Lagrangian Feynman Formulas for Second-Order Parabolic Equations in Bounded and Unbounded Domains

Abstract

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

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

“…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.…”

“…Then by Theorem 5.1 the Lagrangian Feynman formula is valid (cf. [7], [8]) for the semigroup (T C t ) t≥0 generated by C := 1 2 ∆ + b∇ + V :…”

Chernoff approximation of subordinate semigroups

“…In this case the PDO with symbol −H is just a second order differential operator with variable coefficients and the following results are true (see [6], cf. [8]). Assume that there exists α ∈ (0, 1] such thath the closure of (L, C 2,α c (R d )) generates a strongly continuous semigroup (T t ) t 0 on the space C ∞ (R d ).…”

Chernoff Approximation for Semigroups Generated by Killed Feller Processes and Feynman Formulae for Time-Fractional Fokker–Planck–Kolmogorov Equations

“…For example, this method has been used to investigate Schrödinger type evolution equations in [71,66,74,41,30,84,81,83]; stochastic Schrödinger type equations have been studied in [58,57,59,34]. Second order parabolic equations related to diffusions in different geometrical structures (e.g., in Eucliean spaces and their subdomains, Riemannian manifolds and their subdomains, metric graphs, Hilbert spaces) have been studied, e.g., in [19,15,69,14,67,82,70,7,20,90,18,89,17,13,12,86,11,10,85,56]. Evolution equations with non-local operators generating some Markov processes in R d have been considered in [16,19,21,22].…”

“…where p A is given by (18). And the convergence is uniform with respect to x 0 ∈ R d and with respect to t ∈ (0, t * ] for all t * > 0.…”