Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions

Hui Sun; Yangyang Lyu; Hui Sun; Yangyang Lyu

doi:10.3934/math.20241659

AIMS Mathematics

2024, Volume 9, Issue 12: 34838-34862. doi: 10.3934/math.20241659

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Research article

Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions

Hui Sun ,
Yangyang Lyu ^,

School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China

Received: 20 August 2024 Revised: 18 November 2024 Accepted: 05 December 2024 Published: 13 December 2024
MSC : 60F99, 60G60, 60H15

In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on $ \mathbb{R}^d $, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that $ u_0\equiv C $ and $ u_0\in C^\kappa(\mathbb{R}^d) $ with $ \kappa\in(0, 1] $.
- parabolic Anderson model,
- Feynman-Kac formula,
- generalized Gaussian field,
- Brownian bridge,
- Hölder continuity,
- measure-valued initial data
Citation: Hui Sun, Yangyang Lyu. Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions[J]. AIMS Mathematics, 2024, 9(12): 34838-34862. doi: 10.3934/math.20241659

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Abstract

In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on $ \mathbb{R}^d $, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that $ u_0\equiv C $ and $ u_0\in C^\kappa(\mathbb{R}^d) $ with $ \kappa\in(0, 1] $.

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