3. chen.dalang:15:moments*1#
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];