Inverse problems for stochastic partial differential equations: some progresses and open problems (Q6559149)

This survey paper discusses some results on inverse problems of semilinear parabolic and hyperbolic stochastic partial differential equations (SPDEs) with Lipschitz coefficients on bounded domains with \(C^2\)-boundaries driven by a one-dimensional Brownian motion. The authors formulate a collection of inverse problems (P1)--(P7) addressing existence and uniqueness of the inverse mapping, and solve them via Carleman estimates, where they assume a variant of a pseudo-convex condition in the hyperbolic case. The results are mainly based on [\textit{Q. Lü}, Inverse Probl. 28, No. 4, Article ID 045008, 18 p. (2012; Zbl 1236.35213); Inverse Probl. 29, No. 9, Article ID 095011, 22 p. (2013; Zbl 1292.60066); \textit{Q. Lü} and \textit{Z. Yin}, ESAIM, Control Optim. Calc. Var. 26, Paper No. 79, 19 p. (2020; Zbl 1460.93018); \textit{Q. Lü} and \textit{X. Zhang}, Commun. Pure Appl. Math. 68, No. 6, 948--963 (2015; Zbl 1315.60080); \textit{S. Tang} and \textit{X. Zhang}, SIAM J. Control Optim. 48, No. 4, 2191--2216 (2009; Zbl 1203.93027)]. The effect of randomness is discussed carefully throughout the presentation. Finally, the authors discuss several open problems related to inverse problems of SPDEs.