Nonholonomic Relativistic Diffusion and Exact Solutions for Stochastic Einstein Spaces

TL;DR

This paper develops a framework for relativistic stochastic processes on curved spaces, integrating nonholonomic geometry with Einstein gravity to model stochastic gravitational phenomena and solutions.

Contribution

It introduces a novel approach combining nonholonomic geometry with stochastic calculus to construct and analyze stochastic Einstein spaces.

Findings

Formulated Ito and Stratonovich calculus on nonholonomic manifolds

Constructed stochastic Einstein manifolds with off-diagonal metrics

Analyzed conditions for transition between classical and stochastic gravitational processes

Abstract

We develop an approach to the theory nonholonomic relativistic stochastic processes on curved spaces. The Ito and Stratonovich calculus are formulated for spaces with conventional horizontal (holonomic) and vertical (nonholonomic) splitting defined by nonlinear connection structures. Geometric models of relativistic diffusion theory are elaborated for nonholonomic (pseudo) Riemannian manifolds and phase velocity spaces. Applying the anholonomic deformation method, the field equations in Einstein gravity and various modifications are formally integrated in general forms, with generic off-diagonal metrics depending on some classes of generating and integration functions. Choosing random generating functions we can construct various classes of stochastic Einstein manifolds. We show how various types of stochastic gravitational interactions with mixed holonomic/ nonholonomic and random variables can be modelled in explicit form and study their main geometric and stochastic properties. Finally, there are analyzed the conditions when non-random classical gravitational processes transform into stochastic ones and inversely.