Unique Continuation Property for Stochastic Wave Equations

TL;DR

The paper proves that stochastic perturbations can restore unique continuation for linear stochastic wave equations across characteristic surfaces, a property that fails in the deterministic setting. It introduces a stochastic Carleman estimate that leverages Itô quadratic variation to produce a positive energy contribution even on characteristic sets, enabling local UCP results; these extend to equations with stochastic diffusion and stochastic sources, and a global UCP is obtained via an iterative cone-based geometry. The approach reveals a qualitative difference between stochastic and deterministic hyperbolic dynamics and has implications for control and inverse problems in SPDEs. The results hinge on a carefully crafted weight construction with $\psi$, $\phi$, $\ell$, and $\theta$ and a main Carleman inequality (Theorem $\text{th3}$) that drives all UCP conclusions.

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 $Γ$, then it must vanish in a full neighborhood of any point on $Γ$, provided the stochastic diffusion coefficient is non-degenerate. This result stands in sharp contrast to the classical Hörmander-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 Itô 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.

Figures (1)

• Figure 1: Using the local unique continuation property established in Lemma \ref{['lemma6.1']}, we divide the proof of Theorem \ref{['th1']} into the following steps. Step 1. Fix $(t_0,x_0)=(0,0)$. Since $u\equiv0$ in $Q_0(0,0)$ (the yellow region in Figure (a)), Lemma \ref{['lemma6.1']} immediately gives $u\equiv0$ in the gray region of Figure (a). Step 2. Let $(t_0,x_0)$ vary along the boundary $\partial Q_0(0,0)$. Repeating the argument of Step 1 at each such point shows that $u\equiv0$ in the gray region of Figure (b). Step 3. We now apply Lemma \ref{['lemma6.1']} iteratively, sliding the base point $(t_0,x_0)$ along the boundary of the zero region obtained in the previous step. This extends the vanishing property successively to the green domains shown in Figures (c) and (d). Step 4. Finally, the domain $\{(t,x)\mid t\ge T_0\}$ can be covered by the union of the zero regions constructed in the preceding iterations. Together with a backward uniqueness argument, this implies that $u\equiv0$ for all $t\ge0$, which completes the proof of Theorem \ref{['th1']}.

Theorems & Definitions (27)

• Definition 1.1
• Theorem 2.1
• Remark 2.1
• Corollary 2.1
• Remark 2.2
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.3
• Remark 2.5