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

TL;DR

This work analyzes SPDEs on finite tree graphs driven by edge-based space-time white noise, focusing on how graph structure governs the strong Feller property, irreducibility, and long-time behavior. The authors introduce a graph-theoretic null decomposition and S-atoms to reduce the regularity analysis to minimal components and connect noise propagation to zero entries in graph Laplacian eigenfunctions. They derive a sharp bound on the permissible number of noise-free edges, $| obreak obreak obreak | obreak$, ensuring strong Feller and irreducibility: $| obreak obreak ( ext{Γ})| obreak obreak $, and extend the results to nonlinear SPDEs via a Girsanov transformation, showing equivalence with the linear case for these properties. Under a dissipativity condition, they prove the existence and exponential ergodicity of a unique invariant measure, with explicit confirmations on chain, star, and $S$-atom trees; chain graphs require at least one noisy edge, while star graphs require all but one noisy. The methodology blends spectral graph theory, null-space analysis of the Laplacian, and probabilistic transformations to yield rigorous, topology-driven regularity results with practical implications for networked stochastic dynamics.

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.

Figures (24)

• Figure 2: An illustrative example
• Figure 3: Chain graph $L$ with $m$ edges
• Figure 4: Star graph $R$ with $m$ edges
• Figure 5: Strong Feller verification for $\mathcal{S}_T$ on chain graph with $4$ edges. $x$ axis: $\epsilon$, $y$ axis: the difference of \ref{['nonlineardiffernece']}. $N=2^6$, $\tau=2^{-5}$, $T=2^{-1}$, $M_{traj}=500$, $\epsilon \in \{ 10^{-\tfrac{4k}{7}} : k = 0,1,\dots,7\}$.
• Figure 6: Irreducibility verification for $\mathcal{S}_T$ on chain graph with 4 edges. Subfigures from left to right: $(\Psi^{1,l})_{1\le l\le N},\,(\Psi^{2,l})_{1\le l\le N},\,(\Psi^{3,l})_{1\le l\le N},\,(\phi^{1,k})_{0\le k\le N-1}$. $x$ axis: dimension $N$, $y$ axis: reachability probability. $N=2^6,\, \tau=2^{-5},\, T=2^{-1},\, M_{traj}=500$.

Theorems & Definitions (29)

• Lemma 3.1
• Proposition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Lemma 3.6
• Remark 3.7
• Example 1
• Theorem 4.1
• Corollary 4.2