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Interacting Brownian particles and the Wigner law. (English) Zbl 0794.60100
We study interacting diffusing particles governed by the stochastic differential equations \[ dX_ j (t)=\sigma_ n dB_ j (t)-D_ j \varphi_ n (X_ 1,\dots,X_ n) dt, \qquad j=1,2, \dots,n. \] Here the \(B_ j\) are independent Brownian motions in \(\mathbb{R}^ d\), and \[ \varphi_ n (x_ 1,\dots,x_ n)=\alpha_ n {\underset {i\neq j} {\sum \sum}} V(x_ i-x_ j)+\theta_ n \sum_ i U(x_ i). \] The potential \(V\) has a singularity at 0 strong enough to keep the particles apart, and the potential \(U\) 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 of \(d=1\), \(V(x)=-\log | x |\), \(U(x)= x^ 2/2\) 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 as \(t \to \infty\). Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.
Reviewer: L.C.G.Rogers

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
62E20 Asymptotic distribution theory in statistics
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[1] Chan, T.: The Wigner semi-circle law and eigenvalues of matrix-valued diffusions. Probab. Theory Relat. Fields93, 249-272 (1992) · Zbl 0767.60050 · doi:10.1007/BF01195231
[2] Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys.3, 1191-1198 (1962) · Zbl 0111.32703 · doi:10.1063/1.1703862
[3] Ethier, S.N., Kurtz, T.G.: Markov processes: Characterization and convergence. New York: Wiley 1986 · Zbl 0592.60049
[4] McKean, H.P.: Stochastic integrals. New York: Academic Press 1969 · Zbl 0191.46603
[5] Norris, J.R., Rogers, L.C.G., Williams, D.: Brownian motions of ellipsoids. Trans. Am. Math. Soc.294, 757-765 (1986) · Zbl 0613.60072 · doi:10.1090/S0002-9947-1986-0825735-5
[6] Pauwels, E.J., Rogers, L.C.G.: Skew-product decompositions of Brownian motions. In: Durrett, R., Pinsky, M.A. (eds.) Geometry of random motion. (Contemp. Math., vol. 11,73. pp. 237-262) Providence RI: Am. Math. Soc. 1988 · Zbl 0656.58034
[7] Rogers, L.C.G., Williams, D.: Diffusions, Markov processes and martingales vol. II: Itô calculus. Chichester: Wiley 1987 · Zbl 0627.60001
[8] Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979 · Zbl 0426.60069
[9] Sznitman, A.-S.: Topics in propagation of chaos. In: Hennequin, P.L. (ed.) Ecole d’Eté de Probabilités de Saint-Flour XIX. (Lect. Notes Math., vol. 1464, pp. 167-251) Berlin Heidelberg New York: Springer 1991
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