Feynman formulae for Feller semigroups

Butko, Yana A.; Smolyanov, O. G.; Schilling, René L.

doi:10.1134/s1064562410050017

Cited by 14 publications

(17 citation statements)

References 11 publications

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“…In Section 4 we obtain a Lagrangian Feynman formula for a multiplicative perturbation of a Feller semigroup by a function a(·) which is continuous, positive, bounded and bounded away from zero. Note, that analogous Lagrangian Feynman formulas have been proved for some diffusion processes in [8] and have been presented for the Cauchy process in [9]. In Section 5 we consider gradient and bounded Schrödinger perturbations of Feller semigroups and obtain some Hamiltonian and Lagrangian Feynman formulae for them.…”

Section: Introductionmentioning

confidence: 88%

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Lagrangian and Hamiltonian Feynman Formulae for Some Feller Semigroups and Their Perturbations

Butko

Schilling

Smolyanov

2012

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Abstract. 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.Keywords Feynman formulae; Feynman-Kac formulae; approximations of functional integrals, approximations of transition densities.MSC 2010: 47D07, 47D08, 35C99, 60J35, 60J51, 60J60.

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Section: Introductionmentioning

confidence: 88%

“…In this note we prove some Hamiltonian and Lagrangian Feynman formulae for semigroups associated with Feller processes and for perturbations of such semigroups. Several results of the paper have been announced in [9]. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian and Hamiltonian Feynman Formulae for Some Feller Semigroups and Their Perturbations

Butko

Schilling

Smolyanov

2012

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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“…Moreover, by the Leibnitz rule and by Lemma 3 we have F (0)ϕ(x) = A ϕ(x) + V (x)ϕ(x) for each ϕ ∈ D(R m ). Hence, the family (F (t)) t≥0 is Chernoff equivalent to the semigroup e t( A +V ) resolving the Cauchy problem (3). Therefore, the following theorem is true.…”

Section: Lemmamentioning

confidence: 78%

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

2010

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A Feynman formula is a representation of the semigroup, generated by an initialboundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.In the present paper we formulate and prove the Hamiltonian Feynman and FeynmanKac formulae for heat semigroups describing the diffusion of particles with the position-

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“…In this work we obtain the representation of semigroup solutions of the Cauchy problem for the functionaldifferential equation through the Feynman formula (see [4]). It means that although the representation of the evolutionary operator of the Cauchy problem (1) can be defined only in terms of the spectral decomposition (in the simplest situation in terms of the Fourier transform of the solution), nevertheless we obtain an approximation of the evolutionary operator by sequence of n-fold compositions of integrated operators which kernels are elementary functions.…”

Section: Introductionmentioning

confidence: 99%

Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations

Сакбаев¹,

Yaakbarieh²

2012

AJCM

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Feynman formulae for Feller semigroups

Cited by 14 publications

References 11 publications

Lagrangian and Hamiltonian Feynman Formulae for Some Feller Semigroups and Their Perturbations

Lagrangian and Hamiltonian Feynman Formulae for Some Feller Semigroups and Their Perturbations

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

Feynman Formulas Representation of Semigroups Generated by Parabolic Difference-Differential Equations

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