REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS

Author:

KALETA KAMIL1,OLSZEWSKI MARIUSZ1,PIETRUSKA-PAŁUBA KATARZYNA2

Affiliation:

1. Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

2. Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Abstract

For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

Funder

the National Science Center, Poland

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Geometry and Topology,Modeling and Simulation

Cited by 4 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Good labeling property of simple nested fractals;Journal of Fractal Geometry, Mathematics of Fractals and Related Topics;2024-05-28

2. Density of states for the Anderson model on nested fractals;Analysis and Mathematical Physics;2024-03-10

3. Oscillations of BV measures on unbounded nested fractals;Journal of Fractal Geometry;2023-04-14

4. Estimates of the transition densities for the reflected Brownian motion on simple nested fractals;Probability and Mathematical Statistics;2019-12-19

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