  Imposing non-integrable constraints on Ricci flows of (pseudo) Riemannian
metrics we model mutual transforms to, and from, non-Riemannian spaces. Such
evolutions of geometries and physical theories can be modelled for nonholonomic
manifolds and vector/ tangent bundles enabled with fundamental geometric
objects determining Lagrange-Finsler and/or Einstein spaces. Prescribing
corresponding classes of generating functions, we construct different types of
stochastic, fractional, nonholonomic etc models of evolution for nonlinear
dynamical systems, exact solutions of Einstein equations and/or
Lagrange-Finsler configurations. The main result of this paper consists in a
proof of existence of unique and positive solutions of nonlinear diffusion
equations which can be related to stochastic solutions in gravity and Ricci
flow theory. This allows us to formulate stochastic modifications of Perelman's
functionals and prove the main theorems for stochastic Ricci flow evolution. We
show that nonholonomic Ricci flow diffusion can be with self-organized critical
behavior, for gravitational and Lagrange-Finsler systems, and that a
statistical/ thermodynamic analogy to stochastic geometric evolution can be
formulated.
