Interacting Brownian particles and the Wigner law |
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Authors: | L C G Rogers Z Shi |
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Institution: | (1) School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, E1 4NS London, UK |
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Abstract: | Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX
j
(t)=
n
dB
j
(t) –D
jØn(X
1,...,X
n)dt,j=1, 2,...,n. Here theB
jare independent Brownian motions in
d
, and Ø
n
(X
1,...,X
n)=
n
ij
V(X
iX
j) + ni
U(X
1). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x
2/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444 |
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Keywords: | 60K35 60F05 60H10 62E20 |
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