In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show tha...

Let V denote a set of N vertices. To construct a hypergraph process, create a new hyperedge at each event time of a Poisson process; the cardinality K of this hyperedge is random, with generating function ρ(x) def = ∑ ρkx , where P (K = k) = ρk; given K = k, the k vertices appearing in the new hyperedge are selected uniformly at random from V . Assume ρ1 + ρ2 > 0. Hyperedges of cardinality 1 ar...

This paper establishes a process-level large deviations principle for Markov processes in the Euclidean space with a discontinuity in the transition mechanism along a hyper-plane. The transition mechanism of the process is assumed to be continuous on one closed halfspace, and also continuous on the complementary open halfspace. Similar results were recently obtained for discrete time processes ...

In this paper, we study the Martin kernels of general open sets associated with inaccessible points for a large class of purely discontinuous Feller processes in metric measure spaces. AMS 2010 Mathematics Subject Classification: Primary 60J50, 31C40; Secondary 31C35, 60J45, 60J75.

Percolation is a paradigmatic model in disordered systems and has been applied to various natural phenomena. The percolation transition is known as one of the most robust continuous transitions. However, recent extensive studies have revealed that a few models exhibit a discontinuous percolation transition (DPT) in cluster merging processes. Unlike the case of continuous transitions, understand...