Grigorescu, Ilie; Kang, Min Ergodic properties of multidimensional Brownian motion with rebirth. (English) Zbl 1127.60073 Electron. J. Probab. 12, 1299-1322 (2007). Summary: In a bounded open region of the \(d\) dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and determine explicitly the spectrum in dimension one. Two proofs of the exponential ergodicity are given, one using the inverse Laplace transform and properties of analytic semigroups, and the other based on Doeblin’s condition. Both methods admit generalizations to a wide class of processes. Cited in 17 Documents MSC: 60J35 Transition functions, generators and resolvents 60J75 Jump processes (MSC2010) 35K15 Initial value problems for second-order parabolic equations 91B24 Microeconomic theory (price theory and economic markets) 92D10 Genetics and epigenetics 91B28 Finance etc. (MSC2000) Keywords:Green function; Dirichlet Laplacian; exponential ergodicity; inverse Laplace transform; Doeblin’s condition × Cite Format Result Cite Review PDF Full Text: DOI EuDML