Discrete Feynman-Kac approximation for parabolic Anderson model using random walks
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909976999493632 |
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| author | Xia, Panqiu Zheng, Jiayu |
| author_facet | Xia, Panqiu Zheng, Jiayu |
| contents | In this paper, we introduce a natively positive approximation method based on the Feynman-Kac representation using random walks, to approximate the solution to the one-dimensional parabolic Anderson model of Skorokhod type, with either a flat or a Dirac delta initial condition. Assuming the driving noise is a fractional Brownian sheet with Hurst parameters $H \geq \frac{1}{2}$ and $H_* \geq \frac{1}{2}$ in time and space, respectively, we also provide an error analysis of the proposed method. The error in $L^p (Ω)$ norm is of order \[ O \big(h^{\frac{1}{2}[(2H + H_* - 1) \wedge 1] - ε}\big), \] where $h > 0$ is the step size in time (resp. $\sqrt{h}$ in space), and $ε> 0$ can be chosen arbitrarily small. This error order matches the Hölder continuity of the solution in time with a correction order $ε$, making it `almost' optimal. Furthermore, these results provide a quantitative framework for convergence of the partition function of directed polymers in Gaussian environments to the parabolic Anderson model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22844 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discrete Feynman-Kac approximation for parabolic Anderson model using random walks Xia, Panqiu Zheng, Jiayu Probability In this paper, we introduce a natively positive approximation method based on the Feynman-Kac representation using random walks, to approximate the solution to the one-dimensional parabolic Anderson model of Skorokhod type, with either a flat or a Dirac delta initial condition. Assuming the driving noise is a fractional Brownian sheet with Hurst parameters $H \geq \frac{1}{2}$ and $H_* \geq \frac{1}{2}$ in time and space, respectively, we also provide an error analysis of the proposed method. The error in $L^p (Ω)$ norm is of order \[ O \big(h^{\frac{1}{2}[(2H + H_* - 1) \wedge 1] - ε}\big), \] where $h > 0$ is the step size in time (resp. $\sqrt{h}$ in space), and $ε> 0$ can be chosen arbitrarily small. This error order matches the Hölder continuity of the solution in time with a correction order $ε$, making it `almost' optimal. Furthermore, these results provide a quantitative framework for convergence of the partition function of directed polymers in Gaussian environments to the parabolic Anderson model. |
| title | Discrete Feynman-Kac approximation for parabolic Anderson model using random walks |
| topic | Probability |
| url | https://arxiv.org/abs/2512.22844 |