Solving Rough Differential Equations with the Theory of Regularity Structures

Abstract

The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were inspired by the rough path theory. We take a pedagogical approach to facilitate the understanding of this new theory. We recover results of the rough path theory with the regularity structure framework. Hence, we show how to formulate a fixed point problem in the abstract space of modelled distributions to solve the rough differential equations. We also give a proof of the existence of a rough path lift with the theory of regularity structure.

The Sewing lemma for $0 < γ\leq 1$

“…Remark 5.4. A rigorous proof of the latter fact can be found for example in [Bra19,Lemma 3.10] where the author uses wavelets, but let us briefly and informally present an alternative approach in the spirit of the calculations above. Indeed, one wishes to set I t := R (F ) (1 (0∧t,0∨t] ), which is not possible because the indicator function is not a test-function.…”

A stochastic reconstruction theorem

“…At this point, the reconstruction theorem is a purely analytic tool. This is surprising, as it is often applied in SPDEs, see for example [CFG17], [Hai14], [Bay+20], [Bra19] and many more. In all those examples, the reconstruction theorem only uses the analytic properties of the stochastic processes, neglecting their stochastic properties.…”

“…Thus, sewing can indeed be seen as the one-dimensional case of reconstruction. By a result of [Bra19], Lemma 3.10, we can reverse the time derivative and regain I(t) as "f (1 [0,t] )", which is of course only rigorous as the function z from Brault's Lemma, so the arrows in the middle can be reversed.…”

A Sobolev rough path extension theorem via regularity structures

“…We explore an approach based on Hairer's theory of regularity structures [Hai14], which goes back to [FH14], and show that every path with Sobolev regularity α ∈ (1/3, 1/2) and integrability p > 1/α can be lifted to a weakly geometric rough path possessing exactly the same Sobolev regularity. While the rough path lift of a Hölder continuous path is a known and fairly simple application of Hairer's reconstruction theorem ([Hai14, Theorem 3.10]), see [FH14,Proposition 13.23] or [Bra19], lifting a Sobolev path lies outside the current framework of regularity structures and thus requires some serious additional effort. Indeed, we need to use a Sobolev topology on the space of modelled distributions, as introduced in [HL17] and [LPT21b] (see also [HR20]) and additionally to generalize the definition of models from the originally required Hölder bounds to some more general Sobolev bounds.…”