Relativistic analysis of stochastic kinematics

TL;DR

This paper develops a relativistic framework for stochastic kinematics, deriving how diffusivity tensors transform between inertial frames, with applications to Poisson-Kac processes and validation through simulations.

Contribution

It introduces a relativistic transformation theory for diffusivity tensors in stochastic processes, extending classical diffusion analysis to relativistic contexts.

Findings

Diffusivity in a moving frame becomes anisotropic with specific gamma-dependent factors.

The theory is validated through multiple simulation experiments.

Transformations apply to both one-dimensional and higher-dimensional stochastic models.

Abstract

The relativistic analysis of stochastic kinematics is developed in order to determine the transformation of the effective diffusivity tensor in inertial frames. Poisson-Kac stochastic processes are initially considered. For one-dimensional spatial models, the effective diffusion coefficient $D$ measured in a frame $\Sigma$ moving with velocity $w$ with respect to the rest frame of the stochastic process can be expressed as $D= D_0 \, \gamma^{-3}(w)$. Subsequently, higher dimensional processes are analyzed, and it is shown that the diffusivity tensor in a moving frame becomes non-isotropic with $D_\parallel = D_0 \, \gamma^{-3}(w)$, and $D_\perp = D_0 \, \gamma^{-1}(w)$, where $D_\parallel$ and $D_\perp$ are the diffusivities parallel and orthogonal to the velocity of the moving frame. The analysis of discrete Space-Time Diffusion processes permits to obtain a general transformation theory of the tensor diffusivity, confirmed by several different simulation experiments. Several implications of the theory are also addressed and discussed.