Stability Analysis of Degenerate Einstein Model of Brownian Motion

Recent advancements in stochastic processes have uncovered a paradox associated with the Einstein model of Brownian motion of random particles, which diffuse in the media with no boundary . The classical model developed by Einstein provide diffusion coefficient which does not depend on numbers of particles(concentration) and does not degenerate. Based on this model one can predict the propagation speed of particles movement, conflicting with the second law of thermodynamics. We justify that within Einstein paradigm this issue can be resolved. For that we revisited approach proposed by Einstein, and significantly modified his ideas by introducing inverse Kolmogorov equation, with coefficient degenerating as concentration of the particle of interest vanishes. The modified model successfully resolves paradox affiliated to classical Brownian motion model by introducing a concentration-dependent diffusion matrix, establishing a finite propagation speed. Proposed model utilize but of inverse Kolmogorov stochastic parabolic equation and propose sufficient condition (Hypotheses 1.1) for degeneracy of diffusion coefficient, which guarantee finite speed of propagation inside domain of diffusion. This paper outlines the necessary conditions for this property through a counterexample, which provide infinite speed of propagation for the solution of the equation, with diffusion coefficient, which degenerate as concentration vanishes but with lower speed than in (Hypotheses 1.1). The second part focuses on the stability analysis of the solution of the degenerate Einstein model in case when boundary condition are crucial. We considered degenerate Einstein model in the boundary domain with Dirichlet boundary conditions. Our model bridge degenerate Brownian equation in the bulk of media with boundary of the domain. We with detail investigate stability of the problem with perturbed boundary Data, which vanishes with time. A functional dependence is introduced on the solution that satisfies a specific ordinary differential inequality. The investigation explores the solution's dependence on the boundary and initial data of the original problem, demonstrating asymptotic stability under various conditions. These results have practical applications in understanding stochastic processes and its dependence on the boundary Data within bounded domains.