Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability

Abstract

We quantify the asymptotic behaviour of multidimensional drifltess diffusions in domains unbounded in a single direction, with asymptotically normal reflections from the boundary. We identify the critical growth/contraction rates of the domain that separate stability, null recurrence and transience. In the stable case we prove existence and uniqueness of the invariant distribution and establish the polynomial rate of decay of its tail. We also establish matching polynomial upper and lower bounds on the rate of convergence to stationarity in total variation. All exponents are explicit in the model parameters that determine the asymptotics of the growth rate of the domain, the interior covariance, and the reflection vector field. Proofs are probabilistic, and use upper and lower tail bounds for additive functionals up to return times to compact sets, for which we develop novel sub/supermartingale criteria, applicable to general continuous semimartingales. Narrowing domains fall outside of the standard literature, in part because boundary local time can accumulate arbitrarily rapidly. Establishing Feller continuity (essential for characterizing stability) thus requires an extension of the usual approach. Our recurrence/transience classification extends previous work on strictly normal reflections, and expands the range of phenomena observed across all dimensions. For all recurrent cases, we provide quantitative information through upper and lower bounds on tails of return times to compact sets (see https://www.youtube.com/watch?v=oDDDWdbPx74 for a short YouTube video describing the results).

Figures (3)

• Figure 1: A positive-recurrent case ($\beta=-1.2<-1=-\beta_c$): simulation of the normally reflected Brownian motion in an unbounded domain, narrowing sufficiently fast so that (by Theorem \ref{['thm:invariant_distributon']}(b)) the process converges to stationary in total variation with at the rate $t^{-0.1}$ as $t\to\infty$.
• Figure 2: A null-recurrent case ($\beta=0.1<1=\beta_c$): simulation of the normally reflected Brownian motion in an unbounded expanding domain. By Theorem \ref{['thm:return_times']}(b), the tail of the return time decays with rate $t^{-0.45}$ as time $t\to\infty$, making the reflected process "less" recurrent than the modulus of the scalar Brownian motion.
• Figure 3: A transient case ($\beta=0.65>0.5=\beta_c$): the trajectory is a martingale in the interior of the domain, clearly being pushed away from the origin at boundary. The reflection is normal and $\sigma_1^2=0.5$, $\sigma_2^2=1$. By Theorem \ref{['thm:rec_tran']}, normally reflected Brownian motion ($\sigma_1^2=\sigma_2^2=1$) in ${\mathcal{D}}\subset {\mathbb R}^2$ is not transient.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• Remark 2.4
• proof : Proof of Lemma \ref{['lem3.1']}
• Lemma 2.5
• proof