Examples of renormalized SDEs

TL;DR

This paper presents two examples of stochastic processes requiring renormalization to achieve convergence in rough path topology, illustrating connections to renormalization in singular SPDEs.

Contribution

It introduces specific stochastic processes where renormalization of iterated integrals is essential for convergence, linking rough path theory with SPDE renormalization techniques.

Findings

Renormalization is necessary for convergence of certain stochastic processes.

The magnetic Brownian motion example demonstrates dominance effects in small mass limit.

The fractional Brownian motion example captures quadratic variation via stochastic area.

Abstract

We demonstrate two examples of stochastic processes whose lifts to geometric rough paths require a renormalisation procedure to obtain convergence in rough path topologies. Our first example involves a physical Brownian motion subject to a magnetic force which dominates over the friction forces in the small mass limit. Our second example involves a lead-lag process of discretised fractional Brownian motion with Hurst parameter $H \in (1/4,1/2)$, in which the stochastic area captures the quadratic variation of the process. In both examples, a renormalisation of the second iterated integral is needed to ensure convergence of the processes, and we comment on how this procedure mimics negative renormalisation arising in the study of singular SPDEs and regularity structures.