The Feynman-Kac Formula and Semigroups
In this chapter we introduce a class of self-adjoint semigroups naturally attached to the Feynman-Kac formula. Section 1 presents in a regular setting the probabilistic functionals we shall study, and begins the discussion of their functional analytic description. Section 2 introduces the class of potential we shall consider in the sequel. Section 3 studies some properties of the semigroups, which are defined in terms of expectations of Brownian motion functionals already encountered in Section 1. In Section 4 we provide by means of quadratic forms, a functional analytic characterization of the semigroups defined in Section 3. We shall amply use throughout the remaining chapters the bridge between functional analytic and probabilistic point of views developed in this chapter.
KeywordsBrownian Motion Weak Solution Quadratic Form Dirichlet Form Continuous Semigroup
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