
Le Chen and Jingyu Huang
Abstract: In this paper, we study the stochastic heat equation (SHE) on \(\mathbb{R}^d\) subject to a centered Gaussian noise that is white in time and colored in space. We establish the existence and uniqueness of the random field solution in the presence of locally Lipschitz drift and diffusion coefficients, which can have certain superlinear growth. This is a nontrivial extension of the recent work by Dalang, Khoshnevisan and Zhang [DKZ19], where the one-dimensional SHE on \([0,1]\) subject to space-time white noise has been studied.
MSC 2010 subject classifications: Primary. 60H15; Secondary. 35R60.
Keywords: global solution, stochastic heat equation, reaction-diffusion, Dalang’s condition, superlinear growth.
[CH23b] Chen, Le & Huang, Jingyu (2023) ‘Superlinear stochastic heat equation on $mathbb{R}^d$’, Proc. Amer. Math. Soc. 151, 4063–4078. <https://doi.org/10.1090/proc/16436>
@article{chen.huang:23:superlinear,
author = {Chen, Le and Huang, Jingyu},
title = {Superlinear stochastic heat equation on {$\Bbb{R}^d$}},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {151},
year = {2023},
number = {9},
pages = {4063--4078},
issn = {0002-9939},
mrclass = {60H15 (35K57 35R60)},
mrnumber = {4607649},
doi = {10.1090/proc/16436},
url = {https://doi.org/10.1090/proc/16436}
}
References: Balan and Chen [BC18]; Chen and Dalang [CD15b]; Chen and Eisenberg [CE22b]; Chen and Huang [CH19a]; Chen and Kim [CK19]; Conus and Khoshnevisan [CK12]; Da Prato and Zabczyk [DPZ14]; Dalang et al. [DKM+09]; Dalang [Dal99]; Dalang et al. [DKZ19]; Fernández Bonder and Groisman [FBG09]; Foondun and Nualart [FN21]; Huang [Hua17]; Millet and Sanz-Solé [MSS21]; Salins [Sal21]; Sanz-Solé and Sarrà [SSS02]; Walsh [Wal86];
27. chen.khoshnevisan.ea:21:spatial#
[CKNP21b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2023) ‘Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein’s method’, Stoch. Partial Differ. Equ. Anal. Comput. 11, 122–176. <https://doi.org/10.1007/s40072-021-00224-8>
Details
Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein’s method
Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu
Abstract Suppose that \(\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}\) is the solution to a \(d\)-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang’s condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form
where \(g\) is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein’s method for normal approximations. Our results include a central limit theorem for the Hopf-Cole solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.
MSC 2010 subject classifications: Primary. 60H15; Secondary. 60H07, 60F05.
Keywords: stochastic heat equation, ergodicity, central limit theorem, Malliavin calculus, Stein’s method.
[CKNP21b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2023) ‘Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein’s method’, Stoch. Partial Differ. Equ. Anal. Comput. 11, 122–176. <https://doi.org/10.1007/s40072-021-00224-8>
@article{chen.khoshnevisan.ea:23:central,
author = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
title = {Central limit theorems for spatial averages of the stochastic heat equation via {M}alliavin-{S}tein's method},
journal = {Stoch. Partial Differ. Equ. Anal. Comput.},
fjournal = {Stochastic Partial Differential Equations. Analysis and Computations},
volume = {11},
year = {2023},
number = {1},
pages = {122--176},
issn = {2194-0401},
mrclass = {60H15 (35K05 60F05 60H07)},
mrnumber = {4563698},
doi = {10.1007/s40072-021-00224-8},
url = {https://doi.org/10.1007/s40072-021-00224-8}
}
References: Bradley [Bra07]; Burkholder et al. [BDG72]; Carlen and Krée [CK91]; Chen et al. [CHN21]; Chen and Huang [CH19a]; Chen and Kim [CK19]; Chen et al. [CKNP21a]; Chen et al. [CKNP21b]; Chen et al. [CKNP22a]; Conus et al. [CJK12]; Corwin and Ghosal [CG20]; Corwin and Quastel [CQ13]; Cover and Thomas [CT06]; Dalang [Dal99]; Deuschel [Deu88]; Doob [Doo90]; Dym and McKean [DM76]; Esary et al. [EPW67]; Federer [Fed69]; Foondun and Khoshnevisan [FK13]; Helson and Sarason [HS67]; Huang et al. [HLN17b]; Huang et al. [HNV20]; Huang et al. [HNVZ20]; Kardar et al. [KPZ86]; Maruyama [Mar49]; Newman and Wright [NW81]; Newman [New83]; Nourdin and Peccati [NP09]; Nourdin and Peccati [NP12]; Nualart [Nua09]; Nualart and Nualart [NN18]; Nualart and Zheng [NZ20]; Pitt [Pit82]; Prakasa Rao [PR12]; Sanz-Solé and Sarrà [SSS02]; Walsh [Wal86];
26. balan.chen.ea:22:parabolic#
[BCM22] Balan, Raluca, Chen, Le & Ma, Yiping (2022) ‘Parabolic Anderson model with rough noise in space and rough initial conditions’, Electron. Commun. Probab. 27, Paper No. 65, 12. <https://doi.org/10.1214/22-ecp506>
Details
Parabolic Anderson model with rough noise in space and rough initial conditions
Raluca Balan, Le Chen, Yiping Ma
Abstract: In this note, we consider the parabolic Anderson model on \(\mathbb{R}_{+} \times \mathbb{R}\), driven by a Gaussian noise which is fractional in time with index \(H_0>1/2\) and fractional in space with index \(0<H<1/2\) such that \(H_0+H>3/4\). Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all \(p\)-th moments with \(p\ge 2\).
MSC 2010 subject classifications: Primary. 60H15; Secondary. 60H07.
Keywords: stochastic partial differential equations, parabolic Anderson model, Malliavin calculus, rough initial condition, Dirac delta initial condition, rough Gaussian noise.
[BCM22] Balan, Raluca, Chen, Le & Ma, Yiping (2022) ‘Parabolic Anderson model with rough noise in space and rough initial conditions’, Electron. Commun. Probab. 27, Paper No. 65, 12. <https://doi.org/10.1214/22-ecp506>
@article{balan.chen.ea:22:parabolic,
author = {Balan, Raluca and Chen, Le and Ma, Yiping},
title = {Parabolic {A}nderson model with rough noise in space and rough initial conditions},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {27},
year = {2022},
pages = {Paper No. 65, 12},
mrclass = {60H15 (60H07)},
mrnumber = {4529633},
doi = {10.1214/22-ecp506},
url = {https://doi.org/10.1214/22-ecp506}
}
References: Amir et al. [ACQ11]; Balan and Chen [BC18]; Balan et al. [BJQS15]; Chen and Dalang [CD15b]; Chen and Huang [CH19a]; Chen and Kim [CK19]; Chen [Che19]; Chen and Hu [CH21]; Hu et al. [HHL+18]; Hu and Lê [HL19]; Mémin et al. [MMV01]; Nualart [Nua06]; Olver et al. [OLBC10]; Rudin [Rud91];
25. chen.hu:22:holder#
[CH22] Chen, Le & Hu, Guannan (2022) ‘H”older regularity for the nonlinear stochastic time-fractional slow & fast diffusion equations on \(\mathbb{R}^d\)’, Fract. Calc. Appl. Anal. 25, 608–629. <https://doi.org/10.1007/s13540-022-00033-3>
Details
Hölder regularity for the nonlinear stochastic time-fractional slow & fast diffusion equations on \(\mathbb{R}^d\)
Le Chen and Guannan Hu
Abstract In this paper, we study the space-time Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation:
where \(\dot{W}\) is the space-time white noise, \(\alpha\in(0,2]\), \(\beta\in(0,2)\), \(\gamma\ge 0\) and \(\nu>0\). The existence/uniqueness of a random field solution has been obtained in [CHN19] under the condition that \(2(\beta+\gamma)-1-d\beta/\alpha>0\). The Hölder regularity of the solution has been obtained in the same reference, but only for the case \(\beta+\gamma\le 1\). In this paper, we use the idea from the local fractional derivative to establish the Hölder regularity of the solution for all possible cases – \(\beta\in(0,2)\), which in particular recovers the special case in [CHN19] when \(\beta\in (0,1-\gamma]\). As a rather surprising consequence, when \(\gamma=0\), \(\alpha=2\) and \(\beta\) is close to \(2\), the space and time Hölder exponents are both to \(1-\), which is different from the known Hölder exponents for the stochastic wave equation which are \((1/2)-\).
MSC 2010 subject classifications: Primary. 60H15; Secondary. 60G60, 26A33, 60F05.
Keywords: nonlinear stochastic time-fractional slow and fast diffusion equations, local fractional derivative, fractional Taylor theorem, Hölder continuity, Mittag-Leffler function, Fox H-function.
[CH22] Chen, Le & Hu, Guannan (2022) ‘H”older regularity for the nonlinear stochastic time-fractional slow & fast diffusion equations on \(\mathbb{R}^d\)’, Fract. Calc. Appl. Anal. 25, 608–629. <https://doi.org/10.1007/s13540-022-00033-3>
@article{chen.hu:22:holder,
author = {Chen, Le and Hu, Guannan},
title = {H\"{o}lder regularity for the nonlinear stochastic time-fractional slow \& fast diffusion equations on {$\Bbb R^d$}},
journal = {Fract. Calc. Appl. Anal.},
fjournal = {Fractional Calculus and Applied Analysis. An International Journal for Theory and Applications},
volume = {25},
year = {2022},
number = {2},
pages = {608--629},
issn = {1311-0454},
mrclass = {60H15 (26A33 35R11)},
mrnumber = {4437294},
doi = {10.1007/s13540-022-00033-3},
url = {https://doi.org/10.1007/s13540-022-00033-3}
}
References: Ben Adda and Cresson [BAC13]; Chen [Che13]; Chen [Che17a]; Chen and Dalang [CD14a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen et al. [CHN19]; Chen et al. [CHHH17]; Chen et al. [CYZ10]; Chen et al. [CKK15]; Dalang [Dal99]; Erdélyi et al. [EMOT81b]; Foondun and Khoshnevisan [FK09]; Hu and Hu [HH15]; Kilbas and Saigo [KS04]; Kilbas et al. [KST06]; Kolwankar and Gangal [KG96]; Kolwankar and Gangal [KG98]; Liu et al. [LRdS18]; Mijena and Nane [MN15]; Podlubny [Pod99]; Samko et al. [SKM93]; Schneider [Sch96]; Talvila [Tal01]; Walsh [Wal86];
24. chen.khoshnevisan.ea:22:spatial#
[CKNP22b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2022) ‘Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition’, J. Funct. Anal. 282, Paper No. 109290, 35. <https://doi.org/10.1016/j.jfa.2021.109290>
Details
Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition
Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu
Abstract: Let \(\{u(t\,, x)\}_{t >0, x \in\mathbb{R}}\) denote the solution to the parabolic Anderson model with initial condition \(\delta_0\) and driven by space-time white noise on \(\mathbb{R}_+\times\mathbb{R}\), and let \(p_t(x):= (2\pi t)^{-1/2}\exp\{-x^2/(2t)\}\) denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers [CKNP21b, CKNP22a] in order to prove that the random field \(x\mapsto u(t\,,x)/p_t(x)\) is ergodic for every \(t >0\). And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari [HNV20].
MSC 2010 subject classifications: Primary. 60H15; Secondary. 60H07, 60F05.
Keywords: parabolic Anderson model, ergodicity, central limit theorem, Malliavin calculus, Stein’s method, Dirac delta initial condition.
[CKNP22b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2022) ‘Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition’, J. Funct. Anal. 282, Paper No. 109290, 35. <https://doi.org/10.1016/j.jfa.2021.109290>
@article{chen.khoshnevisan.ea:22:spatial,
author = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
title = {Spatial ergodicity and central limit theorems for parabolic {A}nderson model with delta initial condition},
journal = {J. Funct. Anal.},
fjournal = {Journal of Functional Analysis},
volume = {282},
year = {2022},
number = {2},
pages = {Paper No. 109290, 35},
issn = {0022-1236},
mrclass = {60H15 (60F05 60H07)},
mrnumber = {4334682},
mrreviewer = {Ciprian A. Tudor},
doi = {10.1016/j.jfa.2021.109290},
url = {https://doi.org/10.1016/j.jfa.2021.109290}
}
References: Amir et al. [ACQ11]; Balan et al. [BNQSZ22]; Billingsley [Bil99]; Chen and Dalang [CD15b]; Chen et al. [CHN21]; Chen and Huang [CH19a]; Chen et al. [CKNP21b]; Chen et al. [CKNP22a]; Conus et al. [CJK13]; Delgado-Vences et al. [DVNZ20]; Federer [Fed69]; Huang et al. [HNV20]; Huang et al. [HNVZ20]; Nourdin and Peccati [NP09]; Nourdin and Peccati [NP12]; Nualart [Nua09]; Nualart and Nualart [NN18]; Walsh [Wal86];
23. chen.khoshnevisan.ea:21:spatial#
[CKNP21b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2021) ‘Spatial ergodicity for SPDEs via Poincar’e-type inequalities’, Electron. J. Probab. 26, Paper No. 140, 37. <https://doi.org/10.1214/21-ejp690>
Details
Spatial ergodicity for SPDEs via Poincaré-type inequalities
Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu
Abstract: Consider a parabolic stochastic PDE of the form
where \(u=u(t\,,x)\), \(t\ge 0\), \(x\in\mathbb{R}^d\), \(\sigma:\mathbb{R}\to\mathbb{R}\) is Lipschitz continuous and non random, and \(\eta\) is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation \(f\). If, in addition, \(u(0)\equiv1\), then we prove that, under a mild decay condition on \(f\), the process \(x\mapsto u(t\,,x)\) is stationary and ergodic at all times \(t>0\). It has been argued that, when coupled with moment estimates, spatial ergodicity of \(u\) teaches us about the intermittent nature of the solution to such SPDEs [BC95a, Kho14]. Our results provide rigorous justification of such discussions.
Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincaré inequalities. We further showcase the utility of these Poincaré inequalities by: (a) describing conditions that ensure that the random field \(u(t)\) is mixing for every \(t>0\); and by (b) giving a quick proof of a conjecture of Conus et al [CJK12] about the ‘’size’’ of the intermittency islands of \(u\).
The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of Maruyama [Mar49] (see also Dym and McKean [DM76]) in the simple setting where the nonlinear term \(\sigma\) is a constant function.
MSC 2010 subject classifications: Primary. 60H15; Secondary. 37A25, 60H07, 60G10.
Keywords: SPDEs, ergodicity, Malliavin calculus, Poincaré-type inequality.
[CKNP21b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2021) ‘Spatial ergodicity for SPDEs via Poincar’e-type inequalities’, Electron. J. Probab. 26, Paper No. 140, 37. <https://doi.org/10.1214/21-ejp690>
@article{chen.khoshnevisan.ea:21:spatial,
author = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
title = {Spatial ergodicity for {SPDE}s via {P}oincar\'{e}-type inequalities},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {26},
year = {2021},
pages = {Paper No. 140, 37},
mrclass = {60H15 (37A25 60G10 60H07)},
mrnumber = {4346664},
doi = {10.1214/21-ejp690},
url = {https://doi.org/10.1214/21-ejp690}
}
References: Bertini and Cancrini [BC95a]; Burkholder [Bur66]; Burkholder et al. [BDG72]; Burkholder and Gundy [BG70]; Capitaine et al. [CHL97]; Cardon-Weber and Millet [CWM04]; Carlen and Krée [CK91]; Chen et al. [CHN17]; Chen and Huang [CH19b]; Chen [Che16b]; Chen et al. [CHNT17]; Conus et al. [CJK12]; Conus et al. [CJKS13b]; Conus and Khoshnevisan [CK12]; Conus et al. [CJK13]; Dalang [Dal99]; Dalang and Frangos [DF98]; Davis [Dav76]; Doob [Doo90]; Dym and McKean [DM76]; Edgar and Sucheston [ES92]; Foondun and Khoshnevisan [FK09]; Foondun and Khoshnevisan [FK13]; Gaveau and Trauber [GT82]; Hawkes [Haw79]; Hawkes [Haw84]; Hu et al. [HHNT15]; Hu et al. [HHL+17]; Huang et al. [HLN17b]; Huang et al. [HNV20]; Huang et al. [HNVZ20]; Kahane [Kah85]; Karczewska and Zabczyk [KZ01]; Khoshnevisan [Kho02]; Khoshnevisan [Kho14]; Khoshnevisan et al. [KKX17]; Lépingle and Ouvrard [LO73]; Maruyama [Mar49]; Millet and Sanz-Solé [MSS99]; Mueller [Mue91b]; Mueller [Mue09]; Nualart [Nua09]; Ocone [Oco84]; Olver et al. [OLBC10]; Ouvrard [Ouv76]; Peszat [Pes02]; Peszat and Zabczyk [PZ00]; Stein [Ste70]; Walsh [Wal86];
22. chen.khoshnevisan.ea:21:clt#
[CKNP21a] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2021) ‘A CLT for dependent random variables with an application to an infinite system of interacting diffusion processes’, Proc. Amer. Math. Soc. 149, 5367–5384. <https://doi.org/10.1090/proc/15614>
Details
A CLT for dependent random variables with an application to an infinite system of interacting diffusion processes
Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu
Abstract: We present a central limit theorem for stationary random fields that are short-range dependent and asymptotically independent. As an application, we present a central limit theorem for an infinite family of interacting It^o-type diffusion processes.
MSC 2010 subject classifications: Primary. 60F05; Secondary. 60H10, 60J60, 60K35.
Keywords: central limit theorems, stationary processes, short-range dependence, asymptotic independence, interacting diffusions.
[CKNP21a] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2021) ‘A CLT for dependent random variables with an application to an infinite system of interacting diffusion processes’, Proc. Amer. Math. Soc. 149, 5367–5384. <https://doi.org/10.1090/proc/15614>
@article{chen.khoshnevisan.ea:21:clt,
author = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
title = {A {CLT} for dependent random variables with an application to an infinite system of interacting diffusion processes},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {149},
year = {2021},
number = {12},
pages = {5367--5384},
issn = {0002-9939},
mrclass = {60F05},
mrnumber = {4327439},
mrreviewer = {Yoshihiko Maesono},
doi = {10.1090/proc/15614},
url = {https://doi.org/10.1090/proc/15614}
}
References: Bertoin [Ber96]; Bradley [Bra05]; Bradley [Bra07]; Carmona and Molchanov [CM94]; Chen et al. [CKNP21b]; Chen et al. [CKNP22a]; Chen et al. [CKNP22b]; Deuschel [Deu88]; Esary et al. [EPW67]; Federer [Fed69]; Ibragimov [Ibr62]; Jakubowski [Jak91]; Karatzas and Shreve [KS91]; Khoshnevisan and Kim [KK15]; Lahiri [Lah03]; Merlevède et al. [MerlevedePU06]; Newman and Wright [NW81]; Nualart [Nua09]; Rosenblatt [Ros72]; Sanz-Solé and Sarrà [SSS02];
21. chen.khoshnevisan.ea:22:central#
[CKNP22a] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2022) ‘Central limit theorems for parabolic stochastic partial differential equations’, Ann. Inst. Henri Poincar’e Probab. Stat. 58, 1052–1077. <https://doi.org/10.1214/21-aihp1189>
Details
Central limit theorems for parabolic stochastic partial differential equations
Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu
Abstract: Let \(\{u(t\,,x)\}_{t\ge 0, x\in \mathbb{R}^d}\) denote the solution of a \(d\)-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure \(f\) and satisfies Dalang’s condition. We prove two general functional central limit theorems for occupation fields of the form
where \(g\) runs over the class of Lipschitz functions on \(\mathbb{R}^d\) and \(\psi\in L^2(\mathbb{R}^d)\). The proof uses Poincaré-type inequalities, Malliavin calculus, compactness arguments, and Paul Lévy’s classical characterization of Brownian motion as the only mean zero, continuous Lévy process. Our result generalizes central limit theorems of Huang et al [HNV20, HNVZ20] valid when \(g(u)=u\) and \(\psi = \mathbf{1}_{[0,1]^d}\).
MSC 2010 subject classifications: Primary 60H15; Secondary 60F17, 60H07.
Keywords: stochastic heat equation, central limit theorem, Poincaré inequalities, Malliavin calculus, metric entropy.
[CKNP22a] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2022) ‘Central limit theorems for parabolic stochastic partial differential equations’, Ann. Inst. Henri Poincar’e Probab. Stat. 58, 1052–1077. <https://doi.org/10.1214/21-aihp1189>
@article{chen.khoshnevisan.ea:22:central,
author = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
title = {Central limit theorems for parabolic stochastic partial differential equations},
journal = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
fjournal = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques},
volume = {58},
year = {2022},
number = {2},
pages = {1052--1077},
issn = {0246-0203},
mrclass = {60H15 (60F17 60H07)},
mrnumber = {4421618},
doi = {10.1214/21-aihp1189},
url = {https://doi.org/10.1214/21-aihp1189}
}
References: Bertoin [Ber96]; Bradley [Bra07]; Burkholder et al. [BDG72]; Carlen and Krée [CK91]; Chen and Huang [CH19a]; Chen et al. [CKNP21b]; Dalang [Dal99]; Davis [Dav76]; Doob [Doo90]; Dym and McKean [DM76]; Federer [Fed69]; Foondun and Khoshnevisan [FK13]; Huang et al. [HNV20]; Huang et al. [HNVZ20]; Karatzas and Shreve [KS91]; Khoshnevisan [Kho14]; Marcus and Rosen [MR06]; Maruyama [Mar49]; Nualart [Nua09]; Volkonskiui and Rozanov [VR59]; Walsh [Wal86];
20. balan.chen.ea:22:exact#
[BCC22] Balan, Raluca, Chen, Le & Ma, Yiping (2022) ‘Parabolic Anderson model with rough noise in space and rough initial conditions’, Electron. Commun. Probab. 27, Paper No. 65, 12. <https://doi.org/10.1214/22-ecp506>
Details
Exact asymptotics of the stochastic wave equation with time-independent noise
Raluca M. Balan, Le Chen, and Xia Chen
Abstract: In this article, we study the stochastic wave equation in all dimensions \(d\leq 3\), driven by a Gaussian noise \(\dot{W}\) which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the \(p\)-th moment of the solution either when the time is large or when \(p\) is large. For the critical case, that is the case when \(d=3\) and the noise is white, we obtain the exact transition time for the second moment to be finite.
MSC 2010 subject classifications: Primary 60H15; Secondary 60H07, 37H15.
Keywords: stochastic partial differential equations, stochastic wave equation, Malliavin calculus, Lyapunov exponents, exact moment asymptotics, moment blowup.
[BCC22] Balan, Raluca, Chen, Le & Ma, Yiping (2022) ‘Parabolic Anderson model with rough noise in space and rough initial conditions’, Electron. Commun. Probab. 27, Paper No. 65, 12. <https://doi.org/10.1214/22-ecp506>
@article{balan.chen.ea:22:exact,
author = {Balan, Raluca M. and Chen, Le and Chen, Xia},
title = {Exact asymptotics of the stochastic wave equation with time-independent noise},
journal = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
fjournal = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques},
volume = {58},
year = {2022},
number = {3},
pages = {1590--1620},
issn = {0246-0203},
mrclass = {60H15 (37H15 60H07)},
mrnumber = {4452644},
doi = {10.1214/21-aihp1207},
url = {https://doi.org/10.1214/21-aihp1207}
}
References: Balan and Song [BS17]; Balan and Song [BS19]; Bass et al. [BCR09]; Chen and Dalang [CD15a]; Chen [Che07]; Chen [Che17b]; Chen [Che19]; Chen and Li [CL04]; Chen et al. [CLR05]; Chen et al. [CHSX15]; Chen et al. [CDOT21]; Conus and Dalang [CD08]; Dalang [Dal99]; Dalang and Mueller [DM09]; Del Pino and Dolbeault [DPD02]; Gelfand and Vilenkin [GV16]; Gubinelli et al. [GUZ20]; Hairer and Labbé [HL15]; Hu [Hu01]; Hu et al. [HHNT15]; Huang et al. [HLN17b]; Janson [Jan97]; Kardar et al. [KPZ86]; Labbé [Lab13]; Labbé [Lab19]; Lê [Le16]; Nualart [Nua06]; Oh et al. [ORSW21]; Podlubny [Pod99];
19. chen.hu.ea:21:regularity#
[CHN21] Chen, Le, Hu, Yaozhong & Nualart, David (2021) ‘Regularity and strict positivity of densities for the nonlinear stochastic heat equation’, Mem. Amer. Math. Soc. 273, v+102. <https://doi.org/10.1090/memo/1340>
Details
Regularity and strict positivity of densities for the nonlinear stochastic heat equation
Le Chen, Yaozhong Hu, and David Nualart
Abstract: In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove \(u(t,x)\in \mathbb{D}^\infty\) for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: stochastic heat equation, space-time white noise, Malliavin calculus, negative moments, regularity of density, strict positivity of density, measure-valued initial data, parabolic Anderson model.
[CHN21] Chen, Le, Hu, Yaozhong & Nualart, David (2021) ‘Regularity and strict positivity of densities for the nonlinear stochastic heat equation’, Mem. Amer. Math. Soc. 273, v+102. <https://doi.org/10.1090/memo/1340>
@article{chen.hu.ea:21:regularity,
author = {Chen, Le and Hu, Yaozhong and Nualart, David},
title = {Regularity and strict positivity of densities for the nonlinear stochastic heat equation},
journal = {Mem. Amer. Math. Soc.},
fjournal = {Memoirs of the American Mathematical Society},
volume = {273},
year = {2021},
number = {1340},
pages = {v+102},
issn = {0065-9266},
isbn = {978-1-4704-5000-7; 978-1-4704-6809-5},
mrclass = {60H15 (35K05 60G60)},
mrnumber = {4334477},
doi = {10.1090/memo/1340},
url = {https://doi.org/10.1090/memo/1340}
}
References: Bally and Pardoux [BP98]; Bouleau and Hirsch [BH86]; Carmona and Molchanov [CM94]; Chen and Dalang [CD14a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen and Kim [CK17]; Da Prato and Zabczyk [DPZ14]; Debbi and Dozzi [DD05]; Feller [Fel52]; Hu [Hu17]; Hu et al. [HHNS15]; Khoshnevisan [Kho09]; Kilbas et al. [KST06]; Komatsu [Kom84]; Mainardi et al. [MLP01]; Moreno Flores [MF14]; Mueller and Nualart [MN08]; Nualart [Nua95]; Nualart [Nua06]; Nualart [Nua09]; Nualart and Quer-Sardanyons [NQS07]; Nualart [Nua13]; Pardoux and Zhang [PZ93]; Podlubny [Pod99]; Walsh [Wal86]; Zolotarev [Zol86];
18. chen.kim:20:stochastic#
[CK20] Chen, Le & Kim, Kunwoo (2020) ‘Stochastic comparisons for stochastic heat equation’, Electron. J. Probab. 25, Paper No. 140, 38. <https://doi.org/10.1214/20-ejp541>
Details
Stochastic comparisons for stochastic heat equation
Le Chen and Kunwoo Kim
Abstract: We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on \(\mathbb{R}^d\)
where \(\dot{M}\) is a spatially homogeneous Gaussian noise that is white in time and colored in space, and \(\rho\) is a Lipschitz continuous function that vanishes at zero. These results are obtained for rough initial data and under Dalang’s condition, namely,
where \(\hat{f}\) is the spectral measure of the noise. We first show that the nonlinear stochastic heat equation can be approximated by systems of interacting diffusions (SDEs) and then, using those approximations, we establish the comparison principles by comparing either the diffusion coefficient \(\rho\) or the correlation function of the noise \(f\). As corollaries, we obtain Slepian’s inequality for SPDEs and SDEs.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: stochastic heat equation, parabolic Anderson model, infinite dimensional SDE, spatially homogeneous noise, stochastic comparison principle, moment comparison principle, Slepian’s inequality for SPDEs, rough initial data.
[CK20] Chen, Le & Kim, Kunwoo (2020) ‘Stochastic comparisons for stochastic heat equation’, Electron. J. Probab. 25, Paper No. 140, 38. <https://doi.org/10.1214/20-ejp541>
@article{chen.kim:20:stochastic,
author = {Chen, Le and Kim, Kunwoo},
title = {Stochastic comparisons for stochastic heat equation},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {25},
year = {2020},
pages = {Paper No. 140, 38},
mrclass = {60H15 (35R60 60G60)},
mrnumber = {4186259},
mrreviewer = {Vitalii Konarovskyi},
doi = {10.1214/20-ejp541},
url = {https://doi.org/10.1214/20-ejp541}
}
References: Balan and Chen [BC18]; Borodin and Corwin [BC14c]; Carmona and Molchanov [CM94]; Chen and Dalang [CD15b]; Chen and Kim [CK17]; Chen and Kim [CK19]; Chen [Che15]; Chen [Che19]; Cox et al. [CFG96]; Dalang [Dal99]; Ethier and Kurtz [EK86]; Foondun et al. [FJL18]; Foondun and Khoshnevisan [FK13]; Friedman [Fri75]; Geiß and Manthey [GM94]; Huang [Hua17]; Ikeda and Watanabe [IW89]; Joseph et al. [JKM17]; Khoshnevisan et al. [KKX17]; Khoshnevisan et al. [KKX18]; Kim [Kim19]; Mueller [Mue91b]; Revuz and Yor [RY99]; Shiga [Shi94]; Shiga and Shimizu [SS80]; Walsh [Wal86];
17. chen.huang.ea:19:dense#
[CHKK19] Chen, Le, Huang, Jingyu, Khoshnevisan, Davar & Kim, Kunwoo (2019) ‘Dense blowup for parabolic SPDEs’, Electron. J. Probab. 24, Paper No. 118, 33. <https://doi.org/10.1214/19-ejp372>
Details
Dense blowup for parabolic SPDEs
Le Chen, Jingyu Huang, Davar Khoshnevisan, and Kunwoo Kim
Abstract: The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type
such that the solution exists and is unique as a random field in the sense of Dalang [Dal99] and Walsh [Wal86], yet the solution has unbounded oscillations in every open neighborhood of every space-time point. We are not aware of the existence of such a construction in spatial dimensions below \(3\).
En route, it will be proved that when \(\sigma(u)=u\) there exist a large family of parabolic SPDEs whose moment Lyapunov exponents grow at least sub exponentially in its order parameter in the sense that there exist \(A_1,\beta\in(0\,,1)\) such that
This sort of ‘’super intermittency’’ is combined with a local linearization of the solution, and with techniques from Gaussian analysis in order to establish the unbounded oscillations of the sample functions of the solution to our SPDE.
MSC 2010 subject classifications: Primary 35R60, 60H15; Secondary 60G15.
Keywords: stochastic partial differential equations, blowup, intermittency.
