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arXiv · 2509.03760

Inverse Random Source and Cauchy Problems for Semi-Discrete Stochastic Parabolic Equations in Arbitrary Dimensions

Rodrigo Lecaros·Ariel A. Pérez·Manuel F. Prado
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Abstract

In this paper, we study two types of inverse problems for space semi-discrete stochastic parabolic equations in arbitrary dimensions. The first problem concerns a semi-discrete inverse source problem, which involves determining the random source term of the white noise in the semi-discrete stochastic parabolic equation using observation data of the solution at the terminal time and on an arbitrary open spatial subdomain over a time interval. The second problem addresses a semi-discrete Cauchy inverse problem, which involves determining the solution of the stochastic parabolic equation in a space-time subdomain, from measurements of the solution and the trace of its discrete spatial derivative on an arbitrary open subset of the lateral boundary over a time interval. The key tools are three new global Carleman estimates for the semi-discrete stochastic parabolic operator, one for interior observations and two for boundary observations (homogeneous and nonhomogeneous Dirichlet conditions). Applying these Carleman estimates, we obtain Lipschitz and H\"older stability for the first and second inverse problems, respectively.

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Rodrigo Lecaros, Ariel A. Pérez, Manuel F. Prado. 2025-09-03. Inverse Random Source and Cauchy Problems for Semi-Discrete Stochastic Parabolic Equations in Arbitrary Dimensions. https://arxiv.org/abs/2509.03760

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