Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold

Butko, Yana A.

doi:10.1134/s0001434608030024

Cited by 14 publications

(12 citation statements)

References 6 publications

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“…Let now the semigroup (T f t ) t 0 be subordinate to the semigroup (T t ) t 0 with respect to a given convolution semigroup (η t ) t 0 associated to a Bernstein function f defined by a triplet (σ, λ, µ). The statement below follows immediately from Proposition 4.7, Theorems 2.3, 2.4, 3.1, 3.8 and results of [5]. Further, define the families ( F i (t)) t 0 , i = 1, 2, 3, by F i (t)ϕ(x) : = e −tC • S t • F i (t) ϕ(x) = = Γ e −tC(x) K i (A(γ x (t))t, γ x (t), y)ϕ(y)vol Γ (dy) Γ K i (A(γ x (t))t, γ x (t), y)vol Γ (dy)…”

Section: Approximation Of Subordinate Diffusions In a Riemannian Manimentioning

confidence: 80%

“…Denote the inner product of vectors u(x) and v(x) in the tangent space T x Γ as u(x) · v(x). As in [5] define the family (S t ) t 0 on C(Γ) by S t ϕ(x) := ϕ(γ x (t)),…”

Section: Approximation Of Subordinate Diffusions In a Riemannian Manimentioning

confidence: 99%

“…One further advantage of Chernoff approximation is the fact that this method is applicable for a broad class of evolution semigroups corresponding to different types of dynamics on different geometrical structures (see, e.g. [2], [5], [6], [18], [20], [21], [22] and references therein). This method, however, has never before been applied to approximation of subordinate semigroups.…”

Section: Introductionmentioning

confidence: 99%

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Chernoff approximation of subordinate semigroups

Butko

2018

Stoch. Dyn.

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In this note the Chernoff Theorem is used to approximate evolution semigroups constructed by the procedure of subordination. The considered semigroups are subordinate to some original, unknown explicitly but already approximated by the same method, counterparts with respect to subordinators either with known transitional probabilities, or with known and bounded Lévy measure. These results are applied to obtain approximations of semigroups corresponding to subordination of Feller processes, and (Feller type) diffusions in Euclidean spaces, star graphs and Riemannian manifolds. The obtained approximations are based on explicitly given operators and hence can be used for direct calculations and computer modelling. In several cases the obtained approximations are given as iterated integrals of elementary functions and lead to representations of the considered semigroups by Feynman formulae.

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Section: Approximation Of Subordinate Diffusions In a Riemannian Manimentioning

confidence: 80%

“…Denote the inner product of vectors u(x) and v(x) in the tangent space T x Γ as u(x) · v(x). As in [5] define the family (S t ) t 0 on C(Γ) by S t ϕ(x) := ϕ(γ x (t)),…”

Section: Approximation Of Subordinate Diffusions In a Riemannian Manimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Chernoff approximation of subordinate semigroups

Butko

2018

Stoch. Dyn.

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“…Let H(q, D) be a pseudo differential operator with the symbol H(q, p) as in Theorem 2.3. Since H(q, p) is represented by the Lévy-Khintchine type formula (6) we can use Fourier inversion in (5) and find that the integro-differential operator…”

Section: Feller and Lévy Semigroups And Their Generatorsmentioning

confidence: 99%

“…Recently, this method has been successfully applied to obtain Feynman formulae for different classes of problems for evolutionary equations on different geometric structures, see, e.g. [5]- [7], [26], [27], [32] and also to construct some surface measures on infinite dimensional manifolds (see [34]- [38]). This method is based on Chernoff's theorem (see [12] and [33] for the version used here), which is a generalization of the well-known Trotter formula.…”

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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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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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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Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold

Cited by 14 publications

References 6 publications

Chernoff approximation of subordinate semigroups

Chernoff approximation of subordinate semigroups

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

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