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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.

MSC:
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
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