zbMATH — the first resource for mathematics

Feynman-Kac semigroups, ground state diffusions, and large deviations. (English) Zbl 0798.60067
Summary: We study the generalized Schrödinger operator \(-{\mathcal L} + V\), where \({\mathcal L}\) is the generator of a symmetric Markov semigroup \((P_ t)\) on \(L^ 2(E,m)\), and the corresponding Dirichlet form \({\mathcal E}^ V\). By means of the Cramer functional \(\Lambda(V)\), we give necessary and sufficient conditions for \({\mathcal E}^ V\) 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 \(-{\mathcal L} + V\) are also given. By means of large deviations, we find a new condition which ensures the existence of ground state \(\varphi\) of \(-{\mathcal L} + V\) and we construct the ground state process \(Q^ \varphi_ t\), whose generator is given in the diffusion case by \({\mathcal L}_ \varphi = {\mathcal L} + \varphi^{-1} \Gamma (\varphi, \cdot)\), where \(\Gamma\) is the square field operator associated to \({\mathcal L}\). The self-adjointness of \({\mathcal L}_ \varphi\) 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.

60H25 Random operators and equations (aspects of stochastic analysis)
60J45 Probabilistic potential theory
60F10 Large deviations
47D07 Markov semigroups and applications to diffusion processes
31C25 Dirichlet forms
Full Text: DOI