This thesis is concerned with the treatment of problems, arising from various areas of application of Mathematics --- Biology, Fluid Mechanics, Mathematical Physics, Stochastic Dynamics, … --- via computer-assisted proofs. In particular, we are interested in the application of fundamental ideas in the study of stochastic differential equations and their associated partial differential equations to develop tools in Numerical Analysis and solve new problems with the aid of the computer.

First, taking inspiration from the well-established Functional Analysis of weighted Sobolev spaces associated with gradient diffusion processes, we propose a novel but elementary Numerical Analysis of some differential equations which are naturally well-posed on these topologies. In doing so, we propose an elegant solution to the problem of the construction of Sobolev orthogonal polynomials and their use in the resolution of differential equations. Furthermore, we establish a direct link between nonlinear Painlevé recurrence equations and the compactness of embeddings of semi-classically weighted Sobolev spaces. These notions can be applied to both the numerical and rigorous resolution of differential equations via computer-assisted methods: these include Schrödinger-type equations, problems arising from stochastic dynamics and the rigorous computation of (forward) self-similar profiles of semilinear parabolic equations.

Second, we propose a powerful method to give rigorous, tight and explicit bounds on ergodic averages of stochastic flows based on their associated Poisson equation. Our approach is general in the sense that it applies under weak hypoellipticity conditions and outside of perturbative regimes. This allows us to provide a new technique for attacking the fundamental problem of determining the dynamic qualities of a stochastic system via the sign of its Lyapunov exponent. Our method is applied to prove the chaotic nature of both new and classical models, well beyond the prior state-of-the-art.