Unique Continuation Property for Stochastic Wave Equations
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915761226776576 |
|---|---|
| author | Lü, Qi Liao, Zhonghua |
| author_facet | Lü, Qi Liao, Zhonghua |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21854 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unique Continuation Property for Stochastic Wave Equations Lü, Qi Liao, Zhonghua Analysis of PDEs Probability 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. |
| title | Unique Continuation Property for Stochastic Wave Equations |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2601.21854 |