Influences of Numerical Discretizations on Hitting Probabilities for Linear Stochastic Parabolic System

TL;DR

This paper analyzes how standard numerical discretizations affect hitting probabilities in linear stochastic parabolic systems driven by space-time white noise, revealing differences in critical dimensions and convergence properties.

Contribution

It establishes bounds for hitting probabilities of numerical solutions and shows that discretizations alter critical dimensions compared to the exact solution.

Findings

Bounds for hitting probabilities in terms of capacity and Hausdorff measure

Critical dimensions of discretized solutions are half of the exact solution's

Hitting probabilities of numerical solutions may not converge to those of the exact solution

Abstract

This paper investigates the influences of standard numerical discretizations on hitting probabilities for linear stochastic parabolic system driven by space-time white noises. We establish lower and upper bounds for hitting probabilities of the associated numerical solutions of both temporal and spatial semi-discretizations in terms of Bessel-Riesz capacity and Hausdorff measure, respectively. Moreover, the critical dimensions of both temporal and spatial semi-discretizations turn out to be half of those of the exact solution. This reveals that for a large class of Borel sets $A$, the probability of the event that the paths of the numerical solution hit $A$ cannot converge to that of the exact solution.