LAGRANGIAN AND HAMILTONIAN FEYNMAN FORMULAE FOR SOME FELLER SEMIGROUPS AND THEIR PERTURBATIONS

A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of n-fold iterated integrals of some elementary functions as n → ∞. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite-dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman–Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae give an effective tool to calculate functional integrals with respect to probability measures generated by these Feller processes and, in particular, to obtain simulations of Feller processes.

FEYNMAN FORMULAS FOR AN INFINITE-DIMENSIONAL 𝔭-ADIC HEAT TYPE EQUATION

Smolyanov has introduced1 the term "Feynman formula" (in the configuration space) for the representation of a solution of a Cauchy problem by limit of integrals over finite Cartesian products of the domain of the solution when the number of multipliers tends to infinity. In this paper, such formulas (first written by Smolyanov, Shamarov and Kpekpassi in a short note2) are proved for a family of heat type equations where the spatial variable runs over 𝔭-adic space of countable sequences. Equations with 𝔭-adic variables describe, for example, the dynamics of proteins.

EXTENDED FEYNMAN FORMULA FOR THE HARMONIC OSCILLATOR BY THE DISCRETE TIME METHOD

We revisit the extended Feynman formula for the harmonic oscillator beyond and at caustics. The extension has been made by some authors, however, it is not obtained by the discrete formulation of path integral, which we consider the most reliable regularization of it. We derive the result by, especially at caustics, more rigorous method than previous.