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

TL;DR

This paper introduces a positive approximation method based on Feynman-Kac and random walks for the 1D parabolic Anderson model driven by fractional Brownian noise, with error analysis showing near-optimal convergence rates.

Contribution

It presents a novel positive approximation technique using random walks for the parabolic Anderson model with fractional noise, including detailed error bounds.

Findings

Error order matches the solution's Hölder continuity in time.

Method provides a quantitative convergence framework for directed polymers.

Approximation is positive and suitable for fractional Brownian sheet noise.

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 $L^p (\Omega)$ norm is of order \[ O \big(h^{\frac{1}{2}[(2H + H_* - 1) \wedge 1] - \epsilon}\big), \] where $h > 0$ is the step size in time (resp. $\sqrt{h}$ in space), and $\epsilon > 0$ can be chosen arbitrarily small. This error order matches the H\"older continuity of the solution in time with a correction order $\epsilon$, 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.