On ergodic diffusions on continuous graphs whose centered resolvent admits a trace - Miclo, Laurent (Journal of mathematical analysis and applications, 2016)
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.