Stationary solutions of stochastic partial differential equations in the space of tempered distributions

TL;DR

This paper investigates the stationarity of solutions to certain SPDEs in the space of tempered distributions, establishing conditions under which stationary finite-dimensional SDE solutions correspond to stationary SPDE solutions.

Contribution

It extends previous work by showing that stationary solutions of finite-dimensional SDEs lead to stationary solutions of associated SPDEs in the space of tempered distributions under specific conditions.

Findings

Stationary finite-dimensional SDE solutions imply stationary SPDE solutions.

The correspondence holds when initial variables are in a specific set ensuring coefficient relations.

Provides conditions for stationarity transfer between SDEs and SPDEs.

Abstract

In Rajeev (2013), 'Translation invariant diffusion in the space of tempered distributions', it was shown that there is an one to one correspondence between solutions of a class of finite dimensional SDEs and solutions of a class of SPDEs in $\mathcal{S}'$, the space of tempered distributions, driven by the same Brownian motion. There the coefficients $\bar{\sigma}, \bar{b}$ of the finite dimensional SDEs were related to the coefficients of the SPDEs in $\mathcal{S}'$ in a special way, viz. through convolution with the initial value $y$ of the SPDEs. In this paper, we consider the situation where the solutions of the finite dimensional SDEs are stationary and ask whether the corresponding solutions of the equations in $\mathcal{S}'$ are also stationary. We provide an affirmative answer, when the initial random variable takes value in a certain set $\mathcal{C}$, which ensures that the coefficients of the finite dimensional SDEs are related to the coefficients of the SPDEs in the above `special' manner.