Mihály Kovács : On the strong Feller property of the heat equation on quantum graphs with Kirchhoff noise

Speaker
Mihály Kovács , Pázmány Péter Catholic University

Abstract
We consider a so-called quantum graph with standard continuity and Kirchhoff vertex conditions where the Kirchhoff vertex condition is perturbed by Gaussian noise. The strong Feller property of the stochastic parabolic problem is then investigated. We show that the quantum graph setting is very different from the classical one dimensional boundary noise setting, where the transition semigroup is known to be strong Feller. In particular, when the graph is a tree, and there is noise present in all but one of the boundary vertices, then we prove that the transition semigroup associated with the problem is strong Feller at any time T>0. But this turns out to be also a necessary condition for equilateral star graphs.