[CHKK19] Chen, Le, Huang, Jingyu, Khoshnevisan, Davar & Kim, Kunwoo (2019) ‘Dense blowup for parabolic SPDEs’, Electron. J. Probab. 24, Paper No. 118, 33. <https://doi.org/10.1214/19-ejp372>
@article{chen.huang.ea:19:dense,
author = {Chen, Le and Huang, Jingyu and Khoshnevisan, Davar and Kim, Kunwoo},
title = {Dense blowup for parabolic {SPDE}s},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {24},
year = {2019},
pages = {Paper No. 118, 33},
mrclass = {60H15 (35B44 35K15 35K91 35R60 60G15)},
mrnumber = {4029421},
mrreviewer = {Guangqu Zheng},
doi = {10.1214/19-ejp372},
url = {https://doi.org/10.1214/19-ejp372}
}
References: Borell [Bor75]; Chen and Huang [CH19a]; Ciesielski and Taylor [CT62]; Conus [Con13]; Conus et al. [CJK13]; Dalang [Dal99]; Dalang and Frangos [DF98]; Dalang and Lévêque [DL04]; Dalang and Lévêque [DL06]; Dudley [Dud67]; Fernique [Fer71]; Foondun and Khoshnevisan [FK09]; Foondun et al. [FKM15]; Foondun et al. [FKN11]; Hairer [Hai14]; Hairer [Hai13]; Hairer and Pardoux [HP15]; Hu and Nualart [HN09]; Khoshnevisan [Kho02]; Khoshnevisan [Kho14]; Khoshnevisan and Xiao [KX03]; Khoshnevisan et al. [KSXZ13]; Mueller [Mue91b]; Mueller [Mue09]; Mytnik and Perkins [MP03]; Sato [Sat13]; Schilling et al. [SSV10]; Sudakov and Cirelson [SC74]; Walsh [Wal86];
16. chen.hu.ea:19:nonlinear#
[CHN19] Chen, Le, Hu, Yaozhong & Nualart, David (2019) ‘Nonlinear stochastic time-fractional slow and fast diffusion equations on \(\mathbb{R}^d\)’, Stochastic Process. Appl. 129, 5073–5112. <https://doi.org/10.1016/j.spa.2019.01.003>
Details
Nonlinear stochastic time-fractional slow and fast diffusion equations on \(\mathbb{R}^d\)
Le Chen, Yaozhong Hu, David Nualart
Abstract: This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables:
where \(\dot{W}\) is the space-time white noise, \(\alpha\in(0,2]\), \(\beta\in(0,2)\), \(\gamma\ge 0\) and \(\nu>0\). Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition:
In some cases, the initial data can be measures. When \(\beta\in (0,1]\), we prove the sample path regularity of the solution.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic fractional diffusion equations, measure-valued initial data, Hölder continuity, intermittency, the Fox H-function.
[CHN19] Chen, Le, Hu, Yaozhong & Nualart, David (2019) ‘Nonlinear stochastic time-fractional slow and fast diffusion equations on \(\mathbb{R}^d\)’, Stochastic Process. Appl. 129, 5073–5112. <https://doi.org/10.1016/j.spa.2019.01.003>
@article{chen.hu.ea:19:nonlinear,
author = {Chen, Le and Hu, Yaozhong and Nualart, David},
title = {Nonlinear stochastic time-fractional slow and fast diffusion equations on {$\Bbb R^d$}},
journal = {Stochastic Process. Appl.},
fjournal = {Stochastic Processes and their Applications},
volume = {129},
year = {2019},
number = {12},
pages = {5073--5112},
issn = {0304-4149},
mrclass = {60H15 (35R11 35R60 60G60)},
mrnumber = {4025700},
mrreviewer = {Latifa Debbi},
doi = {10.1016/j.spa.2019.01.003},
url = {https://doi.org/10.1016/j.spa.2019.01.003}
}
References: Bertini and Cancrini [BC95a]; Carmona and Molchanov [CM94]; Chen [Che13]; Chen [Che17a]; Chen and Dalang [CD14a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen et al. [CHN21]; Chen and Kim [CK17]; Chen et al. [CHHH17]; Chen et al. [CHSS18]; Chen et al. [CKK15]; Conus et al. [CJKS13a]; Conus et al. [CJKS14]; Dalang [Dal99]; Debbi and Dozzi [DD05]; Diethelm [Die10]; Eidelman and Kochubei [EK04]; Foondun and Khoshnevisan [FK09]; Hu and Hu [HH15]; Kilbas and Saigo [KS04]; Kilbas et al. [KST06]; Kochubeui [Koc90]; Mainardi [Mai10]; Mainardi et al. [MLP01]; Mijena and Nane [MN15]; Mijena and Nane [MN16]; Mueller [Mue91b]; Olver et al. [OLBC10]; Podlubny [Pod99]; Pskhu [Psk09]; Samko et al. [SKM93]; Schneider [Sch96]; Stein and Weiss [SW71]; Walsh [Wal86]; Widder [Wid41]; Wright [Wri40]; Wright [Wri33]; Wright [Wri35];
15. chen.huang:19:comparison#
[CH19a] Chen, Le & Huang, Jingyu (2019) ‘Comparison principle for stochastic heat equation on \(\mathbb{R}^d\)’, Ann. Probab. 47, 989–1035. <https://doi.org/10.1214/18-AOP1277>
Details
Comparison principle for stochastic heat equation on \(\mathbb{R}^d\)
Le Chen and Jingyu Huang
Abstract: We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on \(\mathbb{R}^d\)
for measure-valued initial data, where \(\dot{M}\) is a spatially homogeneous Gaussian noise that is white in time and \(\rho\) is Lipschitz continuous. These results are obtained under the condition that
where \(\hat{f}\) is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang’s condition, i.e., \(\alpha=0\). As some intermediate results, we obtain handy upper bounds for \(L^p(\Omega)\)-moments of \(u(t,x)\) for all \(p\ge 2\), and also prove that \(u\) is a.s. Hölder continuous with order \(\alpha-\epsilon\) in space and \(\alpha/2-\epsilon\) in time for any small \(\epsilon>0\).
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: stochastic heat equation, parabolic Anderson model, space-time Hölder regularity, spatially homogeneous noise, comparison principle, measure-valued initial data.
[CH19a] Chen, Le & Huang, Jingyu (2019) ‘Comparison principle for stochastic heat equation on \(\mathbb{R}^d\)’, Ann. Probab. 47, 989–1035. <https://doi.org/10.1214/18-AOP1277>
@article{chen.huang:19:comparison,
author = {Chen, Le and Huang, Jingyu},
title = {Comparison principle for stochastic heat equation on {$\Bbb R^d$}},
journal = {Ann. Probab.},
fjournal = {The Annals of Probability},
volume = {47},
year = {2019},
number = {2},
pages = {989--1035},
issn = {0091-1798},
mrclass = {60H15 (35B51 35R60 60G60)},
mrnumber = {3916940},
mrreviewer = {Petru A. Cioica-Licht},
doi = {10.1214/18-AOP1277},
url = {https://doi.org/10.1214/18-AOP1277}
}
References: Adams and Fournier [AF03b]; Balan and Chen [BC18]; Carmona and Molchanov [CM94]; Chen and Dalang [CD14a]; Chen and Dalang [CD15b]; Chen and Kim [CK17]; Chen and Kim [CK19]; Chen et al. [CHKK19]; Conus et al. [CJK12]; Dalang [Dal99]; Dalang and Quer-Sardanyons [DQS11]; Dawson and Salehi [DS80]; Foondun and Khoshnevisan [FK13]; Gubinelli and Perkowski [GP17]; Hu et al. [HHN16]; Hu et al. [HHNT15]; Huang [Hua17]; Huang et al. [HLN17a]; Moreno Flores [MF14]; Mueller [Mue91b]; Mueller and Nualart [MN08]; Sanz-Solé and Sarrà [SSS02]; Shiga [Shi94]; Tessitore and Zabczyk [TZ98b]; Walsh [Wal86];
14. chen.kim:19:nonlinear#
[CK19] Chen, Le & Kim, Kunwoo (2019) ‘Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency’, Acta Math. Sci. Ser. B (Engl. Ed.) 39, 645–668. <https://doi.org/10.1007/s10473-019-0303-6>
Details
Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency
Le Chen and Kunwoo Kim
Abstract: In this paper, we study the nonlinear stochastic heat equation in the spatial domain \(\mathbb{R}^d\) subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang’s condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on \(\mathbb{Z}^d\) to that on \(\mathbb{R}^d\). Then we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [CK12] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [HLN17b] using different techniques.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: stochastic heat equation, moment estimates, phase transition, intermittency, intermittency front, measure-valued initial data, moment Lyapunov exponents.
[CK19] Chen, Le & Kim, Kunwoo (2019) ‘Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency’, Acta Math. Sci. Ser. B (Engl. Ed.) 39, 645–668. <https://doi.org/10.1007/s10473-019-0303-6>
@article{chen.kim:19:nonlinear,
author = {Chen, Le and Kim, Kunwoo},
title = {Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency},
journal = {Acta Math. Sci. Ser. B (Engl. Ed.)},
fjournal = {Acta Mathematica Scientia. Series B. English Edition},
volume = {39},
year = {2019},
number = {3},
pages = {645--668},
issn = {0252-9602},
mrclass = {60H15 (35K15 35K58 35R60 60G60)},
mrnumber = {4066498},
doi = {10.1007/s10473-019-0303-6},
url = {https://doi.org/10.1007/s10473-019-0303-6}
}
References: Balan and Chen [BC18]; Carmona and Molchanov [CM94]; Chen [Che13]; Chen and Dalang [CD15b]; Chen et al. [CHN17]; Chen and Huang [CH19a]; Chen and Kim [CK17]; Chen [Che17b]; Conus and Khoshnevisan [CK12]; Dalang [Dal99]; Dalang and Frangos [DF98]; Foondun and Khoshnevisan [FK09]; Foondun and Khoshnevisan [FK13]; Foondun and Khoshnevisan [FK14]; Foondun et al. [FLO17]; Huang [Hua17]; Huang et al. [HLN17b]; Khoshnevisan [Kho14]; Khoshnevisan and Kim [KK15]; Mueller [Mue91b]; Noble [Nob97]; Podlubny [Pod99]; Stein [Ste70]; Tessitore and Zabczyk [TZ98b]; Walsh [Wal86];
13. chen.hu.ea:18:intermittency#
[CHSS18] Chen, Le, Hu, Yaozhong, Kalbasi, Kamran & Nualart, David (2018) ‘Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise’, Probab. Theory Related Fields 171, 431–457. <https://doi.org/10.1007/s00440-017-0783-z>
Details
Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise
Le Chen, Yaozhong Hu, Kamran Kalbasi, and David Nualart
Abstract: This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter \(H\in (0,1/2)\). We establish the Feynman-Kac representation of the solution and use this representation to obtain matching lower and upper bounds for the \(L^p(\Omega)\) moments of the solution.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: stochastic heat equation, Feynman-Kac integral, Feynman-Kac formula, time fractional Gaussian noise, fractional calculus, moment bounds, Lyapunov exponents, intermittency.
