Inverse problems for stochastic partial differential equations

TL;DR

This book provides an overview of recent progress in inverse problems for stochastic PDEs, focusing on second-order stochastic parabolic and hyperbolic equations using Carleman estimates and stochastic calculus.

Contribution

It introduces new approaches and results in inverse problems for stochastic PDEs, emphasizing direct stochastic calculus methods over complex Carleman estimates.

Findings

Results on inverse problems for stochastic parabolic equations

Results on inverse problems for stochastic hyperbolic equations

Encourages further research in stochastic inverse problems

Abstract

This book aims to provide a brief overview of recent advancements in the theory of inverse problems for stochastic partial differential equations. In order to keep the content concise, we will only discuss the inverse problems of two typical classes of stochastic partial differential equations: second-order stochastic parabolic equations and secondorder stochastic hyperbolic equations. The main tool for studying these inverse problem is Carleman estimate. We do not intend to pursue any general treatment of the Carleman estimates themselves and choose direct arguments based on basic stochastic calculus, rather than more general sophisticated methods. As this field is still developing and there are many challenging issues to be addressed, the purpose of this book is not to serve as a comprehensive summary, but rather to spark interest and encourage further exploration in this area among readers. We prefer to present results that, from our perspective, include fresh and promising ideas. In cases where a complete mathematical theory is lacking, we only provide the available results. We do not intend for the current book to be encyclopedic in any sense, and the references are limited.