Ergodic Properties of Multidimensional Brownian Motion with Rebirth
نویسندگان
چکیده
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.
منابع مشابه
Path Collapse for Multidimensional Brownian Motion with Rebirth†
In a bounded open region of the d dimensional Euclidean space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. It was shown that in dimension one coupled paths starting at different points but driven by the same Brownian motion either collapse with probability one or never meet. In higher dimensions, for convex or polyhedral regions the...
متن کاملDiffusion limit for many particles in a periodic stochastic acceleration field
The one-dimensional motion of any number N of particles in the field of many independent waves (with strong spatial correlation) is formulated as a second-order system of stochastic differential equations, driven by two Wiener processes. In the limit of vanishing particle mass m → 0, or equivalently of large noise intensity, we show that the momenta of all N particles converge weakly to N indep...
متن کاملOn the Csáki-Vincze transformation
Csáki and Vincze have defined in 1961 a discrete transformation T which applies to simple random walks and is measure preserving. In this paper, we are interested in ergodic and asymptotic properties of T . We prove that T is exact: ∩ k=1 σ(T (S)) is trivial for each simple random walk S and give a precise description of the lost information at each step k. We then show that, in a suitable scal...
متن کاملErgodicity of hypoelliptic SDEs driven by fractional Brownian motion
We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1 2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander’s condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that the...
متن کاملThe effect of various conductivity and viscosity models considering Brownian motion on nanofluids mixed convection flow and heat transfer
In this paper the effect of using various models for conductivity and viscosity considering Brownian motion of nanoparticles is investigated. This study is numerically conducted inside a cavity full of Water-Al2O3 nanofluid at the case of mixed convection heat transfer. The effect of some parameters such as the nanoparticle volume fraction, Rayleigh, Richardson and Reynolds numbers has been ex...
متن کامل