Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems

Abstract

In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we study two inverse problems for stochastic parabolic equations. One is concerned with a determination problem of the history of a stochastic heat process through the observation at the final time T , for which we obtain a conditional stability estimate. The other is an inverse source problem with observation on the lateral boundary. We derive the uniqueness of the source.2010 Mathematics Subject Classification. Primary 65N21, 60H15.

A weighted identity for stochastic partial differential operators and its applications

“…As far as we know, there exist few works on global Carleman estimates for stochastic partial differential equations. We refer to [17,18,27] for some known results in this respect. However, there is not any known Carleman estimate for general stochastic partial differential operators with complex principal parts.…”

Carleman Estimate and Unique Continuation Property for the Linear Stochastic Korteweg–de Vries Equation

“…P. Gao [6] From ( (2.6) It follows that {y n } ∞ n=1 is a Cauchy sequence that converges strongly in X T .…”

“…In recent years, a great deal of effort has been devoted to studying the controllability of stochastic partial differential equations (see, for instance, [1,6,7,9,10]). The Carleman estimates for the stochastic heat equation, wave equation and Schrödinger equation are complete, but nothing is known for the third-order stochastic dispersion equation.…”

Observability estimate and state observation problems for stochastic hyperbolic equations

“…In [15], the author studied an inverse source problem for stochastic parabolic equations involved in some special domain. It seems that it is hard to generalize the method in [15] for the study of stochastic partial differential equations in general domains.…”

“…In [25], the author proves a boundary observability estimate for the equation (1.2) with (b ij ) 1≤i,j≤n being the identity matrix. More precisely, the author proves that |(z(t), z t (t))| L 2 (Ω,Ft,P ;H 1 15) where z solves the equation (1.2), T satisfies…”