Nonholonomic Relativistic Diffusion and Exact Solutions for Stochastic Einstein Spaces

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.