[CHSS18] Chen, Le, Hu, Yaozhong, Kalbasi, Kamran & Nualart, David (2018) ‘Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise’, Probab. Theory Related Fields 171, 431–457. <https://doi.org/10.1007/s00440-017-0783-z>
@article{chen.hu.ea:18:intermittency,
author = {Chen, Le and Hu, Yaozhong and Kalbasi, Kamran and Nualart, David},
title = {Intermittency for the stochastic heat equation driven by a rough time fractional {G}aussian noise},
journal = {Probab. Theory Related Fields},
fjournal = {Probability Theory and Related Fields},
volume = {171},
year = {2018},
number = {1-2},
pages = {431--457},
issn = {0178-8051},
mrclass = {60H15 (35K15 35R60 60G60)},
mrnumber = {3800837},
mrreviewer = {Isamu D\^{o}ku},
doi = {10.1007/s00440-017-0783-z},
url = {https://doi.org/10.1007/s00440-017-0783-z}
}
References: Balan and Conus [BC16]; Bertini and Cancrini [BC95a]; Carmona and Molchanov [CM94]; Chen and Dalang [CD15b]; Chen et al. [CHN19]; Conus et al. [CJK13]; Foondun and Khoshnevisan [FK09]; Hu [Hu17]; Hu and Lê [HL17]; Hu et al. [HLN12]; Hu and Nualart [HN09]; Hu et al. [HNS09]; Hu et al. [HHNT15]; Kalbasi and Mountford [KM15]; Mémin et al. [MMV01]; Nualart [Nua06]; Olver et al. [OLBC10]; Pipiras and Taqqu [PT01]; Podlubny [Pod99]; Zeldovich et al. [ZRS90];
12. balan.chen:18:parabolic#
[BC18] Balan, Raluca M. & Chen, Le (2018) ‘Parabolic Anderson model with space-time homogeneous Gaussian noise and rough initial condition’, J. Theoret. Probab. 31, 2216–2265. <https://doi.org/10.1007/s10959-017-0772-2>
Details
Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise and Rough Initial Condition
Raluca M. Balan and Le Chen
Abstract: In this article, we study the Parabolic Anderson Model driven by a space-time homogeneous Gaussian noise on \(\mathbb{R}_{+} \times \mathbb{R}^d\), whose covariance kernels in space and time are locally integrable non-negative functions, which are non-negative definite (in the sense of distributions). We assume that the initial condition is given by a signed Borel measure on \(\mathbb{R}^d\), and the spectral measure of the noise satisfies Dalang’s (1999) condition. Under these conditions, we prove that this equation has a unique solution, and we investigate the magnitude of the \(p\)-th moments of the solution, for any \(p \geq 2\). In addition, we show that this solution has a Hölder continuous modification with the same regularity and under the same condition as in the case of the white noise in time, regardless of the temporal covariance function of the noise.
MSC 2010 subject classifications: Primary 60H15; Secondary 60H07, 37H15.
Keywords: stochastic partial differential equations, rough initial conditions, Parabolic Anderson Model, Malliavin calculus, Wiener chaos expansion.
[BC18] Balan, Raluca M. & Chen, Le (2018) ‘Parabolic Anderson model with space-time homogeneous Gaussian noise and rough initial condition’, J. Theoret. Probab. 31, 2216–2265. <https://doi.org/10.1007/s10959-017-0772-2>
@article{balan.chen:18:parabolic,
author = {Balan, Raluca M. and Chen, Le},
title = {Parabolic {A}nderson model with space-time homogeneous {G}aussian noise and rough initial condition},
journal = {J. Theoret. Probab.},
fjournal = {Journal of Theoretical Probability},
volume = {31},
year = {2018},
number = {4},
pages = {2216--2265},
issn = {0894-9840},
mrclass = {60H15 (60H07)},
mrnumber = {3866613},
mrreviewer = {Jan I. Seidler},
doi = {10.1007/s10959-017-0772-2},
url = {https://doi.org/10.1007/s10959-017-0772-2}
}
References: Balan and Conus [BC16]; Balan and Song [BS17]; Borodin and Corwin [BC14b]; Carmona and Molchanov [CM94]; Chen and Dalang [CD14a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen and Huang [CH19a]; Chen and Kim [CK19]; Chen et al. [CHHH17]; Dalang [Dal99]; Dalang and Mueller [DM09]; Dalang and Sanz-Solé [DSS09]; Foondun and Khoshnevisan [FK09]; Hu and Nualart [HN09]; Hu et al. [HHNT15]; Huang et al. [HLN17b]; Janson [Jan97]; Nualart [Nua06]; Olver et al. [OLBC10]; Podlubny [Pod99]; Sanz-Solé and Sarrà [SSS02]; Sanz-Solé and Süß [SSS15];
11. chen.hu.ea:17:two-point#
[CHN17] Chen, Le, Hu, Yaozhong & Nualart, David (2017) ‘Two-point correlation function and Feynman-Kac formula for the stochastic heat equation’, Potential Anal. 46, 779–797. <https://doi.org/10.1007/s11118-016-9601-y>
Details
Two-point correlation function and Feynman-Kac formula for the stochastic heat equation
Le Chen, Yaozhong Hu, David Nualart
Abstract: In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on \(\mathbb{R}\). The bounds for \(p\)-th moments proved in [CD15b] are simplified. We validate the Feynman-Kac formula for the \(p\)-point correlation function of the solutions to this equation with measure-valued initial data.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: stochastic heat equation, two-point correlation function, Feynman-Kac formula, Brownian local time, Malliavin calculus.
[CHN17] Chen, Le, Hu, Yaozhong & Nualart, David (2017) ‘Two-point correlation function and Feynman-Kac formula for the stochastic heat equation’, Potential Anal. 46, 779–797. <https://doi.org/10.1007/s11118-016-9601-y>
@article{chen.hu.ea:17:two-point,
author = {Chen, Le and Hu, Yaozhong and Nualart, David},
title = {Two-point correlation function and {F}eynman-{K}ac formula for the stochastic heat equation},
journal = {Potential Anal.},
fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis},
volume = {46},
year = {2017},
number = {4},
pages = {779--797},
issn = {0926-2601},
mrclass = {60H15 (35K15 35R60 60G60 60H07 60J55)},
mrnumber = {3636598},
mrreviewer = {Robert C. Dalang},
doi = {10.1007/s11118-016-9601-y},
url = {https://doi.org/10.1007/s11118-016-9601-y}
}
References: Airault et al. [ARZ00]; Albeverio et al. [ABD95]; Albeverio et al. [AGHKH05]; Bertini and Cancrini [BC98]; Carmona and Molchanov [CM94]; Chen and Dalang [CD15b]; Chen and Kim [CK17]; Chung and Williams [CW90]; Erdélyi et al. [EMOT81b]; Hu and Nualart [HN09]; Nualart [Nua06]; Nualart and Vives [NV94]; Revuz and Yor [RY99];
10. chen.kim:17:on#
[CK17] Chen, Le & Kim, Kunwoo (2017) ‘On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations’, Ann. Inst. Henri Poincar’e Probab. Stat. 53, 358–388. <https://doi.org/10.1214/15-AIHP719>
Details
On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations
Le Chen and Kunwoo Kim
Abstract: In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on \(\mathbb{R}\) with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller’s comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at \(\pm\infty\). These results generalize a recent work by Moreno Flores [MF14], who proves the strict positivity of the solution to the stochastic heat equation with the delta initial data. As one application, we establish the full intermittency for the equation. As an intermediate step, we prove the Hölder regularity of the solution starting from measure-valued initial data, which generalizes, in some sense, a recent work by Chen and Dalang [CD14a].
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic fractional heat equation, parabolic Anderson model, comparison principle, measure-valued initial data, stable processes.
[CK17] Chen, Le & Kim, Kunwoo (2017) ‘On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations’, Ann. Inst. Henri Poincar’e Probab. Stat. 53, 358–388. <https://doi.org/10.1214/15-AIHP719>
@article{chen.kim:17:on,
author = {Chen, Le and Kim, Kunwoo},
title = {On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations},
journal = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
fjournal = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques},
volume = {53},
year = {2017},
number = {1},
pages = {358--388},
issn = {0246-0203},
mrclass = {60H15 (35B51 35R11 35R60 60G60)},
mrnumber = {3606745},
mrreviewer = {Mohammud Foondun},
doi = {10.1214/15-AIHP719},
url = {https://doi.org/10.1214/15-AIHP719}
}
References: Assing [Ass99]; Bauinov and Simeonov [BS92]; Bertini et al. [BCJL94]; Carmona and Molchanov [CM94]; Chen and Dalang [CD14a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Conus et al. [CJK12]; Conus et al. [CJKS14]; Cox et al. [CFG96]; Debbi [Deb06]; Debbi and Dozzi [DD05]; Foondun and Khoshnevisan [FK09]; Hajek [Haj85]; Ikeda and Watanabe [IW89]; Jacka and Tribe [JT03]; Joseph et al. [JKM17]; Kardar et al. [KPZ86]; Kotelenez [Kot92]; Kunita [Kun90]; Mainardi et al. [MLP01]; Milian [Mil02]; Moreno Flores [MF14]; Mueller [Mue91b]; Mueller [Mue09]; Mueller and Nualart [MN08]; Olver et al. [OLBC10]; Rogers and Williams [RW00]; Shiga [Shi94]; Uchaikin and Zolotarev [UZ99]; Walsh [Wal86]; Zolotarev [Zol86];
9. chen.cranston.ea:17:dissipation#
[CCKK17] Chen, Le, Cranston, Michael, Khoshnevisan, Davar & Kim, Kunwoo (2017) ‘Dissipation and high disorder’, Ann. Probab. 45, 82–99. <https://doi.org/10.1214/15-AOP1040>
Details
Dissipation and high disorder
Le Chen, Michael Cranston, Davar Khoshnevisan, and Kunwoo Kim
Abstract: Given a field \(\{B(x)\}_{x\in\mathbb{Z}^d}\) of independent standard Brownian motions, indexed by \(\mathbb{Z}^d\), the generator of a suitable Markov process on \(\mathbb{Z}^d,\,\,\mathcal{G},\) and sufficiently nice function \(\sigma:[0,\infty)\mapsto[0,\infty),\) we consider the influence of the parameter \(\lambda\) on the behavior of the system,
We show that for any \(\lambda>0\) in dimensions one and two the total mass \(\sum_{x\in\mathbb{Z}^d}u_t(x)\) converges to zero as \(t\to\infty\) while for dimensions greater than two there is a phase transition point \(\lambda_c\in(0,\infty)\) such that for
while for
MSC 2010 subject classifications: Primary: 60J60, 60K35, 60K37, Secondary: 47B80, 60H25.
Keywords: Parabolic Anderson model, strong disorder, stochastic partial differential equations.
