Convergence Rate Estimates for Degenerate Diffusions with Multiplicative Noise via (weak) Hypocoercivity Methods

This thesis concerns itself with the long-term behavior of generalized Langevin dynamics with multiplicative noise, i.e. the solutions to a class of two-component stochastic differential equations in \( \mathbb{R}^{d_1}\times\mathbb{R}^{d_2} \) subject to outer influence induced by potentials \( \Phi \) and \( \Psi \), where the stochastic term is only present in the second component, on which it is dependent. In particular, convergence to an equilibrium defined by an invariant initial distribution \( \mu \) is shown for weak solutions to the generalized Langevin equation obtained via generalized Dirichlet forms, and the convergence rate is estimated by applying hypocoercivity methods relying on weak or classical Poincaré inequalities. As a prerequisite, the space of compactly supported smooth functions is proven to be a domain of essential m-dissipativity for the associated Kolmogorov backward operator on \(L^2(\mu)\). In the second part of the thesis, similar Langevin dynamics are considered, however defined on a product of infinite-dimensional separable Hilbert spaces. The set of finitely based smooth bounded functions is shown to be a domain of essential m-dissipativity for the corresponding Kolmogorov operator \( L \) on \( L^2(\mu) \) for a Gaussian measure \( \mu \), by applying the previous finite-dimensional result to appropriate restrictions of \( L \). Under further bounding conditions on the diffusion coefficient relative to the covariance operators of \( \mu \), hypocoercivity of the generated semigroup is proved, as well as the existence of an associated weakly continuous Markov process which analytically weakly provides a weak solution to the considered Langevin equation