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We analyze infinite-dimensional non-linear degenerate stochastic differentialequations with multiplicative noise. First, essential m-dissipativity of theirassociated Kolmogorov backward generators on $L^2(\mu^{\Phi})$ defined onsmooth finitely based functions is established. Here $\Phi$ is an appropriatepotential and $\mu^{\Phi}$ is the invariant measure with density $e^{-\Phi}$w.r.t. an infinite-dimensional non-degenerate Gaussian measure. Second, we useresolvent methods to construct corresponding right processes with infinitelifetime, solving the martingale problem for the Kolmogorov backwardgenerators. They provide weak solutions, with weakly continuous paths, to thenon-linear degenerate stochastic partial differential equations. Moreover, weidentify the transition semigroup of such a process with the stronglycontinuous contraction semigroup $(T_t)_{t\geq 0}$ generated by thecorresponding Kolmogorov backwards generator. Afterwards, we apply a refinementof the abstract Hilbert space hypocoercivity method, developed by Dolbeaut,Mouhot and Schmeiser, to derive hypocoercivity of $(T_t)_{t\geq 0}$. I.e. wetake domain issues into account and use the formulation in the Kolmogorovbackwards setting worked out by Grothaus and Stilgenbauer. The method enablesus to explicitly compute the constants determining the exponential convergencerate to equilibrium of $(T_t)_{t\geq 0}$. The identification between$(T_t)_{t\geq 0}$ and the transition semigroup of the process yieldsexponential ergodicity of the latter. Finally, we apply our results to secondorder in time stochastic reaction-diffusion and Cahn-Hilliard type equationswith multiplicative noise. More generally, we analyze corresponding couplingsof infinite-dimensional deterministic with stochastic differential equations.

Hypocoercivity for infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise