Hypoellipticity and the Mori–Zwanzig formulation of stochastic differential equations

We develop a thorough mathematical analysis of the effective Mori-Zwanzig equation governing the dynamics of noise-averaged observables in stochastic differential equations driven by multiplicative Gaussian white noise. Such dynamics is generated by a Kolmogorov operator obtained by averaging the Ito 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 us 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 Langevin dynamics of an interacting particle system widely studied in molecular dynamics, and show that such equilibrium state admits an explicit representation.

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