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Discrete Feynman-Kac approximation for parabolic Anderson model using random walks

Panqiu Xia, Jiayu Zheng

TL;DR

This work develops a natively positive, discrete Feynman-Kac scheme to approximate the parabolic Anderson model driven by a fractional Brownian sheet, using random walks to discretize the noise in time and space. It provides rigorous L^p error bounds that reflect the Hölder regularity of the PAM solution, covering both flat and delta initial conditions, and extends from space-time white noise to coloured noises with rates that depend on the Hurst parameters H and H_*. A thorough treatment is given for the white-noise case and its coloured extensions, with delta-initial-condition results proven via a density-matching analysis of the discrete kernels. The paper also connects the approximation to the directed-polymer framework, showing how the partition function in Gaussian environments converges to PAM in distribution and giving Wasserstein-distance rates, thereby offering a quantitative bridge between numerical schemes for SPDEs and polymer models in random media.

Abstract

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.

Discrete Feynman-Kac approximation for parabolic Anderson model using random walks

TL;DR

This work develops a natively positive, discrete Feynman-Kac scheme to approximate the parabolic Anderson model driven by a fractional Brownian sheet, using random walks to discretize the noise in time and space. It provides rigorous L^p error bounds that reflect the Hölder regularity of the PAM solution, covering both flat and delta initial conditions, and extends from space-time white noise to coloured noises with rates that depend on the Hurst parameters H and H_*. A thorough treatment is given for the white-noise case and its coloured extensions, with delta-initial-condition results proven via a density-matching analysis of the discrete kernels. The paper also connects the approximation to the directed-polymer framework, showing how the partition function in Gaussian environments converges to PAM in distribution and giving Wasserstein-distance rates, thereby offering a quantitative bridge between numerical schemes for SPDEs and polymer models in random media.

Abstract

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 and in time and space, respectively, we also provide an error analysis of the proposed method. The error in norm is of order \[ O \big(h^{\frac{1}{2}[(2H + H_* - 1) \wedge 1] - ε}\big), \] where is the step size in time (resp. in space), and can be chosen arbitrarily small. This error order matches the Hölder continuity of the solution in time with a correction order , 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.
Paper Structure (16 sections, 23 theorems, 140 equations, 1 figure)

This paper contains 16 sections, 23 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Discrete Feynman-Kac approximation
  3. Prospects for future work
  4. Organisation of the paper
  5. Preliminaries
  6. Notation
  7. Multiple integrals and chaos expansion
  8. Results on \ref{['pam']}
  9. Technical lemmas
  10. \ref{['pam']} with flat initial condition
  11. White noise
  12. Coloured noises
  13. \ref{['pam']} with delta initial condition
  14. Proof of Theorem \ref{['thm_rate-delta']}
  15. Proof of Lemmas \ref{['lmm-chaos-h-delta']}--\ref{['prop_f-g-delta']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $u$ be the solution to pam with a flat initial condition, namely, $u(0, \cdot) \equiv 1$, driven by a fractional Brownian sheet with Hurst parameters $H \geq \frac{1}{2}$ and $H_* \geq \frac{1}{2}$ in time and space, respectively. Let $u_h = \{u_h(m, n) \coloneqq (m, n) \in \mathbb{Z}_{\geq 0}\t where, for any nonnegative functions $A$ and $B$ defined on the same domain $\rm Dom$, $A(x)\lesssi

Figures (1)

  • Figure 1: Comparison of densities of Brownian motion and rescaled random walk.

Theorems & Definitions (41)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3: Hardy-Littlewood-Sobolev inequality, cf. spl-01-memin-mishura-valkeila
  • Lemma 2.4: Tail probability for normals, cf. ams-41-gordon
  • Lemma 2.5: Bernstein's inequality, cf. jasa-62-bennett
  • Lemma 2.6: Local limit theorem, cf. springer-75-petrov
  • Remark 2.7
  • Corollary 2.8
  • ...and 31 more