Fredholm determinant of an integral operator driven by a diffusion process (Q1009398)

Consider a manifold valued elliptic diffusion process \((X_t)\), two locally Lipschitz positive matrices \(H(t)\) and \(\Gamma(x)\), and the trace class operator \[ f\mapsto K_{X,T} f(s):= \int^T_0 H(s\cap t)\Gamma(X_t)f(t)\,dt. \] The purpose of the author is to differentiate (with respect to \(T> 0\)) the following quantity, of Feynman-Kac type: \[ \lambda(T,x):= \mathbb{E}_x\Biggl[f(X_T)\exp\Biggl(- \int^T_0 W(X_t)\,dt\Biggr)\times \text{det}(\mathbf{1}+ K_{X,T})^p\Biggr]\;(\text{for }p> 0). \] The method uses an auxiliary process \((Z_t)\), in order to apply Feynman-Kac formula to \((t,X_t, Z_t)\). A key tool is an expression for \({d\over dT}\log[\text{det}(\mathbf{1}+ zK_{X,T})]\).