On ergodic diffusions on continuous graphs whose centered resolvent admits a trace

• Published in: Journal of mathematical analysis and applications Vol. 437; no. 2; pp. 737 - 753
• Main Author: Miclo, Laurent
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.05.2016
• Subjects: Diffusions on quantum graphs; Dirichlet forms; Reversibility; Trace class centered resolvents; Diffusions on quantum graphs; Dirichlet forms; Reversibility; Trace class centered resolvents
• ISSN: 0022-247X; 1096-0813; 1096-0813
• DOI: 10.1016/j.jmaa.2016.01.026

We consider ergodic and reversible diffusions on continuous and connected graphs G with a finite number of bifurcation vertices and some rays going to infinity. A necessary and sufficient condition is presented for the spectrum of the associated generator L to be without continuous part and for the sum of the inverses of its eigenvalues (except 0) to be finite. This criterion is easily computable in terms of the coefficients of L and does not depend on the transition kernels at the vertices. Its motivation is that it is conjectured to be also a necessary and sufficient condition for the diffusion to admit strong stationary times whatever its initial distribution (this is known to be true if G is the real line). The above criterion for the centered resolvent to be of trace class is next extended to Markov processes on denumerable connected graphs with only a finite number of vertices of degree larger than or equal to 3.