Hypoellipticity and the Mori-Zwanzig formulation of stochastic differential equations

We develop a thorough analysis of the Mori-Zwanzig (MZ) formulation for stochastic dynamical systems driven by multiplicative white noise. To this end, we first derive a new type of MZ equation, which we call effective Mori-Zwanzig (EMZ) equation, that governs the temporal dynamics of noise-averaged observables. Such dynamics is generated by a Kolmogorov operator obtained by averaging It\^{o}'s representation of the stochastic Liouvillian of the system. Building upon recent work on hypoelliptic operators, we prove that the generator of the EMZ orthogonal dynamics has a spectrum that lies within cusp-shaped region of the complex plane. This allows to rigorously prove that the EMZ memory kernel and fluctuation terms converge exponentially fast in time to an unique equilibrium state. We apply the new theoretical results to the stochastic dynamics of an interacting particle system widely studied in molecular dynamics, and show that such equilibrium state admits an explicit representation.