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Stochastic techniques allow constructing new random processes
from old. For example, given an unrestricted Brownian motion one can con-
struct 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 nec-
essary; all these semigroups are hidden in the unrestricted Brownian motion
semigroup as its subspace semigroups. In other words, in the semigroup the-
ory, 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 unre-
stricted 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 explain-
ing the nature of transmission conditions used in modeling semi-permeable and
permeable membranes.
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