Perturbing Walsh's spider process by infinitesimal drifts
Abstract
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.''
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BibTeX
@misc{bobrowski2026perturbing,
title = {{Perturbing Walsh's spider process by infinitesimal drifts}},
author = {Adam Bobrowski and Andrey Pilipenko and Elżbieta Ratajczyk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06030},
eprint = {2610.06030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06030},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Perturbing Walsh's spider process by infinitesimal drifts
AU - Adam Bobrowski
AU - Andrey Pilipenko
AU - Elżbieta Ratajczyk
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06030
UR - https://arxiv.org/abs/2610.06030
AB - 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.''
ER -