SPASS - Probability, Stochastic Analysis and Statistics Seminar

Stochastic obstacle problems: variational & non variational settings

by Yassine Tahraoui (Scuola Normale Superiore)

Obstacle problems are free boundary  type problems,  well known in the literature of applied mathematics and lead to numerous applications. My aim is to present some results  about the well-posedness and the regularity of  the solution to a "parabolic or hyperbolic"  obstacle problem in the presence of multiplicative noise,   studied in [1, 2]. After showing the well-posedness of such problems, we prove  Lewy-Stampacchia's inequalities, which gives an estimate of the reflected measure generated by the singularities caused by the obstacle near the free boundary.
[1]I. H. Biswas, Y. Tahraoui and G. Vallet: Obstacle problem for a stochastic conservation law and Lewy-Stampacchia inequality.   Journal of Mathematical Analysis and Applications, 527 (1) 127356 (2023)

[2]Y. Tahraoui and G.Vallet: Lewy-Stampacchia's inequality for a stochastic T-monotone obstacle problem.  Stochastic Partial Differential Equations: Analysis and Computations 10, 90-125 (2022).