One-dimensional diffusion processes with moving membrane: partial reflection in combination with jump-like exit of process from membrane

Abstract

By analytical methods we construct the two-parameter Feller semigroup associated with Markov process on a line with moving membrane such that at points on both sides of the membrane it coincides with the ordinary diffusion processes given there, and its behavior after visiting the membrane is determined by one of variants of nonlocal Feller-Wentzell conjugation condition.

On one-dimensional diffusion processes with moving membranes

“…This type of Markov process can serve as a one-dimensional mathematical model of the physical phenomenon of diffusion in media with membranes which are located at some points (cf. [3][4][5]). In our case, such points are the boundaries of the given interval of the line as well as the point of pasting together two given diffusion processes.…”

“…The problem, formulated in such a way, is considered for the first time in bounded curvilinear domains with non-smooth boundaries (cf. [3][4][5][6]). The classical solvability of this problem in the space of continuous functions is established here under some assumptions on its output data by the boundary integral equations method with the use of the fundamental solutions of the uniformly parabolic equations and the associated potentials.…”

One-dimensional Wiener process with the properties of partial reflection and delay

“…The solving of parabolic problems of such kind (see [9,11]) is one of several ways to describe the diffusion process by given Feller-Wentzell boundary condition. Other approaches are reflected in many papers, see, e.g., [4,13,18,20,21], where there are presented results of application of the analytical approach to description of the mentioned class of homogeneous Markov processes based on methods of the semigroup theory and functional analysis in relation to the elliptic boundary value problems, and [1,3,6,14,16,17,19], which partially give the development of methods of stochastic analysis for the construction of such type of processes (see also the references given there).…”

“…Note that the scheme, which is used to solve the problem (1)-( 4), is partially presented in [11], where the same conjugation problem is considered for backward Kolmogotov equation with discontinuous coefficients and for the case when the condition (4) contains the nonlocal term of the integral type, but does not contain the term with derivative of the function with respect to the time variable. Note also that similar problems (with different variants of Feller-Wentzell conjugation condition) were studied in our earlier papers for the cases when S (i) t are finite [10] or semi-infinite [9] rectangular domains.…”

“…account (8)-(11), we find by direct calculation thatsup x∈R R |y − x| 4 P(s, x, t, dy) ≤ C(t − s) 2 , 0 ≤ s < t ≤ T, x)P(s, x, s + ∆s, dy) = 0 for (s, x) ∈ S (i) t , i = 1, 2, γ(s) for x = r(s), x) 2 P(s, x, s + ∆s, dy) = b(s, x) = 1 for (s, x) ∈ S (i)t , i = 1, 2, 0 for x = r(s).…”