Adam Bobrowski

We prove that a snapping-out Brownian motion with large permeability coefficients is a good approximation of Walsh’s spider process on the star-like graph K_{1,k} . Thus, the latter process can be seen as a Brownian motion perturbed by a trace of semi-permeable membrane at the graph’s center. Besides convergence of processes and semigroups we establish, via the Lord Kelvin method of images and analysis of ergodic properties of matrices involved, convergence of cosine families underlying the semigroups and thus gain additional insight into the approximation theorem.

We elaborate on the theorem saying that as permeability coefficients of snapping-out Brownian motions tend to infinity in such a way that their ratio remains constant, these processes converge to a skew Brownian motion. In particular, convergence of the related semigroups, cosine families and projections is discussed.

We provide a rather explicit formula for the resolvent of a concatenation of [Formula: see text] processes in terms of their exit laws and certain probability measures characterizing the way the processes are concatenated. As an application, we prove an averaging principle saying that by concatenating asymptotically splittable processes one can approximate Markov chains.

Following our previous work on complementary boundary conditions, we write Cartesian product of two copies of a space of continuous functions on the real line as the direct sum of two subspaces that are invariant under a cosine family of operators underlying Brownian motion. Both these subspaces are formed by pairs of extensions of continuous functions: in the first subspace the form of these extensions is shaped unequivocally by the transmission conditions describing snapping out Brownian motion, in the second, it is shaped by the transmission conditions of skew Brownian motion with certain degree of stickiness. In this sense, the above transmission conditions are complementary to each other.