On recovering solutions for SPDEs from their averages

TL;DR

This paper introduces a new boundary value problem for linear parabolic SPDEs where the initial condition is replaced by an average over time and probability space, enabling solution recovery when initial data is unknown.

Contribution

It establishes well-posedness, existence, and regularity results for this novel boundary value problem involving averages, expanding methods for solving forward and backward SPDEs.

Findings

Proves existence and uniqueness of solutions under the new boundary conditions.

Demonstrates the possibility of recovering solutions from observable averages.

Provides regularity results for solutions of the modified SPDE problem.

Abstract

We study linear stochastic partial differential equations of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution either over time and probabilistic space for forward SPDEs and over time for backward SPDEs. Well-posedness, existence, uniqueness, and a regularity of the solution for this new problem are obtained. In particular, this can be considered as a possibility to recover a solution of a forward SPDE in a setting where its values at the initial time are unknown, and where the average of the solution over time and probability space is observable, as well as the input processes.