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A nonlinear stochastic differential equation involving the Hilbert transform. (English) Zbl 0935.60095
Starting from the study of systems of \(N\) interacting Brownian particles under a potential with logarithmic singularity at 0, E. Cépa and D. Lépingle [Probab. Theory Relat. Fields 107, No. 4, 429-449 (1997; Zbl 0883.60089)] have proved the weak convergence of the empirical distributions to the unique probability measure-valued function \((\mu_t)\) solving weakly a nonlinear second-order integro-partial differential equation, the so-called McKean-Vlasov equation. In the present paper the authors show that this equation \[ \partial_t u=1/2 \sigma^2\partial^2_xu-2\lambda \partial_x \biggl(u(t,x) {\mathcal H}\bigl(u(t,.) \bigr)(x)\biggr),\quad t\geq 0,\;u(t,x)dt \to \delta_0\;(t\to 0), \] involving the Hilbert transform \({\mathcal H}(u(s,.))(x)= \lim_{\varepsilon\to 0}\int_{|x-y|> \varepsilon}(x-y)^{-1} u(t,y)dy\) admits a unique classical solution which as well as its Hilbert transform are real analytic in \(R^*_+\times R\). To this equation they associate the stochastic differential equation \[ dX_t=\sigma dB_t+2\lambda{\mathcal H}\bigl(u(t,.) \bigr) (X_t) dt, \quad X_0=0. \] Existence and uniqueness of a strong solution \((X_t)\) are proved as well as that \(u(t,.)\) is the density of \(X_t\) w.r.t. the Lebesgue measure. As a conclusion the authors are able to adapt the singular drift scheme by Sznitman to link systems of particles and nonlinear processes.
Reviewer: R.Buckdahn (Brest)

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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