Transience and recurrence of Markov processes

Abstract

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Conditional gauge theorem for non-local Feynman-Kac transforms

“…Let E be a Lusin space (i.e., a space that is homeomorphic to a Borel subset of a compact metric space) and B(E) be the Borel σ -algebra on E, and let m be a σ -finite measure on B(E) with supp[m] = E. Let X = ( , M, M t , X t , P x , x ∈ E) be a Borel standard process on E, which is transient in the sense of [23]. Here, a Borel standard process on Lusin space E is a strong Markov process satisfying the following conditions: (i) it is right continuous, (ii) it has no branching points, (iii) its resolvents map Borel functions into Borel functions and (iv) it is quasi-left continuous on (0, ζ), where ζ is the lifetime of the process.…”

Ergodicity for the Navier‐Stokes equation with degenerate random forcing: Finite‐dimensional approximation

“…The main idea for proving (2) is to prove that the Fokker-Planck operator associated with (1.4 N )-(1.5 N ) is hypoelliptic. Since the work of Harris, there has been extensive work extending his basic result to continuous-time Markov processes [9,13,16,22]. In particular, it is proven in [1,9] that a diffusion process satisfying the continuous analogue of Harris' condition has at most one invariant probability measure.…”

A stability conjecture on bandwidth sharing networks

“…As in the countable state space theory, it has been shown that Harris recurrence is sufficient for a Markov process to have a stationary measure, see [13]. If this measure is a finite measure, then a stationary distribution exists, and our process is said to be positive Harris recurrent.…”