Butko

pp. 42-52

The paper describes an approach to solve initial and initial-boundary value problems for evolution equations. This approach is based on representation of solutions of such equations by limits of n-fold iterated integrals when n tends to infinity (such representations are called Feynman formulae). These formulae allow to calculate solutions of evolution equations directly, are suitable for approximation of transition probabilities of stochastic processes, can be used for computer modeling of classical, quantum and stochastic dynamics. In the present note Feynman formulae are constructed both for evolution semigroups, obtained by additive and multiplicative perturbations of generators of some original semigroups, and for the Cauchy-Dirichlet initial-boundary value problem for a differential equation of parabolic type. In particular, the paper presents Feynman formulae for the Cauchy and the Cauchy-Dirichlet problems for a second order parabolic equation with variable coefficients and for the Cauchy problem for Schrödinger equation.

The Cauchy problem for a parabolic partial differential equation with biharmonic operator and additive perturbation is considered in this note. Such equations are used in different  domains of physics, chemistry, biology, and computer sciences.  The solution of the considered problem is represented by Feynman formulae, i.e. by limits of iterated integrals of elementary functions when multiplicity if integrals tends to infinity.  The main part of these formulae is proved  with the help of Chernoff's theorem; some formulae are obtained on the base of the Yosida approximations.  Different types of Feynman formulae are presented in this work: Lagrangin and Hamiltonian. Lagrangian Feynman formulae are suitable for computer modeling of the considered dynamics. Hamiltonian Feynman formulae are related to some phase space Feynman path integrals; such integrals are important objects in quantum physics.

In the present paper a new method of investigation and description of linear dynamics was considered. This method was based on representations of corresponding evolution semigroups (or, what is the same, representations of solutions of the corresponding equations) with Feynman formulas, i.e. with limits of finite multiple integrals when n tends to infinity. Sometimes one succeeds to get Feynman formulas containing only integrals of elementary functions. Such Feynman formulas allow to calculate solutions of evolution equations directly, to approximate transition probabilities of stochastic processes, to model stochastic and quantum dynamics numerically. The limits in Feynman formulas agree with some functional integrals with respect to probability measures or Feynman type pseudomeasures. Nowadays, functional integrals (or path integrals) play one of the central roles in mathematical apparatus of theoretical physics; they are important objects of quantum field theory, especially in the theory of gauge fields. To solve a variety of problems it is worth to apply Hamiltonian formalism of quantum mechanics and to deal with (Hamiltonian) Feynman path integrals in phase space. There are many different approaches to define such integrals mathematically rigorously. And different classes of integrable functions arise in the frame of each approach. In this paper the approach of Smolyanov and his coauthors was used. This approach allows to connect Feynman path integrals in pahse space with Hamiltonian Feynman formulas for evolution semigroups. This method was actively used last decade to describe different types of dynamics in domains of Euclidean spaces and Riemannian manifolds, in infinite dimensional linear and non-linear spaces, to investigate p-adic analogues of equations of mathematical physics. The present work is expository, it brings together some of the results of recent articles of the author (joint with Boettcher, Grothaus, Schilling, Smolyanov), in which the method of Feynman formulas was subsequently developed to investigate Feller semigroups and the connection of such formulas with Feynman path integrals in phase space was studied. In this paper Feynman formulas for Feller semigroups and semigroups generated with different quantizations of a quadratic Hamilton function were obtained; a construction of Feynman path integral in phase space was introduced; Feynman path integrals in phase space for Feller semigroups and semigroups generated with different quantizations of a quadratic Hamilton function were presented.

Class of parabolic second order equations, produced by different quantization types of one classical system’s quadratic Hamiltonian, was considered. Solution of Cauchy-Dirichlet problem for considered class of equations on the interval was represented as Hamiltonian Feynman’s formula, that is as a limit of finite-multiplicity elementary integrals when multiplicity approached infinity. New formula for direct computation of the solution to formulated problem and computer simulation of corresponding dynamics was obtained in this work. Connection between differential operators corresponding to different quantization types of quadratic Hamiltonian and connection of obtained Hamiltonian Feynman’s formula with Feynman’s path integrals in phase space were also discussed in the article.

Cauchy-Neyman problem for parabolic equation on half-line with variable coefficients depending on coordinate was considered.Solution of this problem was presented as a limit of elementary multiple integrals, containing coefficients of equation and initial conditions, when multiple approached infinity. Such formulas were called “Feynman’s formulas”. Similar presentations of evolution equations’ solutions could be used for direct computation and computer simulation of researched dynamics. Among other things, limits of finite-multiplicity integrals in Feynman’s formulas agreed with some functional integrals in some probability measures on set of trajectories in the areas where equations were considered. By this means, Feynman’s formulas allowed to approximate functional integrals and, consequently, transition probabilities (which usually weren’t expressed in terms of elementary functions) of corresponding random processes. Method for obtaining Feynman’s formulas for evolution equations has been proposed in the works of Smolyanov O. G. and his co-authors in 1999 – 2003 years. This method was based on the application of Chernoff’s theorem and allowed to obtain Feynman’s and Feynman-Katz’s formulas for extensive class of evolution equations on different geometric structures.

A new method of describing linear dynamics is considered.  This method is based on representations of corresponding evolution semigroups (or, what is the same, representations of solutions of corresponding evolution equations) by Feynman formulae, i.e. by limits of n-fold iterated integrals, when tends to infinity. Green functions of many initial-boundary value problems are not known explicitly, whereas   it is possible to obtain Feynman formulae containing only elementary functions as integrands for some of these problems. Such Feynman formulae, that allow to calculate solutions of evolution equations directly, to approximate  transition probabilities of stochastic processes, are useful  for computer modeling of quantum and stochastic dynamics. The notion “Feynman formula” (in this context) and the method to obtain such formulae were introduced in works of Smolyanov and his coauthors in the late  nineties. In the last decade it has been actively applied to describe different types of dynamics in domains of Euclidean spaces and Riemannian manifolds, in infinite dimensional linear and non-linear spaces. In the present note the multiplicative perturbations of generators of strongly continuous semigroups on a Banach space of some continuous functions are considered.  A Feynman formula   is obtained for the semigroups with   perturbed generators.  Therefore, a new formula is given for the description and the investigation of the perturbed dynamics. Also some particular examples, when the obtained Feynman formula contains only elementary functions as integrands, are considered in this note.