7 papers
Perturbing Walsh's spider process by infinitesimal drifts
Adam Bobrowski, Andrey Pilipenko, Elżbieta Ratajczyk
We consider Walsh's process on a star graph perturbed by the fact that its paths need to additionally pass through semi-permeable soft membranes, each located on a different edge a…
On Portenko's approximation of skew Brownian motion
Adam Bobrowski, Andrey Pilipenko
From the perspective of the theory of operator semigroups, we reflect back on the classical theorem of Portenko devoted to approximation of skew Brownian motion. The theorem says t…
Analytic and stochastic description of Brownian motions on star graphs
Adam Bobrowski, Andrey Pilipenko
We provide a detailed description of all possible Feller processes on infinite} star graphs with finite number of edges, processes that while away from the graph's center behave li…
A limit theorem for certain Feller semigroups, and the distribution of the local time for Brownian motion when it exits an interval
Adam Bobrowski, Andrey Pilipenko
We consider local singular perturbations of a one-dimensional Laplace operator from the point of view of semigroup theory. Under certain assumptions, we prove the convergence of th…
A kinetic model approximation of Walsh's spider process on the infinite star-like graph
Adam Bobrowski, Elżbieta Ratajczyk
We consider processes of deterministic motions on copies of the star-like graph with edges which are perturbed by two stochastic mechanisms: one caused by in…
From snapping out Brownian motions to Walsh's spider processes on star-like graphs
Adam Bobrowski, Elżbieta Ratajczyk
By analyzing matrices involved, 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…
Approximation of skew Brownian motion by snapping-out Brownian motions
Adam Bobrowski, Elżbieta Ratajczyk
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 proces…