Asymptotic properties of stochastic partial differential equations in the sublinear regime

Chen, Le and Xia, Panqiu 2026. Asymptotic properties of stochastic partial differential equations in the sublinear regime. The Annals of Probability 54 (4) , pp. 1686-1714. 10.1214/24-AOP1741

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Abstract

In this paper we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish nontrivial moment upper bounds and almost sure spatial asymptotic properties for solutions. These results shed light on the smoothing intermittency effect under weak diffusion (i.e., sublinear growth) previously observed by Zeldovich et al. (Proc. Natl. Acad. Sci. USA 84 (1987) 6323–6325). The sample-path spatial asymptotics obtained in this paper partially bridge a gap in earlier works of Conus et al. (Ann. Probab. 41 (2013) 2225–2260; Probab. Theory Related Fields 156 (2013) 483–533), 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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Schools > Mathematics
Publisher:	Institute of Mathematical Statistics
ISSN:	0091-1798
Date of First Compliant Deposit:	19 November 2024
Date of Acceptance:	31 October 2024
Last Modified:	03 Aug 2026 06:30
URI:	https://orca.cardiff.ac.uk/id/eprint/174135

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