Hitting Probabilities for Approximations of Systems of Stochastic Heat Equations

Hitting probability is a fundamental concept in probabilistic potential theory, with applications in physics, probability, and partial differential equations. For systems of stochastic heat equations driven by space–time white noise, this chapter establishes lower and upper bounds for hitting probabilities of the numerical approximations in terms of Bessel–Riesz capacity and Hausdorff measure, respectively. It turns out that standard spatial and temporal discretization schemes may fail to preserve the critical dimension associated with the original hitting probability. For comparison, we also analyze the hitting probabilities of numerical approximations for systems of stochastic ordinary differential equations, showing that the original critical dimension is preserved.