Unique Continuation Property for Stochastic Wave Equations

TL;DR

This paper proves that stochastic wave equations can restore the unique continuation property across characteristic surfaces, a phenomenon that fails in deterministic cases, using a novel stochastic Carleman estimate.

Contribution

It introduces the first proof of UCP for stochastic wave equations across characteristic surfaces, contrasting the deterministic failure, via a new stochastic Carleman estimate.

Findings

UCP holds for stochastic wave equations across characteristic hypersurfaces.

The stochastic Carleman estimate reveals a positive energy contribution from Itô diffusion.

Results differ qualitatively from deterministic wave equations, impacting control and inverse problems.

Abstract

This paper establishes a fundamental and surprising phenomenon in the theory of stochastic wave equations: the restoration of the unique continuation property (UCP) across characteristic hypersurfaces, a property that is known to fail generically in the deterministic setting. We prove that if a solution to a linear stochastic wave equation vanishes on one side of a characteristic surface $\Gamma$, then it must vanish in a full neighborhood of any point on $\Gamma$, provided the stochastic diffusion coefficient is non-degenerate. This result stands in sharp contrast to the classical H\"ormander-type counterexamples for deterministic waves. Furthermore, we extend the UCP to equations with non-homogeneous stochastic sources and establish a global unique continuation result from the interior of an arbitrarily narrow characteristic cone. Our proofs rely on a novel stochastic Carleman estimate, where the It\^o diffusion term introduces a crucial positive energy contribution that is absent in deterministic models. These findings demonstrate a qualitative difference between deterministic and stochastic hyperbolic dynamics and open new avenues for control theory and inverse problems in stochastic setting.