19. chen.hu.ea:21:regularity#
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 The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model.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];