Free propagators and Feynman-Kac propagators
speaker: Archil Gulisashvili (Ohio University)abstract: Propagators or evolution families are two-parameter relatives of semigroups. They satisfy the flow condition (forward propagators) or the backward flow condition (backward propagators). Important examples of propagators are the so-called free propagators, which are integral operators generated by backward transition probability functions (forward free propagators), or by transition probability functions (backward free propagators). More examples of propagators are celebrated Feynman-Kac propagators, which are perturbations of free propagators by time-dependent functions or measures. Feynman-Kac propagators have important applications in Mathematical Physics, Probability Theory, and Partial Differential Equations. In the lecture, we will introduce classes of time-dependent measures, generalizing the Kato class on Rn, and explain the elements of the theory of Feynman-Kac propagators associated with such measures. Most of the lecture will be devoted to the inheritance problem for Feynman-Kac propagators, more precisely, we will explain in what form the properties of free propagators are inherited by their Feynman-Kac perturbations. Among the properties discussed in the lecture are the $Lp$-boundedness, the (Lp-Lq)-smoothing property, and the boundedness in various spaces of continuous functions (the Feller property, the Feller-Dynkin property, and the $BUC$-property).
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