An inverse potential problem for the stochastic heat equation with space-time noise

TL;DR

This work tackles the inverse problem of identifying the covariance operator $Q$ of a space-time Gaussian random potential driving the stochastic heat equation with Dirichlet boundary conditions. By formulating the second-moment correlation $\theta(t,u_0)=\mathbb{E}[u(t,u_0)\otimes u(t,u_0)]$ and constructing finite-difference-like quantities $\theta^{i,j}(t)$, the authors prove that $Q$ is uniquely determined from $\{\theta^{i,j}(t_0)\}$ at a fixed time $t_0>0$, under a regularity condition $\Lambda_{\gamma,Q}<\infty$. The key method combines a tensor-product representation, the spectral analysis of the operator $A_0+H_Q$, and semigroup theory to recover $H_Q$ and hence the eigenvalues $\lambda_k^2$ that define $Q$. The results provide a rigorous pathway for inverse problems in stochastic diffusion, and the framework may guide extensions to time-dependent or wave-type systems in future work.

Abstract

This paper investigates an inverse potential problem for the stochastic heat equation driven by space-time Gaussian noise, which is spatially colored and temporally white. The objective is to determine the covariance operator of the random potential. We establish that the covariance operator can be uniquely identified from the correlation of the mild solution to the stochastic heat equation at a final time, where the initial conditions are specified by a complete orthonormal basis. The analysis relies on characterizing a tensor product structure inherent in the problem and utilizing the monotonicity properties of the operators associated with the system.

Theorems & Definitions (20)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof