On the strong Feller property of the heat equation on quantum graphs with Kirchoff noise
Mohamed Fkirine, Mihály Kovács, Eszter Sikolya
TL;DR
This work analyzes the strong Feller property for the heat equation on quantum graphs with Kirchhoff noise at vertices, revealing a marked departure from the classical 1D boundary-noise setting. By recasting the problem as a boundary-control system and exploiting null-controllability and wave-equation controllability on metric trees (via ST-active sets), the authors derive conditions under which strong Feller holds, notably for trees when noise is present at all boundary vertices except possibly one. They show this condition is necessary in certain equilateral star graphs and provide counterexamples where loops eliminate the strong Feller property. Additional results address invariant-measure existence and solution regularity, demonstrating that these long-term and regularity properties can be obtained without requiring strong Feller in the network setting, thereby enriching the understanding of stochastic diffusion on quantum graphs and its control-theoretic implications.
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. 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, by giving examples and counterexamples to the strong Feller property. In particular, when the graph is a tree, and there is noise present in all of the boundary vertices except one, then the transition semigroup associated with the problem is strong Feller at any time T > 0. This turns out to be also a necessary condition for equilateral star graphs. We also comment on the existence and uniqueness of the invariant measure and the regularity of the solution.