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

Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider ergodic and reversible diffusions on continuous and connected graphs $\GG$ 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 $\GG$ is the real line).\par 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.

Mots clés: Diffusions on quantum graphs; trace class centered resolvents; reversibility; Dirichlet forms

Domaines: Probabilités [math.PR]

Laurent Miclo. On ergodic diffusions on continuous graphs whose centered resolvent admits a trace. Journal of Mathematical Analysis and Applications, 2016, 437 (2), pp.737-753. ⟨10.1016/j.jmaa.2016.01.026⟩. ⟨hal-00957019v1⟩