Feynman-Kac Semigroups, Ground State Diffusions, and Large Deviations

We study the generalized Schrödinger operator −L + V, where L is the generator of a symmetric Markov semigroup (Pt) on L2(E, m), and the corresponding Dirichlet form EV. By means of the Cramer functional Λ(V), we give necessary and sufficient conditions for EV to be lower bounded and for the Feynman-Kac semigroup (PVt) to be bounded. Some sufficient conditions for the essential self-adjointness of −L + V are also given. By means of large deviations, we find a new condition which ensures the existence of ground state φ of −L + V and we construct the ground state process Qφt, whose generator is given in the diffusion case by Lφ = L + φ−1Γ(φ, · ), where Γ is the square field operator associated to L. The self-adjointness of Lφ is discussed. As applications, we consider perturbation of the semigroups of second quantization on an abstract Wiener space, the time evolution of euclidean quantum fields, and stochastic quantization.