A Renormalization Group Approach to Connect Discrete- and Continuous-Time Descriptions of Gaussian Processes

Identifying correct discretization schemes of continuous stochastic processes is an important task, which is needed to infer model parameters from experimental observations. Motivated by the observation that consistent discretizations of continuous models should be invariant under temporal coarse graining, we derive an explicit Renormalization Group transformation on linear stochastic time series and show that the Renormalization Group fixed points correspond to discretizations of naturally occuring physical dynamics. Our fixed point analysis explains why standard embedding procedures do not allow for reconstructing hidden Markov dynamics, and why the Euler-Maruyama scheme applied to underdamped Langevin equations works for numerical integration, but not to derive the likelihood of a partially observed process in the context of parametric inference.