An inverse potential problem for the stochastic heat equation with space-time noise

TL;DR

This paper addresses the inverse problem of identifying the covariance operator of a random potential in a stochastic heat equation driven by space-time noise, demonstrating unique identifiability from solution correlations.

Contribution

It introduces a method to uniquely determine the covariance operator from the correlation of the solution at a final time, leveraging tensor product structures and operator monotonicity.

Findings

Covariance operator can be uniquely identified from solution correlation.

The approach relies on tensor product structure and monotonicity properties.

Results apply to stochastic heat equations with spatially colored, temporally white noise.

Abstract

This paper investigates an inverse potential problem for the stochastic heat equation driven by space-time Gaussian noise, which is spatially colored and temporally white. The objective is to determine the covariance operator of the random potential. We establish that the covariance operator can be uniquely identified from the correlation of the mild solution to the stochastic heat equation at a final time, where the initial conditions are specified by a complete orthonormal basis. The analysis relies on characterizing a tensor product structure inherent in the problem and utilizing the monotonicity properties of the operators associated with the system.