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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.

##### 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) 60J60 Diffusion processes
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