Calderon-Type Uniqueness Theorem for Stochastic Partial Differential Equations

TL;DR

This paper establishes a Calderón-type uniqueness theorem for stochastic PDEs using stochastic pseudo-differential operators and a novel Carleman estimate, advancing the theoretical understanding of stochastic inverse problems.

Contribution

It introduces stochastic pseudo-differential operators and proves a new Carleman estimate, leading to a uniqueness theorem for stochastic PDEs.

Findings

Proves a Calderón-type uniqueness theorem for stochastic PDEs.

Develops the concept and properties of stochastic pseudo-differential operators.

Establishes a new Carleman-type estimate for stochastic equations.

Abstract

In this Note, we present a Calder\'on-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations. To this aim, we introduce the concept of stochastic pseudo-differential operators, and establish their boundedness and other fundamental properties. The proof of our uniqueness theorem is based on a new Carleman-type estimate.