[Paper Review] Asymptotic behavior of a relativistic diffusion in Robertson-Walker space-times

Jürgen Angst|arXiv (Cornell University)|Jan 1, 2014

Geometric Analysis and Curvature Flows20 references4 citations

TL;DR

This paper analyzes the long-time asymptotic behavior of relativistic diffusion on Robertson-Walker space-times, proving that the diffusion's spatial projection almost surely converges to a random point on the causal boundary as it approaches explosion time. The tangent vector's limiting behavior is shown to depend on the time function's finiteness and the scale factor's growth rate, with distinct dynamics in flat, hyperbolic, and spherical spatial sections.

ABSTRACT

We determine the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a Robertson-Walker space-time. We prove in particular that when approaching the explosion time of the diffusion, its projection on the base manifold almost surely converges to a random point of the causal boundary and we also describe the behavior of the tangent vector in the neighborhood of this limiting point. 1

Motivation & Objective

• To understand the long-term behavior of relativistic diffusion in a broad class of Lorentzian manifolds, specifically Robertson-Walker space-times.
• To determine whether the diffusion process converges almost surely to a limiting point in the causal boundary of the base manifold.
• To characterize the asymptotic behavior of the tangent vector in the unitary tangent bundle as the process approaches explosion time.
• To relate the limiting dynamics to geometric properties of the space-time, such as the finiteness of the time interval and the growth of the scale factor.

Proposed method

• Uses the relativistic diffusion process defined on the future-directed unitary tangent bundle of a Lorentz manifold, as constructed by Franchi and Le Jan.
• Applies stochastic differential equations on the unitary tangent bundle, with dynamics driven by a time-changed Brownian motion on the unit sphere in the tangent space.
• Employs a time change via the clock $ A_s = ∫_0^s \frac{a_u}{\alpha^2(t_u)} du $ to simplify the SDEs governing the spatial and velocity components.
• Analyzes the complex process $ z_s = x_s + i y_s $, where $ y_s = \dot{x}_s / |\dot{x}_s| $, to study oscillatory behavior in the compact spatial case.
• Applies convergence results for stochastic integrals with finite quadratic variation to show almost sure convergence of $ I_s $, $ J_s $, and $ K_s $.
• Derives asymptotic expressions for $ x_s $ and $ \dot{x}_s / |\dot{x}_s| $ using the solution $ z_s = z_0 e^{-iA_s} + i e^{-iA_s} I_s $, leading to circular motion on the sphere in the compact case.

Experimental results

Research questions

• RQ1 Does the spatial projection of relativistic diffusion in a Robertson-Walker space-time converge almost surely to a point on the causal boundary?
• RQ2 How does the asymptotic behavior of the tangent vector depend on the time interval's finiteness and the scale factor's growth rate?
• RQ3 What is the limiting behavior of the diffusion in the case of compact spatial sections, such as $ \mathbb{S}^d $, and how does it differ from flat or hyperbolic cases?
• RQ4 Can the Poisson boundary of the relativistic diffusion be geometrically characterized via the causal boundary of the space-time?
• RQ5 What role does the time-changed Brownian motion play in determining the long-term dynamics of the diffusion process?

Key findings

• The spatial component $ \xi_s $ of the relativistic diffusion almost surely converges to a random point $ \xi_\infty $ on the causal boundary $ \partial\mathcal{M}_c^+ $ as $ s \to \tau $, the explosion time.
• In the flat case $ M = \mathbb{R}^d $, the normalized velocity $ \dot{x}_s / |\dot{x}_s| $ converges almost surely to a random direction in $ \mathbb{S}^{d-1} $, consistent with a preferred direction in Minkowski space.
• In the hyperbolic case $ M = \mathbb{H}^d $, the normalized velocity converges almost surely to $ (1, \theta_\infty) $, indicating alignment with a random light-like direction.
• In the spherical case $ M = \mathbb{S}^d $, both $ x_s $ and $ \dot{x}_s / |\dot{x}_s| $ asymptotically describe a random great circle on the sphere, with $ x_s $ and $ \dot{x}_s / |\dot{x}_s| $ rotating around orthogonal unit vectors $ U_\infty $ and $ V_\infty $.
• The limiting behavior in the spherical case arises from the solution $ z_s = (z_0 + iI_\infty)e^{-iA_s} + o(1) $, where $ I_\infty $ is the almost sure limit of the stochastic integral $ I_s $, and $ A_s \to \infty $.
• The total variation of the drift term and the quadratic variation of the martingale part both converge almost surely, ensuring the convergence of the integral components in the spherical case.