New semigroups from old: An approach to Feller boundary conditions

Adam Bobrowski

doi:10.3934/dcdss.2023084

Article Contents

2024, Volume 17, Issue 5&6: 2108-2140. Doi: 10.3934/dcdss.2023084

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New semigroups from old: An approach to Feller boundary conditions

Adam Bobrowski

Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland

Dedicated to J.A. Goldstein, a prolific mathematician and a kind friend

Received: September 2022

Early access: April 2023

Published: May 2024

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Abstract

Stochastic techniques allow constructing new random processes from old. For example, given an unrestricted Brownian motion one can construct reflecting, elastic, killed, and stopped Brownian motions. On the other hand, from the functional-analytic point of view, the semigroups that describe these new processes seem to need to be obtained independently, using the Hille–Yosida theorem. The aim of this article is to show that this is not necessary; all these semigroups are hidden in the unrestricted Brownian motion semigroup as its subspace semigroups. In other words, in the semigroup theory, subspace semigroups play the role of a number of advanced techniques of stochastic analysis.

To exemplify this, we exhibit a number of invariant subspaces for the unrestricted Brownian motion semigroup, construct the corresponding semigroups and show how they are related to the semigroups describing Brownian motions on the half-line with various types of boundary behavior at $ x = 0 $, including the Brownian motions named above. This analysis is then applied to explaining the nature of transmission conditions used in modeling semi-permeable and permeable membranes.

Keywords:

Invariant subspaces,
cosine families,
Feller–Wentzel boundary and transmission conditions,
local times,
skew Brownian motion,
snapping-out Brownian motion,
Feller semigroups and resolvents.

Mathematics Subject Classification: Primary: 35B06, 47D07, 47D09, 60J35, 60J65; Secondary: 60J55, 60J70.

Citation:

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Figure 1. Free basic types of behavior of BM at the boundary — (B) reflection, (C) killing (with removing), and (D) stopping

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Figure 2. Three extreme cases of boundary conditions

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Figure 3. Elementary exit BM (on the left) versus elastic BM (on the right) — both are undefined after a sufficiently long time is spent at the boundary

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Figure 4. A new construction of reflecting Brownian motion starting at $ 0 $: the original path of an unrestricted Brownian (A), its maximum (B), and the difference between the two (C)

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Figure 5. Skew BM is constructed from reflecting BM as follows: each excursion from zero to $ (0, \infty) $ is with probability $ 1- \alpha $ reflected to $ (-\infty, 0) $. In the particular path depicted above, it is the sixth, the eighth, the tenth and the twelfth excursions that are reflected, the remaining ones are left intact

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