Anderson model with Lévy potential (Q1906544)

The equation \[ du_t = L_t u_t dt + u_td \zeta_t, \quad u_0 = \varphi \] is considered, where \(L_t\) is the infinitesimal generator of a time inhomogeneous Feller process and \(\zeta\) is a family of discontinuous semimartingales. The result is that, under conditions where a solution to the equation exists, it has Doléan-Feynman-Kac representation \[ u(t,x) = E_x \left[ \varphi (w_{0,t}) \exp \left\{ \int^t_0 d \zeta^c_s (w_{s,t}) - {1 \over 2} \left \langle \int^t_0 d \zeta^c_. (w_{.,t}) \right \rangle_t \right\} \prod_{0 < s \leq t} \bigl( 1 + \Delta \zeta_s (w_{s,t}) \bigr) \right] \] where \(\zeta^c\) denotes the continuous part of \(\zeta\) and \(\Delta \zeta\) denotes the jumps. \(E_x\) is expectation with respect to the time inhomogeneous Feller process, the trajectories of which are denoted by \(w_{s,t}\). This result is a trivial extension of the usual Doléan's representation for solutions to the stochastic O.D.E. \(dH_t = H_t dM_t\) where \(M\) is a discontinuous semimartingale.