  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.
