Winding asymptotics of reflected Brownian motion in a three-dimensional cone in the null-recurrent regime

Toby and Werner (1995) proved Cauchy-type winding limit laws for obliquely reflected Brownian motion in bounded planar domains and for transient reflected Brownian motion in three-dimensional cones with radially homogeneous reflection: in the transient cone case the winding numbers around lines through the vertex, normalized by $\log t$, converge in law to Cauchy variables, based on an almost-sure clock asymptotic $\int_1^t |Z_s|^{-2}\,ds \sim c\log t$ derived from the ergodicity of an associated reflected Ornstein-Uhlenbeck process. They left the null-recurrent (critical) regime, in which the harmonic exponent $\alpha$ of the cone equals zero, unresolved, conjecturing that there the clock grows on the scale $(\log t)^2$ with a stable limit of index $1/2$. This problem asks for the long-time winding asymptotics in that regime: determine whether the renormalized clock converges in law (in particular whether the $(\log t)^2$ stable $(1/2)$ conjecture is correct), and derive the induced limit law for the winding numbers around lines through the vertex. The problem remains open.

Background

Reflected Brownian motion (RBM) is the diffusion obtained by confining Brownian motion to a domain and pushing it back at the boundary: in a planar domain $D$ with a $C^1$ reflection field $u$ on $\partial D$ pointing inward, it is the continuous semimartingale $X_t = B_t + \int_0^t u(X_s)\,dL_s$, where $B$ is planar Brownian motion and $L$ is a boundary local time that increases only when $X$ sits on the boundary. The winding number of a planar path around a point $z$ is the total continuous change of the polar argument of $X_t - z$; in three dimensions one winds around a line, tracking the angular motion in the plane orthogonal to it. For free planar Brownian motion, Spitzer (1958) proved the archetypal result: the winding angle around a fixed point, normalized by $\log t$, converges in law to a multiple of a Cauchy random variable (the symmetric, heavy-tailed stable law of index 1). The mechanism is the skew-product decomposition, in which the angular motion is a one-dimensional Brownian motion run at the clock $\int_0^t |Z_s|^{-2}\,ds$, an additive functional that accumulates when the path is near the winding target; Cauchy limits reflect the heavy-tailed fluctuations of this clock.

Toby and Werner (1995) extended this circle of ideas to reflected motions. In a bounded planar domain with smooth boundary and oblique reflection they showed that $(\varphi_t^1,\dots,\varphi_t^p)/t$ converges in law to $(a_i C_i + b_i)$ with independent standard Cauchy variables $C_i$, where $a_i = 0$ when the winding point lies outside the domain and the drifts $b_i$ are ergodic averages of the boundary push. For a three-dimensional cone $\Omega$ with vertex $0$ whose angular cross-section has $C^2$ boundary, and a radially homogeneous reflection field $v$ ($v(r\omega) = v(\omega)$) with strictly positive inward normal component, they proved the analogue at the $\log t$ scale for the windings $\theta_t^i$ around lines through the vertex, in the transient regime. Transience is characterized there by the existence of a positive harmonic function $\Phi(r\omega) = r^{\alpha}\psi(\omega)$ in $\Omega$ satisfying the conormal boundary condition $\langle v, \nabla\Phi\rangle = 0$ on $\partial\Omega\setminus\{0\}$ with exponent $\alpha < 0$. Their proof rests on three pillars: a skew product $Z_t = |Z_t|\,Y\bigl(\int_0^t |Z_s|^{-2}\,ds\bigr)$ with $Y$ a reflected Brownian motion on the spherical cross-section; the fact that the logarithmically rescaled process $\tilde Z_u = e^{-u/2}Z(e^u)$ is a stationary, Harris-recurrent (recurrent in the measure-theoretic sense that supports ergodic theorems for Markov processes) reflected Ornstein-Uhlenbeck diffusion (the Ornstein-Uhlenbeck process being the Gaussian diffusion with generator $\tfrac12\Delta - \tfrac12 x\cdot\nabla$); and the resulting almost-sure clock asymptotic $A(t)/\log t \to c \in (0,\infty)$ for the angular clock $A(t) = \int_1^t |Z_s|^{-2}\,ds$, with $c$ equal to the invariant expectation $\mathbb{E}(|Z_1|^{-2})$.

