منابع مشابه
Range of Brownian motion with drift
Let (Bδ(t))t≥0 be a Brownian motion starting at 0 with drift δ > 0. Define by induction S1 = − inf t≥0 Bδ(t), ρ1 the last time such that Bδ(ρ1) = −S1, S2 = sup 0≤t≤ρ1 Bδ(t), ρ2 the last time such that Bδ(ρ2) = S2 and so on. Setting Ak = Sk + Sk+1 ; k ≥ 1, we compute the law of (A1, · · · , Ak) and the distribution of ((Bδ(t + ρl) − Bδ(ρl) ; 0 ≤ t ≤ ρl−1 − ρl))2≤l≤k for any k ≥ 2, conditionally ...
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Consider a strong Markov process X° that has continuous sample paths in Rd (d > 2) and the following two properties. (1) Away from the origin X° behaves like Brownian motion with a polar drift given in spherical polar coordinates by n{8)/2r. Here /i is a bounded Borel measurable function on the unit sphere in RJ, with average value Ji. (2) X° is absorbed at the origin. It is shown that X° reach...
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Consider 3 dimensional Brownian motion started on the unit sphere {|x| = 1} with initial density ρ. Let ρt be the first hitting density on the sphere {|x| = t + 1}, t > 0. Then the linear operators Tt defined by Tt ρ = ρt form a semigroup with an infinitesimal generator which is approximately the square root of the Laplacian. This paper studies the analogous situation for Brownian motion with a...
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dXt = dWt + dAt , where Wt is d-dimensional Brownian motion with d ≥ 2 and the ith component of At is a process of bounded variation that stands in the same relationship to a measure πi as ∫ t 0 f (Xs)ds does to the measure f (x)dx. We prove weak existence and uniqueness for the above stochastic differential equation when the measures πi are members of the Kato class Kd−1. As a typical example,...
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Let B = (Bt)t≥0 be a standard Brownian motion started at zero, and let μ ∈ R be a given and fixed constant. Set B t = Bt+μt and S t = max 0≤s≤t B s for t ≥ 0 . Then the process: (x ∨ Sμ)−Bμ = ((x ∨ S t )−B t )t≥0 realises an explicit construction of the reflecting Brownian motion with drift −μ started at x in R+ . Moreover, if the latter process is denoted by Z = (Z t )t≥0 , then the classic Lé...
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ژورنال
عنوان ژورنال: Journal of Theoretical Probability
سال: 2006
ISSN: 0894-9840,1572-9230
DOI: 10.1007/s10959-006-0012-7