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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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