Convergence of Density Approximations for Stochastic Heat Equation

TL;DR

This paper proves the convergence of density approximations for the stochastic heat equation in both uniform and total variation metrics, establishing specific convergence rates and marking a first in SPDE density approximation results.

Contribution

It demonstrates the convergence of density approximations for the stochastic heat equation in both uniform and total variation distances, with precise convergence rates.

Findings

Density approximations converge in uniform topology with order 1/2 (nonlinear) and nearly 1 (linear).

Distributions of approximations converge to the original in total variation distance.

First known result on density approximation convergence for SPDEs.

Abstract

This paper investigates the convergence of density approximations for stochastic heat equation in both uniform convergence topology and total variation distance. The convergence order of the densities in uniform convergence topology is shown to be exactly $1/2$ in the nonlinear case and nearly $1$ in the linear case. This result implies that the distributions of the approximations always converge to the distribution of the origin equation in total variation distance. As far as we know, this is the first result on the convergence of density approximations to the stochastic partial differential equation.