Chernoff approximation of subordinate semigroups

Abstract

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.

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

“…Analogous results hold true for distributed order time-fractional Fokker-Planck-Kolmogorov equations with non-local operators L considered in Subsection 3.2 and in [16,21] uniformly in x ∈ G and locally uniformly in t ∈ [0, ∞). Here the family (F o (t)) t≥0 has been constructed from the family (F (t)) t≥0 given in (17)…”

“…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]. Evolution equations with the Vladimirov operator (this operator is a p-adic analogue of the Laplace operator) have been investigated in [79,80,78,77,76].…”

“…On the other hand, representations of solutions of evolution equations in the form of Feynman formulae can be obtained by different methods, not necessarily via the Chernoff Theorem. And such Feynman formulae may have no relations to any Chernoff approximation, or their relations may be quite indirect (see, e.g., Feynman formulae (16), (27), (31) below).…”

“…To this aim, we need some preparations. 4 Note, that some families (F (t)) t≥0 , which are Chernoff equivalent to certain Feller semigroups, are constructed, e.g., in [16,21,22].…”

“…Therefore, the approximations f n constructed in Theorem 4.1 can be used simultaneousely to approximate path integrals appearing in different stochastic representations of the same function f (t, x) and to approximate solutions of corresponding time-non-fractional evolution equations of hihger order. Obviousely, approximations f n similar to those of Theorem 4.1 can be constructed also for subordinate semigroups discussed in Section 3.1 of [16]. Namely, assume that a semigroup (T f t ) t≥0 is subordinate to a given semigroup (T t ) t≥0 on a Banach space (X, ⋅ X ) with respect to a given convolution semigroup (η t ) t≥0 associated to a Bernstein function f , i.e.…”