Discrete Feynman-Kac Approximation for Parabolic Anderson Model
Analysis
This paper introduces a novel, positive approximation method for the parabolic Anderson model, leveraging the Feynman-Kac representation and random walks. The key contribution is an error analysis for the approximation, demonstrating a convergence rate that is nearly optimal, matching the Hölder continuity of the solution. This work is significant because it provides a quantitative framework for understanding the convergence of directed polymers to the parabolic Anderson model, a crucial connection in statistical physics.
Key Takeaways
- •Introduces a positive approximation method for the parabolic Anderson model using Feynman-Kac representation and random walks.
- •Provides an error analysis with a nearly optimal convergence rate.
- •Offers a quantitative framework for the convergence of directed polymers to the parabolic Anderson model.
Reference
“The error in $L^p (Ω)$ norm is of order \[ O ig(h^{ rac{1}{2}[(2H + H_* - 1) \wedge 1] - ε}ig), \] where $h > 0$ is the step size in time (resp. $\sqrt{h}$ in space), and $ε> 0$ can be chosen arbitrarily small.”