[Paper Review] Poisson boundary of a relativistic diffusion in curved space-times: an example

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

Geometric Analysis and Curvature Flows11 references4 citations

TL;DR

This paper establishes that the Poisson boundary of a relativistic diffusion in a spatially flat, fast-expanding Robertson-Walker space-time with exponential scale factor growth is generated by the spatial limit point $x_\infty$ of the diffusion's trajectory. Using a novel probabilistic devissage method based on subdiffusions, shift-couplings, and invariance under spatial translations, the authors prove that the invariant sigma field of the full diffusion coincides almost surely with $\sigma(x_\infty)$, identifying the Poisson boundary with the causal boundary $\partial\mathcal{M}_c^+$, which is isomorphic to $\mathbb{R}^3$. This is the first such computation in a non-constant curvature Lorentzian setting.

ABSTRACT

We study in details the long-time asymptotic behavior of a relativistic diffusion taking values in the unitary tangent bundle of a curved Lorentzian manifold, namely a spatially flat and fast expanding Robertson-Walker space-time. We prove in particular that the Poisson boundary of the diffusion can be identified with the causal boundary of the underlying manifold.

Motivation & Objective

• To determine the Poisson boundary of relativistic diffusion in curved Lorentzian manifolds, particularly in non-constant curvature settings.
• To extend the understanding of long-time asymptotics of relativistic diffusion beyond flat spacetimes like Minkowski space.
• To develop a general probabilistic method for computing Poisson boundaries of diffusions from those of their subdiffusions.
• To establish that the invariant sigma field of the full relativistic diffusion is generated by the spatial limit point $x_\infty$ in exponential-growth RW space-times.
• To provide a new framework for Poisson boundary computation in hypoelliptic, non-symmetric settings using equivariance and continuity of harmonic functions.

Proposed method

• Utilizes the relativistic diffusion process on the unit tangent bundle $T^1\mathcal{M}$ of a Robertson-Walker space-time with exponential scale factor growth.
• Identifies two key subdiffusions: $(t_s, \dot{t}_s)$ and $(t_s, \dot{t}_s, \dot{x}_s/|\dot{x}_s|)$, which inherit symmetries from the base manifold.
• Applies successive shift-coupling techniques to prove that the Poisson boundaries of these subdiffusions are trivial.
• Employs an abstract conditioning argument based on the hypoellipticity of the infinitesimal generator and equivariance under Euclidean spatial translations.
• Uses approximate identities and regularized conditional expectations to show that bounded harmonic functions on the full state space are constant in the spatial variable, leading to the conclusion that the invariant sigma field is generated by $x_\infty$.
• Leverages continuity of $\mathcal{G}$-harmonic functions (due to hypoellipticity) to pass from regularized to pointwise convergence in the final step.

Experimental results

Research questions

• RQ1 What is the Poisson boundary of relativistic diffusion in a curved, non-constant curvature Lorentzian spacetime such as a fast-expanding Robertson-Walker universe?
• RQ2 Can the invariant sigma field of the full relativistic diffusion process be identified with the sigma field generated by the spatial limit point $x_\infty$?
• RQ3 How can one compute the Poisson boundary of a diffusion in a hypoelliptic, non-symmetric setting where classical methods like Doob’s transform or Lie group symmetries fail?
• RQ4 To what extent can the Poisson boundary of a full diffusion be deduced from the Poisson boundaries of its subdiffusions in geometrically symmetric settings?
• RQ5 What general conditions allow a devissage-style reduction from a full diffusion to a subdiffusion in order to compute the Poisson boundary?

Key findings

• The relativistic diffusion almost surely converges to a random point $x_\infty$ in $\mathbb{R}^3$ as time $s \to \infty$, corresponding to the causal boundary $\partial\mathcal{M}_c^+$ of the spacetime.
• The invariant sigma field of the full diffusion process coincides almost surely with $\sigma(x_\infty)$, meaning that $x_\infty$ fully determines the asymptotic behavior of the process.
• The Poisson boundary of the relativistic diffusion is identified with the causal boundary $\partial\mathcal{M}_c^+$, which is isomorphic to $\mathbb{R}^3$ in the exponential-growth RW case.
• The proof establishes that bounded harmonic functions on the full state space $E \times \mathbb{R}^3$ are constant in the spatial variable $x$, implying that the only asymptotic information is carried by $x_\infty$.
• The method demonstrates that the invariant sigma field of the full process is generated by the invariant sigma field of the subdiffusion $(t_s, \dot{t}_s, \dot{x}_s/|\dot{x}_s|)$ and the additional random variable $x_\infty$, via a novel conditioning argument.
• The result provides the first explicit computation of the Poisson boundary for a relativistic diffusion in a non-constant curvature Lorentzian manifold, extending prior results in flat spacetimes.