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Parabolic Anderson model with colored noise on torus

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arxiv 2308.10802 v1 pith:GY67MOBN submitted 2023-08-21 math.PR

Le Chen , Cheng Ouyang , William Vickery This is my paper · ORCID
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abstract

We construct an intrinsic family of Gaussian noises on $d$-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 torus in the It\^{o} 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 H\"{o}lder continuity in space and time. The study of the toy model of $\mathbb{T}^d$ in the present paper is a first step towards our effort in understanding how geometry and topology play an role in the behavior of stochastic PDEs on general (compact) manifolds.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A stochastic heat equation with non-locally Lipschitz coefficients

    math.PR 2025-07 conditional novelty 6.0 of 10

    For the stochastic heat equation on the torus with coefficients growing like u|log u|^A near zero (A<1 for drift, A<1/4 for noise), a unique global strictly positive mild solution exists.

  2. On ergodic properties of stochastic PDEs

    math.PR 2024-12 conditional novelty 6.0 of 10

    For the parabolic Anderson model in dimension d≥3 with weak multiplicative noise, the solution converges in distribution to a limiting random field, extending earlier ergodicity results to broader noises and initial c...

  3. Moment estimates for the stochastic heat equation on Cartan-Hadamard manifolds

    math.PR 2024-11 reject novelty 6.0 of 10

    For the Parabolic Anderson model on Cartan-Hadamard manifolds, the paper establishes a curvature-dependent Dalang condition, exponential moment upper bounds with a spectral-gap term, and asymptotically matching lower bounds.

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