A non-linear stochastic differential equation involving the Hilbert transform

We consider a non-linear stochastic differential equation which involves th e Hilbert transform, X-t = sigma . B-t + 2 lambda integral(0)(t) Hu(s, X-s) ds. In the previous equation, u(t, .) is the density of mu(t), the lax of X-t, and H represents the Hilbert transform in the space variable. In order to define correctly the solutions, we first study the associated non-linea r second-order integro-partial differential equation which can be reduced t o the holomorphic Burgers equation. The real analyticity of solutions allow s us to prove existence and uniqueness of the non-linear diffusion process. This stochastic differential equation has been introduced when studying th e limit of systems of Brownian particles with electrostatic repulsion when the number of particles increases to infinity. More precisely, it has been show that the empirical measure process tends to the unique solution mu = ( mu(t))(t greater than or equal to 0) of the non-linear second-order integro -partial differential, equation studied here. (C) 1999 Academic Press.