Diffusion Processes and their Sample Paths.

1968 ◽  
Vol 75 (1) ◽  
pp. 99
Author(s):  
G. F. Newell ◽  
K. Ito ◽  
H. P. McKean
Keyword(s):  
Diffusion Processes ◽  
Sample Paths

Diffusion Processes and Their Sample Paths.

1996 ◽  
Vol 91 (436) ◽  
pp. 1754
Author(s):  
PALE ◽  
Kiyosi Ito ◽  
Henry P. McKean
Keyword(s):  
Diffusion Processes ◽  
Sample Paths

Transition functions, paths and path integrals

1991 ◽  
Vol 434 (1890) ◽  
pp. 41-63 ◽  
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Nonlinear Dynamical Systems ◽  
Diffusion Theory ◽  
Main Theme ◽  
Nonlinear Dynamical ◽  
Transition Functions ◽  
Sample Paths ◽  
Itô Calculus ◽  
Expository Paper

The main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory. Section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which Kolmogorov achieved via PDE theory (the theory of partial differential equations) and his criterion for path continuity. Section 2 describes how Itô calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process X by solving an SDE (stochastic differential equation) driven by brownian motion. (Of course, SDEs have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems.) Though §2 begins heuristically, the mathematics is then tightened up. This paper is, after all, a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation. Once Kolmogorov’s precise language is available, §2 then takes a quick sight-seeing trip through some of the great developments by Doob, Itô and their successors. (In an age in which so many do simulations of Itô equations, I have explained precisely in the briefest possible fashion what the exact theory is. It is easy and usable.) You will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems, and of what the Malliavin calculus is about. You will, however, be advised on guide-books on these and other areas (including physicists’ favourites: large deviations, measure-valued diffusions, etc.), so that you can later explore at your leisure. Confession : the paper consists very largely of selected tracks (remixed!) from the album (Rogers & Williams 1987 Diffusions, Markov processes and martingales ; Chichester: Wiley). Tributes to Kolmogorov’s work in probability and statistics have appeared in (every book ever written on probability and in) Ann. Probability 17 (1989), 815-964, Ann. Statist. 18 (1990), 987-1031, Bull. Lond. math. Soc. . 22 (1990), 31-100, Teor. Veroyatnost Primenen 34 (1989), no. 1, Usp. mat. Nauk 43 (1988), no. 6. The official biography by Shiryaev will be a wonderful volume. For me, this paper is a further expression of my thanks to Kolmogorov and (as he would have wished) to Lévy, Doob and Itô too.

Meridional Circulation, Turbulence, and Diffusion Processes in Stars

1976 ◽  
Vol 32 ◽  
pp. 109-116 ◽  
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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