Fokker-Planck equation for stochastic heat equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912563656130560 |
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| author | Meng, Qingyan Duan, Jinqiao Wei, Jinlong Kloeden, Peter E. |
| author_facet | Meng, Qingyan Duan, Jinqiao Wei, Jinlong Kloeden, Peter E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fokker-Planck equation for stochastic heat equations Meng, Qingyan Duan, Jinqiao Wei, Jinlong Kloeden, Peter E. Probability 35Q84, 35R60, 60H15, 97K50 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. |
| title | Fokker-Planck equation for stochastic heat equations |
| topic | Probability 35Q84, 35R60, 60H15, 97K50 |
| url | https://arxiv.org/abs/2509.01283 |