[CCKK17] Chen, Le, Cranston, Michael, Khoshnevisan, Davar & Kim, Kunwoo (2017) ‘Dissipation and high disorder’, Ann. Probab. 45, 82–99. <https://doi.org/10.1214/15-AOP1040>
@article{chen.cranston.ea:17:dissipation,
author = {Chen, Le and Cranston, Michael and Khoshnevisan, Davar and Kim, Kunwoo},
title = {Dissipation and high disorder},
journal = {Ann. Probab.},
fjournal = {The Annals of Probability},
volume = {45},
year = {2017},
number = {1},
pages = {82--99},
issn = {0091-1798},
mrclass = {60J60 (47B80 60H25 60K35 60K37)},
mrnumber = {3601646},
mrreviewer = {Khoa L\^{e}},
doi = {10.1214/15-AOP1040},
url = {https://doi.org/10.1214/15-AOP1040}
}
References: Borodin and Corwin [BC14c]; Carmona and Hu [CH06]; Carmona et al. [CKM01]; Carmona and Molchanov [CM94]; Chung and Fuchs [CF51]; Cox et al. [CFG96]; Cranston et al. [CMS02]; Dalang and Mueller [DM03]; Dawson and Perkins [DP12]; Foondun and Khoshnevisan [FK09]; Georgiou et al. [GJKS15]; Hoeffding [Hoe63]; Liggett [Lig05]; Mueller [Mue91b]; Mueller [Mue09]; Mueller and Nualart [MN08]; Mueller and Tribe [MT04]; Shiga [Shi92]; Shiga and Shimizu [SS80]; Spitzer [Spi81]; Walsh [Wal86];
8. chen.khoshnevisan.ea:17:boundedness#
[CKK17] Chen, Le, Khoshnevisan, Davar & Kim, Kunwoo (2017) ‘A boundedness trichotomy for the stochastic heat equation’, Ann. Inst. Henri Poincar’e Probab. Stat. 53, 1991–2004. <https://doi.org/10.1214/16-AIHP780>
Details
A boundedness trichotomy for the stochastic heat equation
Le Chen, Davar Khoshnevisan, and Kunwoo Kim
Abstract: We consider the stochastic heat equation with a multiplicative white noise forcing term under standard ‘’intermitency conditions’’. The main finding of this paper is that, under mild regularity hypotheses, the a.s.-boundedness of the solution \(x\mapsto u(t\,,x)\) can be characterized generically by the decay rate, at \(\pm\infty\), of the initial function \(u_0\). More specifically, we prove that there are three generic boundedness regimes, depending on the numerical value of
MSC 2010 subject classifications: Primary 60H15. Secondary 35R60.
Keywords: stochastic heat equation.
[CKK17] Chen, Le, Khoshnevisan, Davar & Kim, Kunwoo (2017) ‘A boundedness trichotomy for the stochastic heat equation’, Ann. Inst. Henri Poincar’e Probab. Stat. 53, 1991–2004. <https://doi.org/10.1214/16-AIHP780>
@article{chen.khoshnevisan.ea:17:boundedness,
author = {Chen, Le and Khoshnevisan, Davar and Kim, Kunwoo},
title = {A boundedness trichotomy for the stochastic heat equation},
journal = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.},
fjournal = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques},
volume = {53},
year = {2017},
number = {4},
pages = {1991--2004},
issn = {0246-0203},
mrclass = {60H15 (35R60)},
mrnumber = {3729644},
mrreviewer = {Paul Andr\'{e} Razafimandimby},
doi = {10.1214/16-AIHP780},
url = {https://doi.org/10.1214/16-AIHP780}
}
References: Chen and Dalang [CD14a]; Chen and Dalang [CD15b]; Conus and Khoshnevisan [CK12]; Dareiotis and Gerencsér [DG15]; Foondun and Khoshnevisan [FK09]; Foondun and Khoshnevisan [FK10]; Joseph et al. [JKM17]; Khoshnevisan [Kho14]; Mueller [Mue91b]; Mueller [Mue09]; Shiga [Shi92]; Walsh [Wal86];
7. chen:17:nonlinear#
[Che17a] Chen, Le (2017) ‘Nonlinear stochastic time-fractional diffusion equations on \(\mathbb{R}^d\): moments, H”older regularity and intermittency’, Trans. Amer. Math. Soc. 369, 8497–8535. <https://doi.org/10.1090/tran/6951>
Details
Nonlinear stochastic time-fractional diffusion equations on** \(\mathbb{R}\): moments, Hölder regularity and intermittency
Le Chen
Abstract: We study the nonlinear stochastic time-fractional diffusion equations in the spatial domain \(\mathbb{R}\), driven by multiplicative space-time white noise. The fractional index \(\beta\) varies continuously from \(0\) to \(2\). The case \(\beta=1\) (resp. \(\beta=2\)) corresponds to the stochastic heat (resp. wave) equation. The cases \(\beta\in \:]0,1[\:\) and \(\beta\in \:]1,2[\:\) are called slow diffusion equations and fast diffusion equations, respectively. Existence and uniqueness of random field solutions with measure-valued initial data, such as the Dirac delta measure, are established. Upper bounds on all \(p\)-th moments \((p\ge 2)\) are obtained, which are expressed using a kernel function \(\mathcal{K}(t,x)\). The second moment is sharp. We obtain the Hölder continuity of the solution for the slow diffusion equations when the initial data is a bounded function. We prove the weak intermittency for both slow and fast diffusion equations. In this study, we introduce a special function, the two-parameter Mainardi functions, which are generalizations of the one-parameter Mainardi functions.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic time-fractional diffusion equations, Anderson model, measure-valued initial data, Hölder continuity, intermittency, two-parameter Mainardi function.
[Che17a] Chen, Le (2017) ‘Nonlinear stochastic time-fractional diffusion equations on \(\mathbb{R}^d\): moments, H”older regularity and intermittency’, Trans. Amer. Math. Soc. 369, 8497–8535. <https://doi.org/10.1090/tran/6951>
@article{chen:17:nonlinear,
author = {Chen, Le},
title = {Nonlinear stochastic time-fractional diffusion equations on {$\Bbb{R}$}: moments, {H}\"{o}lder regularity and intermittency},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the American Mathematical Society},
volume = {369},
year = {2017},
number = {12},
pages = {8497--8535},
issn = {0002-9947},
mrclass = {60H15 (35R11 35R60 60G60)},
mrnumber = {3710633},
mrreviewer = {Feng-Yu Wang},
doi = {10.1090/tran/6951},
url = {https://doi.org/10.1090/tran/6951}
}
References: Aki and Richards [AR09]; Bertini and Cancrini [BC95a]; Carmona and Molchanov [CM94]; Chen [Che13]; Chen and Dalang [CD14a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen and Kim [CK17]; Conus et al. [CJKS13a]; Conus et al. [CJKS14]; Craiem et al. [CRA+08]; Debbi and Dozzi [DD05]; Diethelm [Die10]; Dimitrienko [Dim11]; Doi and Edwards [DE86]; Ferry [Fer80]; Foondun and Khoshnevisan [FK09]; Kardar et al. [KPZ86]; Klausner [Kla12]; Lemaitre and Jean-Louis [LJL90]; Magin [Mag10]; Mainardi [Mai10]; Mainardi et al. [MLP01]; Mijena and Nane [MN15]; Mijena and Nane [MN16]; Oldham et al. [OMS09]; Olver et al. [OLBC10]; Perkins and Lach [PL11]; Podlubny [Pod99]; Polyanin [Pol02]; Schneider [Sch96]; Umarov and Saydamatov [US06]; Walsh [Wal86]; Widder [Wid41]; Wright [Wri40]; Wright [Wri35];
6. chen.hu.ea:17:space-time#
[CHHH17] Chen, Le, Hu, Guannan, Hu, Yaozhong & Huang, Jingyu (2017) ‘Space-time fractional diffusions in Gaussian noisy environment’, Stochastics 89, 171–206. <https://doi.org/10.1080/17442508.2016.1146282>
Details
Space-time fractional diffusions in Gaussian noisy environment
Le Chen, Guannan Hu, Yaozhong Hu, and Jingyu Huang
Abstract: This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables:
where \(\dot{W}\) is the space-time white noise, \(\alpha\in(0,2]\), \(\beta\in(0,2)\), \(\gamma\ge 0\) and \(\nu>0\). Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition:
In some cases, the initial data can be measures. When \(\beta\in (0,1]\), we prove the sample path regularity of the solution.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic fractional diffusion equations, measure-valued initial data, Hölder continuity, intermittency, the Fox H-function.
[CHHH17] Chen, Le, Hu, Guannan, Hu, Yaozhong & Huang, Jingyu (2017) ‘Space-time fractional diffusions in Gaussian noisy environment’, Stochastics 89, 171–206. <https://doi.org/10.1080/17442508.2016.1146282>
@article{chen.hu.ea:17:space-time,
author = {Chen, Le and Hu, Guannan and Hu, Yaozhong and Huang, Jingyu},
title = {Space-time fractional diffusions in {G}aussian noisy environment},
journal = {Stochastics},
fjournal = {Stochastics. An International Journal of Probability and Stochastic Processes},
volume = {89},
year = {2017},
number = {1},
pages = {171--206},
issn = {1744-2508},
mrclass = {60H15 (60G22)},
mrnumber = {3574699},
mrreviewer = {Xiliang Fan},
doi = {10.1080/17442508.2016.1146282},
url = {https://doi.org/10.1080/17442508.2016.1146282}
}
References: Bertini and Cancrini [BC95a]; Carmona and Molchanov [CM94]; Chen [Che13]; Chen [Che17a]; Chen and Dalang [CD14a]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Dalang [CD15c]; Chen et al. [CHN21]; Chen and Kim [CK17]; Chen et al. [CHHH17]; Chen et al. [CHSS18]; Chen et al. [CKK15]; Conus et al. [CJKS13a]; Conus et al. [CJKS14]; Dalang [Dal99]; Debbi and Dozzi [DD05]; Diethelm [Die10]; Eidelman and Kochubei [EK04]; Foondun and Khoshnevisan [FK09]; Hu and Hu [HH15]; Kilbas and Saigo [KS04]; Kilbas et al. [KST06]; Kochubeui [Koc90]; Mainardi [Mai10]; Mainardi et al. [MLP01]; Mijena and Nane [MN15]; Mijena and Nane [MN16]; Mueller [Mue91b]; Olver et al. [OLBC10]; Podlubny [Pod99]; Pskhu [Psk09]; Samko et al. [SKM93]; Schneider [Sch96]; Stein and Weiss [SW71]; Walsh [Wal86]; Widder [Wid41]; Wright [Wri40]; Wright [Wri33]; Wright [Wri35];
5. chen.khoshnevisan.ea:16:decorrelation#
[CKK16] Chen, Le, Khoshnevisan, Davar & Kim, Kunwoo (2016) ‘Decorrelation of total mass via energy’, Potential Anal. 45, 157–166. <https://doi.org/10.1007/s11118-016-9540-7>
Details
Decorrelation of total mass via energy
Le Chen, Davar Khoshnevisan, and Kunwoo Kim
Abstract: The main result of this small note is a quantified version of the assertion that if \(u\) and \(v\) solve two nonlinear stochastic heat equations, and if the mutual energy between the initial states of the two stochastic PDEs is small, then the total masses of the two systems are nearly uncorrelated for a very long time. One of the consequences of this fact is that a stochastic heat equation with regular coefficients is a finite system if and only if the initial state is integrable.
