arXiv subjects

Elżbieta Ratajczyk

Publications and source records attributed to Elżbieta Ratajczyk.

5 recordsLinked to original sources

arXiv2026

Perturbing Walsh's spider process by infinitesimal drifts

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 at a small distance from the center, and each having its own permeability coefficient. We describe the limit processes as the distances converge to $0$, in terms of the characteristics of the approximating ones. The main finding is that the limit process is still of Walsh type, but its parameters need to be appropriately modified. The paper is a close companion to A. Bobrowski and A. Pilipenko ``On Portenko's approximation of skew Brownian motion'' (arXiv:2601.20440), and to A. Pilipenko and G. Vilmart, ``Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits.''

math.PR↗

arXiv2024

A kinetic model approximation of Walsh's spider process on the infinite star-like graph

We consider processes of deterministic motions on $k$ copies of the star-like graph $S_k= K_{1,k}$ with $k$ edges which are perturbed by two stochastic mechanisms: one caused by interfaces located at the graphs' centers, the other describing jumps between different copies of the same edge. We prove that diffusing scaling of these processes leads in the limit to the Walsh's spider process on $S_k$.

math.PR↗

arXiv2024

From snapping out Brownian motions to Walsh's spider processes on star-like graphs

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 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.

math.PR↗

arXiv2024

Approximation of skew Brownian motion by snapping-out Brownian motions

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.

math.PR↗

arXiv2023

On pairs of complementary transmission conditions and on approximation of skew Brownian motion by snapping-out Brownian motions

Following our previous work on `perpendicular' boundary conditions, we show that transmission conditions \[ f'(0-)=α(f(0+)-f(0-)), \quad f'(0+)=β(f(0+)-f(0-)),\] describing so-called snapping out Brownian motions on the real line, are in a sense complementary to the transmission conditions \[f(0-)=-f(0+), \quad f''(0+) =αf'(0-)+βf'(0+). \] As an application of the analysis leading to this result, we also provide a deeper semigroup-theoretic insight into the theorem saying that as the coefficients $α$ and $β$ tend to infinity but their ratio remains constant, the snapping-out Brownian motions converge to a skew Brownian motion. In particular, the transmission condition \[ αf'(0+) = βf'(0-), \] that characterizes the skew Brownian motion turns out to be complementary to \[ f(0-) = - f(0+), βf'(0+)=- αf'(0-). \]

math.PR↗