A Non-linear Stochastic Differential Equation Involving the Hilbert Transform

We consider a non-linear stochastic differential equation which involves the Hilbert transform, Xt=σ·Bt+2λ ∫t0 Hu(s, Xs) ds. In the previous equation, u(t, ·) is the density of μt, the lax of Xt, and H represents the Hilbert transform in the space variable. In order to define correctly the solutions, we first study the associated non-linear second-order integro-partial differential equation which can be reduced to the holomorphic Burgers equation. The real analyticity of solutions allows us to prove existence and uniqueness of the non-linear diffusion process. This stochastic differential equation has been introduced when studying the 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 μ=(μt)t⩾0 of the non-linear second-order integro-partial differential, equation studied here.