Fokker-Planck equation for stochastic heat equations

Qingyan Meng; Jinqiao Duan; Jinlong Wei; Peter E. Kloeden

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arXiv · 2509.01283

Fokker-Planck equation for stochastic heat equations

Qingyan Meng·Jinqiao Duan·Jinlong Wei·Peter E. Kloeden

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Abstract

This work is devoted to the study of the Fokker--Planck equation for a stochastic heat equation with an additive $Q$-Wiener noise and non-homogeneous boundary conditions. We explicitly construct the probability density function and establish the associated Fokker--Planck equation by applying the eigenfunction expansion technique. Moreover, the Feynman--Kac formula is used to obtain the probabilistic representation of the solution. The analysis is further extended to cases with multiplicative noise involving nonlocal diffusion operators under homogeneous boundary conditions, as well as the corresponding Kardar--Parisi--Zhang (KPZ) equation. Notably, the evolution of the probability density function for the stochastic heat equation depends critically on the spatial location.

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BibTeX RIS

Qingyan Meng, Jinqiao Duan, Jinlong Wei, Peter E. Kloeden. 2025-09-01. Fokker-Planck equation for stochastic heat equations. https://arxiv.org/abs/2509.01283

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