Infinite dimensional Kolmogorov operators with time dependent drift coefficients
The main object of this paper is the Kolmogorov operator in \([0,T] \times H\), where \(H\) is a separable Hilbert space with norm \(|\cdot|\) and inner product \(\langle \cdot,\cdot\rangle\) and \(T>0\) is fixed, defined by \[ L_0u=D_tu+N(t)u, \] with \(N(t)u(t,x)=\tfrac 12\text{Tr}[CD^2_xu(t,x)]+ \langle x,A*D_xu(t,x)\rangle +\langle F(t,x\rangle,D_xu(t,x)\rangle.\) Here, \(A:D(A)\subset H\to H\) is the infinitesimal generator of a \(C_0\) semigroup \(e^{tA}\) in \(H\), \(A*\) is the adjoint of \(A\), \(C\) is a positive linear operator on \(H\), and the mapping \(F:D(F)\subset[0,T]\times H\to H\) such that \(F(t,\cdot)\) is quasi-dissipative for all \(t\in [0,T]\) (see paper definitions). \(D_t\) and \(D_x\) denote the derivatives in \(t\) and \(x\), respectively.
scientific article; zbMATH DE number 5357846