University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Some existence and uniqueness result for infinite dimensional Fokker--Planck equations

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

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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_0F,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 $K_0F$ is the Kolmogorov operator $$ K_0Fu(t,x)=D_tu(t,x)+ rac12mbox{Tr};[BB*D2_xu(t,x)]+langle Ax+F(t,x),D_xu(t,x) angle $$ 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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