Barriers, exit time and survival probability for unimodal Lévy processes

Abstract

We give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal Lévy processes in smooth domains.

Extension and trace for nonlocal operators

“…Here the change of the order of integration is justified because Lϕ is bounded on U and E x τ U < ∞, cf., e.g., [12,41]. As usual, we let p U denote the transition density of the process killed upon leaving U .…”

Martin Kernels for Markov Processes with Jumps

“…This is based on estimates obtained recently in [9,10,18,20] and will be studied in detail in [19]. Other extensions can be obtained by allowing the Lévy kernel to depend on x or restricting it to a domain, as described in the following two examples.…”

On Estimates of Transition Density for Subordinate Brownian Motions with Gaussian Components in C1,1-open Sets

“…Case (2): Suppose that |x − y| ≥ √ t/8, t ∈ (0, T 0 ], and x and y are contained in two distinct connected components of D. By (3.5) and Lemma 3.4, we have the same conclusion in (3.6). Case (3): Suppose that |x − y| < √ t/8 and t ∈ (0, T 0 ].…”

“…Proof of Theorem 1.5 (2) and Theorem 1.6 (2). Since two proofs are almost identical, we just prove Theorem 1.5 (2). First note that the distance between two distinct connected components of D is at least R 0 .…”