A Ray-Knight theorem for symmetric Markov processes

Local Times for Spectrally Negative Lévy Processes

“…By the Ray-Knight theorem, see for example [18], the local times of Brownian motion are identical in law to a squared Beseel process. A similar relationship was shown in [4] between symmetric Markov processes and a family of squares of Gaussian processes. For general Markov process, the representation the local times involves a so called permanental process.…”

On the Favorite Points of Symmetric Lévy Processes

“…We recall a generalized second Ray-Knight Theorem due to Eisenbaum et al [10]; see also Marcus and Rosen [18]. Let X = {X t , t ≥ 0} be a strongly symmetric Borel right process with values in R with continuous α-potential densities u α (x, y).…”

“…As we have mentioned earlier, we will write the probability P η as P and its expectation as E. It follows from Theorem 6.1 of Eisenbaum et al [10] that, for a recurrent symmetric Lévy process with characteristic exponent ψ(λ), the associated Gaussian process η is centered with stationary increments such that its covariance u T 0 (x, y) is given by…”

“…We state the generalized second Ray-Knight Theorem for transient symmetric Markov processes due to Eisenbaum et al [10], see also [18,Thm. 8.2.3].…”

“…Later, Bass et al [3] proved that when X is a symmetric stable Lévy process in R of index α ∈ (1, 2), its favorite points process is still transient. They appealed to a generalized Ray-Knight Theorem [10] and made a clever use of Slepian's Lemma. Marcus [16] adopted the approach of [3] and extended their result to a large class of symmetric Lévy processes.…”