Quantifying the effect of graph structure on strong Feller property of SPDEs

TL;DR

This paper explores how the structure of finite tree graphs affects the strong Feller property of SPDEs, introducing a graph-based null decomposition method to analyze eigenfunctions and establish conditions for ergodicity.

Contribution

It introduces a novel graph-based null decomposition approach to analyze the strong Feller property of SPDEs on tree graphs, linking graph structure to stochastic properties.

Findings

Addition of noise to any single edge suffices for chain graphs.

At most one noise-free edge is allowed in star graphs for properties to hold.

Existence and exponential ergodicity of a unique invariant measure are established.

Abstract

This paper investigates how the structure of the underlying graph influences the behavior of stochastic partial differential equations (SPDEs) on finite tree graphs, where each edge is driven by space-time white noise. We first introduce a novel graph-based null decomposition approach to analyzing the strong Feller property of the Markov semigroup generated by SPDEs on tree graphs. By examining the positions of zero entries in eigenfunctions of the graph Laplacian operator, we establish a sharp upper bound on the number of noise-free edges that ensures both the strong Feller property and irreducibility. Interestingly, we find that the addition of noise to any single edge is sufficient for chain graphs, whereas for star graphs, at most one edge can remain noise-free without compromising the system's properties. Furthermore, under a dissipative condition, we prove the existence and exponential ergodicity of a unique invariant measure.