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

Adam Bobrowski, Elżbieta Ratajczyk

Abstract

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.

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

Abstract

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 . Thus, the latter process can be seen as a Brownian motion perturbed by a trace of semi-permeable membrane at the graph's center.

Paper Structure

This paper contains 14 sections, 10 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Sticky snapping-out Brownian motion
  3. Walsh's sticky process
  4. The main result
  5. Two generation theorems
  6. The main convergence theorem
  7. A sketch of the theory of convergence of semigroups and bird's-eye view of our results
  8. The main theorem
  9. Convergence outside of $C(K_{1,k})$
  10. Introductory remarks
  11. Definition of $M(\mathpzc c)$
  12. Cosine families
  13. Lord Kelvin's method of images and the generation theorem
  14. Convergence of cosine families

Key Result

Lemma 2.1

Let $A_i,B_i,C_i, i \in \mathcal{K}$ be given constants such that $A_i>0$. Then, for any $\varepsilon > 0$ there is precisely one solution $(D_i(\varepsilon))_{i\in \mathcal{K}}\in \mathbb{R}^k$ to the system Moreover, the limits $\lim_{\varepsilon \to 0} D_i(\varepsilon), i \in \mathcal{K}$ exist and are finite.

Figures (3)

  • Figure 1: Sticky snapping out Brownian motion is a Feller process on $k$ copies of $[0,\infty]$ (here $k =3$), which on the $i$th copy behaves like a one-dimensional sticky Brownian motion with stickiness coefficient $a_i/b_i$. After spending enough time at $(0,i)$ the process jumps to one of the points $(0,j), j\not =i$ to continue its motion on the corresponding copy of $[0,\infty]$, and so on. Times between jumps are governed by parameters $c_i$.
  • Figure 2: The infinite star-like graph $K_{1,k}$ with $k=8$ edges. Walsh's sticky process on $K_{1,k}$ is a Feller process whose behavior at the graph's center is characterized by the boundary condition visible above; outside of the center, on each of the edges, the process behaves like a standard one-dimensional Brownian motion.
  • Figure 3: State-space collapse. As permeability coefficients $c_i$ become infinite, times spent at the points $(i,0), i \in \mathcal{K}$ before jumps become shorter and shorter. As a result, in the limit all these points are lumped together and $S_k$ becomes $K_{1,k}$ (here $k =5$).

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Theorem 4.1
  • ...and 8 more