MSC 2010 subject classifications: Primary 60H15, 60H25; Secondary 35R60, 60K37, 60J30, 60B15.
Keywords: stochastic heat equation, finite particle systems, total mass, mutual energy.
[CKK16] Chen, Le, Khoshnevisan, Davar & Kim, Kunwoo (2016) ‘Decorrelation of total mass via energy’, Potential Anal. 45, 157–166. <https://doi.org/10.1007/s11118-016-9540-7>
@article{chen.khoshnevisan.ea:16:decorrelation,
author = {Chen, Le and Khoshnevisan, Davar and Kim, Kunwoo},
title = {Decorrelation of total mass via energy},
journal = {Potential Anal.},
fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis},
volume = {45},
year = {2016},
number = {1},
pages = {157--166},
issn = {0926-2601},
mrclass = {60H15 (35K10 35R60 60B15 60H25 60K37)},
mrnumber = {3511809},
mrreviewer = {Ya. \={I}. B\={i}lopol{\cprime}s{\cprime}ka},
doi = {10.1007/s11118-016-9540-7},
url = {https://doi.org/10.1007/s11118-016-9540-7}
}
References: Chen and Dalang [CD15b]; Chen and Kim [CK17]; Dalang [Dal99]; Dalang and Mueller [DM03]; Donoho and Stark [DS89]; Foondun and Khoshnevisan [FK09]; Fukushima et al. [FOT94]; Khoshnevisan [Kho09]; Liggett [Lig05]; Mueller [Mue91b]; Mueller [Mue09]; Spitzer [Spi81]; Walsh [Wal86];
4. chen.dalang:15:moment#
[CD15a] Chen, Le & Dalang, Robert C. (2015) ‘Moment bounds and asymptotics for the stochastic wave equation’, Stochastic Process. Appl. 125, 1605–1628. <https://doi.org/10.1016/j.spa.2014.11.009>
Details
Moment bounds and asymptotics for the stochastic wave equation
Le Chen and Robert C. Dalang
Abstract: We consider the stochastic wave equation on the real line driven by space-time white noise and with irregular initial data. We give bounds on higher moments and, for the hyperbolic Anderson model, explicit formulas for second moments. These bounds imply weak intermittency and allow us to obtain sharp bounds on growth indices for certain classes of initial conditions with unbounded support. system if and only if the initial state is integrable.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic wave equation, hyperbolic Anderson model, intermittency, growth indices.
[CD15a] Chen, Le & Dalang, Robert C. (2015) ‘Moment bounds and asymptotics for the stochastic wave equation’, Stochastic Process. Appl. 125, 1605–1628. <https://doi.org/10.1016/j.spa.2014.11.009>
@article{chen.dalang:15:moment,
author = {Chen, Le and Dalang, Robert C.},
title = {Moment bounds and asymptotics for the stochastic wave equation},
journal = {Stochastic Process. Appl.},
fjournal = {Stochastic Processes and their Applications},
volume = {125},
year = {2015},
number = {4},
pages = {1605--1628},
issn = {0304-4149},
mrclass = {60H15 (35R60 60G60)},
mrnumber = {3310358},
mrreviewer = {Martin Ondrej\'{a}t},
doi = {10.1016/j.spa.2014.11.009},
url = {https://doi.org/10.1016/j.spa.2014.11.009}
}
References: Balan and Conus [BC16]; Brzezniak and Ondreját [BO11]; Brzezniak and Ondreját [BO07]; Cairoli and Walsh [CW75]; Carmona and Nualart [CN88a]; Carmona and Nualart [CN88b]; Carmona and Molchanov [CM94]; Chen [Che13]; Chen and Dalang [CD14b]; Chen and Dalang [CD15b]; Chow [Cho02]; Conus and Dalang [CD08]; Conus and Khoshnevisan [CK12]; Conus et al. [CJKS13a]; Dalang et al. [DKM+09]; Dalang [Dal99]; Dalang [Dal09]; Dalang and Frangos [DF98]; Dalang and Mueller [DM09]; Dalang et al. [DMT08]; Dalang and Quer-Sardanyons [DQS11]; Dalang and Sanz-Solé [DSS09]; Erdélyi et al. [EMOT81a]; Foondun and Khoshnevisan [FK09]; Kevorkian [Kev00]; Millet and Morien [MM01]; Millet and Sanz-Solé [MSS99]; Nualart and Quer-Sardanyons [NQS07]; Olver et al. [OLBC10]; Ondreját [Ond10a]; Ondreját [Ond10b]; Orsingher [Ors82]; Peszat [Pes02]; Peszat and Zabczyk [PZ97]; Quer-Sardanyons and Sanz-Solé [QSSS04]; Sanz-Solé and Sarrà [SSS00]; Walsh [Wal86]; Watson [Wat95];
3. chen.dalang:15:moments*1#
[CD15c] Chen, Le & Dalang, Robert C. (2015) ‘Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation’, Stoch. Partial Differ. Equ. Anal. Comput. 3, 360–397. <https://doi.org/10.1007/s40072-015-0054-x>
Details
Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation
Le Chen and Robert C. Dalang
Abstract: We study the nonlinear fractional stochastic heat equation in the spatial domain \(\mathbb{R}\) driven by space-time white noise. The initial condition is taken to be a measure on \(\mathbb{R}\), such as the Dirac delta function, but this measure may also have non-compact support. Existence and uniqueness, as well as upper and lower bounds on all \(p\)-th moments \((p\ge 2)\), are obtained. These bounds are uniform in the spatial variable, which answers an open problem mentioned in Conus and Khoshnevisan [CK12]. We improve the weak intermittency statement by Foondun and Khoshnevisan [FK09], and we show that the growth indices (of linear type) introduced in [CK12] are infinite. We introduce the notion of ‘’growth indices of exponential type” in order to characterize the manner in which high peaks propagate away from the origin, and we show that the presence of a fractional differential operator leads to significantly different behavior compared with the standard stochastic heat equation.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear fractional stochastic heat equation, parabolic Anderson model, rough initial data, intermittency, growth indices, stable processes.
[CD15c] Chen, Le & Dalang, Robert C. (2015) ‘Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation’, Stoch. Partial Differ. Equ. Anal. Comput. 3, 360–397. <https://doi.org/10.1007/s40072-015-0054-x>
@article{chen.dalang:15:moments*1,
author = {Chen, Le and Dalang, Robert C.},
title = {Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation},
journal = {Stoch. Partial Differ. Equ. Anal. Comput.},
fjournal = {Stochastic Partial Differential Equations. Analysis and Computations},
volume = {3},
year = {2015},
number = {3},
pages = {360--397},
issn = {2194-0401},
mrclass = {60H15 (35R11 35R60 60G60)},
mrnumber = {3383450},
mrreviewer = {Jan I. Seidler},
doi = {10.1007/s40072-015-0054-x},
url = {https://doi.org/10.1007/s40072-015-0054-x}
}
References: Balan and Conus [BC14a]; Balan and Conus [BC16]; Bertini et al. [BCJL94]; Carlen and Krée [CK91]; Carmona and Molchanov [CM94]; Chen and Dalang [CD15a]; Chen and Dalang [CD15b]; Chen and Kim [CK17]; Conus and Khoshnevisan [CK12]; Conus et al. [CJKS14]; Dalang [Dal99]; Debbi and Dozzi [DD05]; Erdélyi et al. [EMOT81b]; Foondun and Khoshnevisan [FK09]; Gawronski [Gaw84]; Khoshnevisan [Kho14]; Lukacs [Luk70]; Mainardi et al. [MLP01]; Oldham et al. [OMS09]; Olver et al. [OLBC10]; Podlubny [Pod99]; Sato [Sat13]; Uchaikin and Zolotarev [UZ99]; Walsh [Wal86]; Yosida [Yos95]; Zolotarev [Zol86];
2. chen.dalang:15:moments#
[CD15b] Chen, Le & Dalang, Robert C. (2015) ‘Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions’, Ann. Probab. 43, 3006–3051. <https://doi.org/10.1214/14-AOP954>
Details
Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions
Le Chen and Robert C. Dalang
Abstract: We study the nonlinear stochastic heat equation in the spatial domain \(\mathbb{R}\), driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on \(\mathbb{R}\), such as the Dirac delta function, but this measure may also have non-compact support and even be non-tempered (for instance with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall’s lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all \(p\)-th moments \((p\ge 2)\) are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when \(p=2\). We determine the growth indices introduced by Conus and Khoshnevisan [CK12].
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic heat equation, parabolic Anderson model, rough initial data, growth indices.
[CD15b] Chen, Le & Dalang, Robert C. (2015) ‘Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions’, Ann. Probab. 43, 3006–3051. <https://doi.org/10.1214/14-AOP954>
@article{chen.dalang:15:moments,
author = {Chen, Le and Dalang, Robert C.},
title = {Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions},
journal = {Ann. Probab.},
fjournal = {The Annals of Probability},
volume = {43},
year = {2015},
number = {6},
pages = {3006--3051},
issn = {0091-1798},
mrclass = {60H15 (35R60 60G60)},
mrnumber = {3433576},
mrreviewer = {Mathew Joseph},
doi = {10.1214/14-AOP954},
url = {https://doi.org/10.1214/14-AOP954}
}
References: Adams and Fournier [AF03b]; Amir et al. [ACQ11]; Bertini and Cancrini [BC95a]; Borodin and Corwin [BC14b]; Carmona and Molchanov [CM94]; Chen [Che13]; Chung and Williams [CW90]; Conus and Khoshnevisan [CK10]; Conus and Khoshnevisan [CK12]; Conus et al. [CJKS14]; Cranston et al. [CMS02]; Dalang and Frangos [DF98]; Dalang et al. [DKN07]; Dalang et al. [DKN09]; Dalang and Mueller [DM09]; Dalang et al. [DMT08]; Erdélyi et al. [EMOT54]; Foondun and Khoshnevisan [FK09]; John [Joh91]; Khoshnevisan [Kho09]; Mueller [Mue91b]; Mytnik et al. [MPS06]; Olver et al. [OLBC10]; Pospíšil and Tribe [PT07]; Sanz-Solé and Sarrà [SSS00]; Sanz-Solé and Sarrà [SSS02]; Shiga [Shi94]; Walsh [Wal86]; Zeldovich et al. [ZRS90];
1. chen.dalang:14:holder-continuity#
[CD14a] Chen, Le & Dalang, Robert C. (2014) ‘Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions’, Stoch. Partial Differ. Equ. Anal. Comput. 2, 316–352. <https://doi.org/10.1007/s40072-014-0034-6>
Details
Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions
Le Chen and Robert C. Dalang
Abstract: We study space-time regularity of the solution of the nonlinear stochastic heat equation in one spatial dimension driven by space-time white noise, with a rough initial condition. This initial condition is a locally finite measure \(\mu\) with, possibly, exponentially growing tails. We show how this regularity depends, in a neighborhood of \(t=0\), on the regularity of the initial condition. On compact sets in which \(t>0\), the classical Hölder-continuity exponents \(\frac{1}{4}-\) in time and \(\frac{1}{2}-\) in space remain valid. However, on compact sets that include \(t=0\), the Hölder continuity of the solution is \(\left(\frac{\alpha}{2}\wedge \frac{1}{4}\right)-\) in time and \(\left(\alpha\wedge \frac{1}{2}\right)-\) in space, provided \(\mu\) is absolutely continuous with an \(\alpha\)-Hölder continuous density.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic heat equation, rough initial data, sample path Hölder continuity, moments of increments.
