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On Portenko's approximation of skew Brownian motion

Adam Bobrowski, Andrey Pilipenko

TL;DR

The paper develops a semigroup–theoretic, singular-perturbation framework to justify Portenko’s drift-concentration approximation of skew Brownian motion and extends it to Walsh spider processes on finite and infinite star graphs. By analyzing Sturm–Liouville problems and their resolvents, it establishes strong convergence of approximating semigroups to skew Brownian and Walsh semigroups, with explicit parameter transformations: $\widetilde{p} = \frac{p}{p+(1-p)e^{-2\alpha}}$ for skew Brownian motion and $\widetilde{p}_i = \frac{p_i e^{2\alpha_i}}{\sum_j p_j e^{2\alpha_j}}$ for Walsh spiders. The work provides constructive representations via eigenfunctions, Green kernels, and excursions of reflected Brownian motions, and extends the results from finite intervals to finite graphs and then to infinite graphs, linking to Le Gall’s perspective on membranes and regular drift limits. This yields a unified methodology for deriving boundary/center conditions from localized drift perturbations and clarifies how microscopic drifts translate into macroscopic boundary behavior on graphs. The results have implications for diffusion on networks and the rigorous understanding of singular perturbations in stochastic processes on graphs.

Abstract

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 that by concentrating the power of drift of a diffusion process around a point one obtains an equivalent of a semi-permeable membrane at this point, described by skew Brownian motion's boundary condition. We prove convergence of the corresponding Feller semigroups and in doing so, generalize Portenko's theorem to the case of the Walsh processes on star graphs. Our analysis leads through singular perturbations of Sturm--Liouville equations, and reveals that as a result of Portenko-type approximation parameters of Walsh processes are transformed in a simple and elegant manner.

On Portenko's approximation of skew Brownian motion

TL;DR

The paper develops a semigroup–theoretic, singular-perturbation framework to justify Portenko’s drift-concentration approximation of skew Brownian motion and extends it to Walsh spider processes on finite and infinite star graphs. By analyzing Sturm–Liouville problems and their resolvents, it establishes strong convergence of approximating semigroups to skew Brownian and Walsh semigroups, with explicit parameter transformations: for skew Brownian motion and for Walsh spiders. The work provides constructive representations via eigenfunctions, Green kernels, and excursions of reflected Brownian motions, and extends the results from finite intervals to finite graphs and then to infinite graphs, linking to Le Gall’s perspective on membranes and regular drift limits. This yields a unified methodology for deriving boundary/center conditions from localized drift perturbations and clarifies how microscopic drifts translate into macroscopic boundary behavior on graphs. The results have implications for diffusion on networks and the rigorous understanding of singular perturbations in stochastic processes on graphs.

Abstract

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 that by concentrating the power of drift of a diffusion process around a point one obtains an equivalent of a semi-permeable membrane at this point, described by skew Brownian motion's boundary condition. We prove convergence of the corresponding Feller semigroups and in doing so, generalize Portenko's theorem to the case of the Walsh processes on star graphs. Our analysis leads through singular perturbations of Sturm--Liouville equations, and reveals that as a result of Portenko-type approximation parameters of Walsh processes are transformed in a simple and elegant manner.
Paper Structure (15 sections, 18 theorems, 99 equations, 2 figures)

This paper contains 15 sections, 18 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Portenko-type approximation of skew Brownian motion
  3. Proof of Theorem \ref{['thm:glo']}
  4. The existence and convergence of eigenvectors of related Sturm--Liouville problems
  5. Convergence of resolvents
  6. A singular perturbation of Walsh's spider process
  7. Portenko-type approximation of Walsh's spider process
  8. Proof of Theorem \ref{['thm:glo2']}.
  9. The process generated by $A_{\textnormal{Walsh}}$ can be constructed by means of excursions of the reflected Brownian motion on $[0,r]$
  10. A version on the infinite graph
  11. Construction of approximating processes
  12. $X_\varepsilon$s as Feller processes
  13. Generators of $X_\varepsilon$
  14. Convergence of $X_\varepsilon$, as $\varepsilon \to 0$
  15. Convergence of resolvents

Key Result

Theorem 2.1

Let $\left ( \mathrm e^{t A_\varepsilon} \right)_{t \ge 0}$ be the semigroup generated by $A_\varepsilon$. Then strongly and uniformly with respect to $t$ in compact subsets of $[0,\infty)$, where $\left ( \mathrm e^{t A_{\textnormal{skew}}} \right)_{t \ge 0}$ is the semigroup related to the skew Brownian motion with parameter (which is no smaller than $p$) and reflecting barriers at $-r$ and $r

Figures (2)

  • Figure 1: Solutions to the Sturm--Liouville eigenvalue problems \ref{['eac:1']}--\ref{['eac:2']} with $r=\lambda =1, \gamma =\frac{1}{4}$ and $a_\varepsilon$s obtained via \ref{['zero']} from $a$ defined by $a(x)=\mathrm e^{\frac{1}{2} x}[x<0]+\mathrm e^{-5x}[x\ge 0]$. Since $\widetilde{\gamma} = \gamma \mathrm e^{-2\alpha}$, is significantly smaller than $\gamma$, the limit function (labelled $\varepsilon=0$) has a manifest tip at $0$, reflecting the transformed transmission condition $k'(0+)=\widetilde{\gamma} k(0-)$.
  • Figure 2: Star graph $K_{1,{\mathpzc k}}$ with ${\mathpzc k}=9$ edges, each of them with length $r$.

Theorems & Definitions (37)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof : Proof of Proposition \ref{['prop:1']}
  • ...and 27 more