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

Abstract

We study the generalized Schrödinger operator − L + V, where L is the generator of a symmetric Markov semigroup ( P t ) on L 2( E, m), and the corresponding Dirichlet form E V . By means of the Cramer functional Λ( V), we give necessary and sufficient conditions for E V to be lower bounded and for the Feynman-Kac semigroup ( P V t ) 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 Q φ t , whose generator is given in the diffusion case by L φ = L + φ −1 Γ( φ, · ), where Γ is the square field operator associated to L . The self-adjointness of L φ 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.