[CD14a] Chen, Le & Dalang, Robert C. (2014) ‘Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions’, Stoch. Partial Differ. Equ. Anal. Comput. 2, 316–352. <https://doi.org/10.1007/s40072-014-0034-6>
@article{chen.dalang:14:holder-continuity, author = {Chen, Le and Dalang, Robert C.}, title = {H\"{o}lder-continuity for the nonlinear stochastic heat equation with rough initial conditions}, journal = {Stoch. Partial Differ. Equ. Anal. Comput.}, fjournal = {Stochastic Partial Differential Equations. Analysis and Computations}, volume = {2}, year = {2014}, number = {3}, pages = {316--352}, issn = {2194-0401}, mrclass = {60H15 (35B65 35K15 35R06 35R60 60G60)}, mrnumber = {3255231}, mrreviewer = {Anna Karczewska}, doi = {10.1007/s40072-014-0034-6}, url = {https://doi.org/10.1007/s40072-014-0034-6} }
References: Bally et al. [BMSS95]; Bertini and Cancrini [BC95a]; Brzezniak [Brz97]; Chen [Che13]; Chen and Dalang [CD15b]; Conus and Khoshnevisan [CK10]; Conus and Khoshnevisan [CK12]; Conus et al. [CJKS14]; Dalang [Dal09]; Dalang et al. [DKN07]; Dalang et al. [DKN09]; Foondun and Khoshnevisan [FK09]; Friedman [Fri63]; Gelfand and Vilenkin [GV16]; Khoshnevisan [Kho09]; Kunita [Kun90]; Kuo [Kuo06]; Olver et al. [OLBC10]; Peszat and Seidler [PS98]; Pospíšil and Tribe [PT07]; Revuz and Yor [RY99]; Sanz-Solé and Sarrà [SSS00]; Sanz-Solé and Sarrà [SSS02]; Seidler [Sei93]; Shiga [Shi94]; Stein [Ste70]; Walsh [Wal86];
Ph.D. Thesis#
0. chen:13:moments#
[Che13] Chen, Le (2013) ‘Moments, Intermittency, and Growth Indices for Nonlinear Stochastic PDE’s with Rough Initial Conditions’, EPFL Ph.D. Thesis. <https://doi.org/10.5075/epfl-thesis-5712>
Details
Moments, Intermittency, and Growth Indices for Nonlinear Stochastic PDE’s with Rough Initial Conditions
Le Chen
The École polytechnique fédérale de Lausanne
(EPFL)
Switzerland
Thesis commitee:
Prof. K. Krieger (EPFL, president)
Prof. Robert Dalang (EPFL, thesis director/supervisor)
Prof. Thomas Mountford (EPFL)
Prof. Davar Khoshnevisan (University of Utah, USA)
Prof. Roger Tribe (University of Warwick, UK)
Abstract: In this thesis, we study several stochastic partial differential equations (SPDE’s) in the spatial domain \(\mathbb{R}\), driven by multiplicative space-time white noise. We are interested in how rough and unbounded initial data affect the random field solution and the asymptotic properties of this solution.
We first study the nonlinear stochastic heat equation. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on \(\mathbb{R}\), such as the Dirac delta function, but this measure may also have non-compact support and even be non-tempered (for instance with exponentially growing tails). Existence and uniqueness is proved without appealing to Gronwall’s lemma, by keeping tight control over moments in the Picard iteration scheme. Upper and lower bounds on all \(p\)-th moments \((p\ge 2)\) are obtained. These bounds become equalities for the parabolic Anderson model when \(p=2\). We determine the growth indices introduced by Conus and Khoshnevisan [CK12] and, despite the irregular initial conditions, we establish Hölder continuity of the solution for \(t>0\).
In order to study a wider class of SPDE’s, we consider a more general problem, consisting in a stochastic integral equation of space-time convolution type. We give a set of assumptions which guarantee that the stochastic integral equation in question has a unique random field solution, with moment formulas and sample path continuity properties. As a first application, we show how certain properties of an extra potential term in the nonlinear stochastic heat equation influence the admissible initial data. As a second application, we investigate the nonlinear stochastic wave equation on \(\mathbb{R}_+\times\mathbb{R}\). All the properties obtained for the stochastic heat equation – moment formulas, growth indices, Hölder continuity, etc. – are also obtained for the stochastic wave equation.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: nonlinear stochastic heat equation, nonlinear stochastic wave equation, parabolic Anderson model, hyperbolic Anderson model, rough initial data, Hölder continuity, Lyapunov exponents, growth indices.
[Che13] Chen, Le (2013) ‘Moments, Intermittency, and Growth Indices for Nonlinear Stochastic PDE’s with Rough Initial Conditions’, EPFL Ph.D. Thesis. <https://doi.org/10.5075/epfl-thesis-5712>
@article{chen:13:moments,
title = {Moments, Intermittency, and Growth Indices for Nonlinear Stochastic PDE's with Rough Initial Conditions},
author = {Chen, Le},
journaltitle = {EPFL Ph.D. Thesis},
institution = {MATHAA},
publisher = {EPFL},
address = {Lausanne},
year = {2013},
url = {http://infoscience.epfl.ch/record/185885},
doi = {10.5075/epfl-thesis-5712}
}
References:
Unpublished notes#
-3. chen:16:third#
[Che16a] Le Chen (2016) ‘The third moment for the parabolic Anderson model’, Preprint arXiv:1609.01005
Details
The third moment for the parabolic Anderson model
Le Chen
University of Kansas
Abstract: In this paper, we study the parabolic Anderson model starting from the Dirac delta initial data:
where \(\dot{W}\) denotes the space-time white noise. By evaluating the threefold contour integral in the third moment formula by Borodin and Corwin [BC14c], we obtain some explicit formulas for \(\mathbb{E}[u(t,x)^3]\). One application of these formulas is given to show the exact phase transition for the intermittency front of order three. These moment formulas enable us to give another characterization for the intermittency fronts based on some integrability conditions, from which we are able to obtain some almost sure results.
MSC 2010 subject classifications: Primary 60H15; Secondary 35R60.
Keywords: stochastic heat equation, parabolic Anderson model, Dirac delta initial condition, space-time white noise, moment formula, intermittency fronts, growth indices
[Che16a] Le Chen (2016) ‘The third moment for the parabolic Anderson model’, Preprint arXiv:1609.01005
@article{chen:16:third,
title = {The third moment for the parabolic Anderson model},
author = {Le Chen},
year = {2016},
month = {September},
journal = {Preprint arXiv:1609.01005},
url = {https://www.arxiv.org/abs/1609.01005}
}
Figures:
\(λ=3\)#
\(λ=4\)#
\(λ=5\)#
\(λ=6\)#
\(λ=7\)#
\(λ=8\)#
References: Bertini and Cancrini [BC95a]; Borodin and Corwin [BC14c]; Carmona and Molchanov [CM94]; Chen and Dalang [CD15b]; Chen et al. [CHN17]; Chen and Kim [CK17]; Chen and Kim [CK19]; Chen [Che15]; Foondun and Khoshnevisan [FK09]; Kardar [Kar87]; König [Kon16]; Olver et al. [OLBC10]; Walsh [Wal86];
-4. chen.huang:19:regularity#
[CH19b] Le Chen & Jingyu Huang (2019) ‘Regularity and strict positivity of densities for the stochastic heat equation on \(\mathbb{R}^d\)’, Preprint arXiv:1902.02382
Details
Regularity and strict positivity of densities for the stochastic heat equation on \(\mathbb{R}^d\)
Le Chen and Jingyu Huang
Abstract: In this paper, we study the stochastic heat equation with a general multiplicative Gaussian noise that is white in time and colored in space. Both regularity and strict positivity of the densities of the solution have been established. The difficulty, and hence the contribution, of the paper lie in three aspects: rough initial conditions, degenerate diffusion coefficient, and weakest possible assumptions on the correlation function of the noise. In particular, our results cover the parabolic Anderson model starting from a Dirac delta initial measure.
MSC 2010 subject classifications: Primary 60H15; Secondary 35R60, 60G60.
Keywords: stochastic heat equation, parabolic Anderson model, Malliavin calculus, negative moments, regularity of density, strict positivity of density, measure-valued initial conditions, spatially colored noise.
[CH19b] Le Chen & Jingyu Huang (2019) ‘Regularity and strict positivity of densities for the stochastic heat equation on \(\mathbb{R}^d\)’, Preprint arXiv:1902.02382
@article{chen.huang:19:regularity,
title = {Regularity and strict positivity of densities for the stochastic heat equation on $\mathbb{R}^d$},
author = {Le Chen and Jingyu Huang},
year = {2019},
month = {February},
journal = {Preprint arXiv:1902.02382},
url = {https://www.arxiv.org/abs/1902.02382}
}
References: Amir et al. [ACQ11]; Balan and Chen [BC18]; Bally and Pardoux [BP98]; Bertini and Giacomin [BG97]; Bouleau and Hirsch [BH91]; Cannizzaro et al. [CFG17]; Carmona and Molchanov [CM94]; Chen and Dalang [CD15b]; Chen et al. [CHN21]; Chen and Huang [CH19a]; Chen and Kim [CK19]; Dalang et al. [DKM+09]; Dalang [Dal99]; Dalang and Quer-Sardanyons [DQS11]; Foondun and Khoshnevisan [FK13]; Grafakos [Gra14]; Hairer [Hai14]; Hairer [Hai13]; Hu et al. [HHNS15]; Mueller and Nualart [MN08]; Nualart [Nua09]; Nualart and Quer-Sardanyons [NQS07]; Nualart [Nua13]; Olver et al. [OLBC10]; Pardoux and Zhang [PZ93]; Quer-Sardanyons and Sanz-Solé [QSSS04]; Walsh [Wal86];