Some existence and uniqueness result for infinite dimensional Fokker--Planck equations

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• Da Prato, G (Scuola Normale Superiore di Pisa)
• Monday 10 September 2012, 09:50-10:40
• Seminar Room 1, Newton Institute.

If you have a question about this talk, please contact Mustapha Amrani.

Stochastic Partial Differential Equations (SPDEs)

We are here concerned with a Fokker—Planck equation in a separable Hilbert space $H$ of the form egin{equation} label{e1} int_{0}^{Tint_H mathcal K_0}F,u(t,x),mu_t(dx)dt=-int_H u(0,x),zeta(dx),quadorall;uinmathcal E nd{equation} The unknown is a probability kernel $(mu_t)_{tin [0,T]}$. Here $K_0^{F$ is the Kolmogorov operator $$ K_0}Fu(t,x)=D_tu(t,x)+rac12mbox{Tr};[BB^*D2_xu(t,x)]+langle Ax+F(t,x),D_xu(t,x) angle $$ where $A:D(A) ubset H o H$ is self-adjoint, $F:[0,T] imes D(F) o H$ is nonlinear and $mathcal E$ is a space of suitable test functions. $K_0^F$ is related to the stochastic PDE egin{equation} label{e2} dX=(AX+F(t,X))dt+BdW(t) X(0)=x. nd{equation} We present some existence and uniqueness results for equation (1) both when problem (2) is well posed and when it is not.

This talk is part of the Isaac Newton Institute Seminar Series series.

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