Другие журналы

Butko
On fundamental solutions, transition densities and fractional derivatives
Engineering Education # 01, January 2015 DOI: 10.7463/0115.0754986 pp. 4252
Feynman formulae for evolution semigroups
Engineering Education # 03, March 2014 DOI: 10.7463/0314.0701581 The paper describes an approach to solve initial and initialboundary value problems for evolution equations. This approach is based on representation of solutions of such equations by limits of nfold 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 CauchyDirichlet initialboundary value problem for a differential equation of parabolic type. In particular, the paper presents Feynman formulae for the Cauchy and the CauchyDirichlet problems for a second order parabolic equation with variable coefficients and for the Cauchy problem for Schrödinger equation.
Feynman formulae for a parabolic equation with biharmonic differential operator on a configuration space
Engineering Education # 08, August 2012 DOI: 10.7463/0812.0445534 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.
7730569/315838 Representations of evolution semigroups with Feynman formulas and Feynman path integrals in phase space
Engineering Education # 02, February 2012 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 nonlinear spaces, to investigate padic 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.
7730569/251251 Feinmann’s formulas for class of parabolic equations, corresponding to tauquantization of quadratic Hamiltonian
Engineering Education # 11, November 2011 Class of parabolic second order equations, produced by different quantization types of one classical system’s quadratic Hamiltonian, was considered. Solution of CauchyDirichlet problem for considered class of equations on the interval was represented as Hamiltonian Feynman’s formula, that is as a limit of finitemultiplicity 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.
7730569/246219 Presentation of CauchyNeyman problem’s solution for parabolic equation on halfline with Langrangian Feinmann’s formula
Engineering Education # 11, November 2011 CauchyNeyman problem for parabolic equation on halfline 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 finitemultiplicity 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 coauthors in 1999 – 2003 years. This method was based on the application of Chernoff’s theorem and allowed to obtain Feynman’s and FeynmanKatz’s formulas for extensive class of evolution equations on different geometric structures.
7730569/239563 Feynman formula for semigroups with multiplicatively perturbed generators
Engineering Education # 10, October 2011 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 nfold iterated integrals, when tends to infinity. Green functions of many initialboundary 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 nonlinear 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.



