New semigroups from old: An approach to Feller boundary conditions

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.