The transient argument breaks down exactly in the critical, null-recurrent regime $\alpha = 0$: no positive harmonic $r^{\alpha}\psi$ with $\alpha < 0$ exists, yet the motion keeps returning toward the vertex; such critical reflection fields arise by continuity between transient and recurrent configurations of the same geometry. There the log-rescaled process is null-recurrent — recurrent, in that it returns to every neighborhood of the vertex, but carrying no finite invariant probability measure, so that time averages need not converge — rather than ergodic, so no almost-sure clock linear in $\log t$ is available. By analogy with the classical planar critical picture, Toby and Werner conjectured that the clock grows roughly like $(\log t)^2$ times a stable random variable of index $1/2$ (a Lévy-distributed random variable, the hitting-time law of one-dimensional Brownian motion, with density decaying like $x^{-3/2}e^{-c/x}$), but they did not prove a winding limit theorem in this regime. The companion study of DeBlassie, Hobson, Housworth and Toby (1996) determined escape-rate asymptotics for transient reflected Brownian motion in wedges and cones, which governs the radial behavior but not the angular clock or the windings.

Problem Statement

Let $\Omega$ be an open connected cone in $\mathbb{R}^3$ with vertex $0$ whose angular cross-section $\hat\Omega = \{\omega \in \Omega : |\omega| = 1\}$ has boundary a finite union of $C^2$ Jordan curves (simple closed curves), and let $v$ be a $C^1$ radially homogeneous reflection vector field on $\partial\Omega\setminus\{0\}$ (i.e. $v(r\omega) = v(\omega)$ for $r > 0$) with strictly positive inward normal component. Let $Z$ be the reflected Brownian motion in $\Omega$ with reflection field $v$, in the null-recurrent (critical) regime in which the harmonic exponent $\alpha$ of the cone equals $0$: there is no positive harmonic function $\Phi(r\omega) = r^{\alpha}\psi(\omega)$ with conormal boundary condition $\langle v, \nabla\Phi\rangle = 0$ on $\partial\Omega\setminus\{0\}$ and exponent $\alpha < 0$, and the logarithmically rescaled process $e^{-u/2}Z(e^u)$ is null-recurrent. Determine the long-time asymptotic behavior of:

(1) the angular clock

$$A(t) = \int_1^t |Z_s|^{-2}\,ds,$$

in particular whether $A(t)/(\log t)^2$ converges in law as $t \to \infty$, and whether the limit is (a constant multiple of) a stable random variable of index $1/2$, as conjectured by Toby and Werner;

(2) the winding numbers $\theta_t^i$ of $Z$ around lines $L^i$ through the vertex — the correct normalization and the limit law, including its type, its dependence on the cone geometry and reflection field, and its behavior for lines not meeting the cone.

References

1. Ellen Toby and Wendelin Werner, "On windings of multidimensional reflected brownian motion", Stochastics and Stochastic Reports 55 (1995), no. 3-4, pp. 315-327. DOI: 10.1080/17442509508834030, https://doi.org/10.1080/17442509508834030 (primary source; cf. Theorem 5.2 of Geng and Iyer, ALEA 18 (2021))
2. Frank Spitzer, "Some theorems concerning 2-dimensional Brownian motion", Transactions of the American Mathematical Society 87 (1958), no. 1, pp. 187-197. DOI: 10.1090/S0002-9947-1958-0104296-5, https://doi.org/10.1090/S0002-9947-1958-0104296-5
3. R. Dante DeBlassie, David Hobson, Elizabeth Ann Housworth, and Ellen H. Toby, "Escape rates for transient reflected brownian motion in wedges and cones", Stochastics and Stochastic Reports 57 (1996), no. 3-4, pp. 199-211. DOI: 10.1080/17442509608834060, https://doi.org/10.1080/17442509608834060
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