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Diffusing particles with electrostatic repulsion. (English) Zbl 0883.60089
Summary: We study a diffusion model of an interacting particles system with general drift and diffusion coefficients, and electrostatic inter-particles repulsion. More precisely, the finite particle system is shown to be well defined thanks to recent results on multivalued stochastic differential equations [see the first author, in: Séminaire de probabilités XXIX. Lect. Notes Math. 1613, 86-107 (1995; Zbl 0833.60079)], and then we consider the behaviour of this system when the number of particles \(N\) goes to infinity (through the empirical measure process). In the particular case of affine drift and constant diffusion coefficient, we prove that a limiting measure-valued process exists and is the unique solution of a deterministic PDE. Our treatment of the convergence problem (as \(N\uparrow\infty\)) is partly similar to that of T. Chan [Probab. Theory Relat. Fields 93, No. 2, 249-272 (1992; Zbl 0767.60050)] and L. C. G. Rogers and Z. Shi [ibid. 95, No. 4, 555-570 (1993; Zbl 0794.60100)], except we consider here a more general case allowing collisions between particles, which leads to a second-order limiting PDE.

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)
60J60 Diffusion processes
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