Titlebook: Seminar on Stochastic Processes, 1988; E. Çinlar,K. L. Chung,J. Glover Book 1989 Birkhäuser Boston 1989 Brownian excursion.Brownian motion

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Innocence

,Reminiscences of some of Paul Lévy’s ideas in Brownian Motion and in Markov Chains,We begin with a resume. Let {. ≥ 0} be a semigroup of stochastic matrices with elements p.(., where . is a countable set, satisfying the condition .. It is known that .(0) = . exists and . The state . is called . if . < +∞, and . if . = +∞ (Lévy’s terminology). The matrix .) is called . when equality holds in (3) for all ..

HPA533

constellation

GOUGE

The Independence of Hitting Times and Hitting Positions to Spheres for Drifted Brownian Motions,A drifted Brownian motion X. is a diffusion process on .. whose infinitesimal generator has the form . where ... is the usual Laplace operator and . is a smooth vector field on ... When b ≡ 0, X. becomes the usual n-dimensional Brownian motion.

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A Maximal Inequality,Let X be a uniformly integrable, cadlag non-negative regular supermartingale. Such a process X has the representation . where A. is continuous and increasing on the half open interval [0,∞), A. = 0 and A may assign mass to ∞ which is just A. - A. where .. Then we have the maximal inequality.

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Right Brownian Motion and Representation of Initial Problem,Let {.: t > 0} be the right Brownian motion on [0, ∞) determined by the transition density: for . ∈ [0,∞). . This is a Markov process having the tendency moving to the right direction. 0 can be a starting point, but is never reached, i.e., {0} is a polar set.