Diffusion Processes in Random Environments

Denote by αt(μ)the probability law ofAt(μ)= ∫0texp(2(Bs+μs))dsfor a Brownian motion {Bs,s≥ 0}. It is well known that αt(μ)is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt(0)and αt(1), as well as an equally remarkable relationship between αt(μ)and αt(ν)for two different drifts μ and ν. In this paper, hinging on previous results about αt(μ), we give different proofs of Dufresne's results and present extensions of them for theprocesses{At(μ),t≥ 0}.

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On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts

Advances in Applied Probability ◽

10.1239/aap/1046366105 ◽

2003 ◽

Vol 35 (1) ◽

pp. 184-206 ◽

Cited By ~ 12

Author(s):

Hiroyuki Matsumoto ◽

Marc Yor

Keyword(s):

Brownian Motion ◽

Stochastic Analysis ◽

Diffusion Processes ◽

Mathematical Finance ◽

Hyperbolic Spaces ◽

Random Environments ◽

Brownian Motions

Denote by αt(μ) the probability law of At(μ) = ∫0texp(2(Bs+μ s))ds for a Brownian motion {Bs, s ≥ 0}. It is well known that αt(μ) is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt(0) and αt(1), as well as an equally remarkable relationship between αt(μ) and αt(ν) for two different drifts μ and ν. In this paper, hinging on previous results about αt(μ), we give different proofs of Dufresne's results and present extensions of them for the processes {At(μ), t ≥ 0}.

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Meridional Circulation, Turbulence, and Diffusion Processes in Stars

International Astronomical Union Colloquium ◽

10.1017/s0252921100080477 ◽

1976 ◽

Vol 32 ◽

pp. 109-116 ◽

Cited By ~ 1

Author(s):

S. Vauclair

Keyword(s):

Convection Zone ◽

Diffusion Processes ◽

Meridional Circulation ◽

Diffusion Time ◽

Work In Progress ◽

First Results ◽

Stellar Envelopes ◽

And Diffusion ◽

Turbulence And Diffusion ◽

Hr Diagram

This paper gives the first results of a work in progress, in collaboration with G. Michaud and G. Vauclair. It is a first attempt to compute the effects of meridional circulation and turbulence on diffusion processes in stellar envelopes. Computations have been made for a 2 Mʘstar, which lies in the Am - δ Scuti region of the HR diagram.Let us recall that in Am stars diffusion cannot occur between the two outer convection zones, contrary to what was assumed by Watson (1970, 1971) and Smith (1971), since they are linked by overshooting (Latour, 1972; Toomre et al., 1975). But diffusion may occur at the bottom of the second convection zone. According to Vauclair et al. (1974), the second convection zone, due to He II ionization, disappears after a time equal to the helium diffusion time, and then diffusion may happen at the bottom of the first convection zone, so that the arguments by Watson and Smith are preserved.

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