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

TL;DR

This paper introduces a Renormalization Group method for Gaussian processes that clarifies discretization invariance, explains limitations of delay embedding, and proposes effective Markovian discretizations for complex stochastic systems.

Contribution

It develops an RG framework for Gaussian time series, linking fixed points to discretizations of SDEs, and offers insights into model inference and reconstruction.

Findings

RG fixed points correspond to linear SDE discretizations

Standard delay embedding fails for partially observed systems

Proposes effective Markovian discretization for underdamped processes

Abstract

Discretization of continuous stochastic processes is needed to numerically simulate them or to infer models from experimental time series. However, depending on the nature of the process, the same discretization scheme, if not accurate enough, may perform very differently for the two tasks. Exact discretizations, which work equally well at any scale, are characterized by the property of invariance under coarse-graining. Motivated by this observation, we build an explicit Renormalization Group approach for Gaussian time series generated by auto-regressive models. We show that the RG fixed points correspond to discretizations of linear SDEs, and only come in the form of first order Markov processes or non-Markovian ones. This fact provides an alternative explanation of why standard delay-vector embedding procedures fail in reconstructing partially observed noise-driven systems. We also suggest a possible effective Markovian discretization for the inference of partially observed underdamped equilibrium processes based on the exploitation of the Einstein relation.