# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Cépa, E.; Lépingle, D., Diffusing particles with electrostatic repulsion, Probab. theory related fields, 107, 429-449, (1997) · Zbl 0883.60089 [2] Chan, T., The Wigner semi-circle law and eigenvalues of matrix-valued diffusions, Probab. theory related fields, 93, 249-272, (1992) · Zbl 0767.60050 [3] McKean, H.P., Stochastic integrals, (1969), Academic Press New York · Zbl 0191.46603 [4] Revuz, D.; Yor, M., Continuous martingales and Brownian motion, (1991), Springer-Verlag New York/Berlin · Zbl 0731.60002 [5] Rogers, L.C.G.; Shi, Z., Interacting Brownian particles and the Wigner law, Probab. theory related fields, 95, 555-571, (1993) · Zbl 0794.60100 [6] Roynette, B.; Vallois, P., Instabilité de certaines EDS non-linéaires, J. funct. anal., 130, 2, (1995) · Zbl 0821.60063 [7] Rudin, W., Real and complex analysis, (1966), McGraw-Hill New York · Zbl 0148.02904 [8] A. S. Sznitman, Topics in propagation of chaos, in École d’été de probabilités de Saint-Flour, XIX, 1991, pp. 167-251.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.