Direct and Inverse Problems for PDEs with Random Coefficients - Abstract
We consider parabolic evolution problems with random coefficients formulated in a full space and time weak sense. Having strict uniform bounds for the spatial operator, the almost sure existence and uniqueness of a solution is inherited from the deterministic case by a pathwise treatment. Now we relax this restriction of having uniform bounds and allow them to be random variables instead. That means, that the elliptic operator does not need to be bounded form above and below by strict constants, but rather by random variables which may depend on the stochastic parameter. The most interesting cases are the extreme situations when both the coercivity constant tends to zero and the continuity constant to infinity, which are covered under these assumptions. Depending on the number of existing moments for these bounds, we can prove existence of $p$-moments for the solution. Moreover, we go along similar lines and show existence of $p$-moments for Petrov-Galerkin solutions and also prove quasi-optimality in suitable random $L_p$-spaces.