A Rough Functional Breuer-Major Theorem

TL;DR

This paper extends the functional Breuer-Major theorem to rough path spaces, employing advanced Malliavin calculus, combinatorial analysis, and a bespoke Slutsky's lemma to establish convergence results for rough path functionals.

Contribution

It introduces a novel extension of the Breuer-Major theorem to rough paths, combining Malliavin calculus and combinatorial techniques for convergence analysis.

Findings

Convergence of finite-dimensional distributions to a Stratonovich Brownian rough path.

Explicit computation of moments showing convergence to a corrected Brownian rough path.

Development of a new approach overcoming the lack of hypercontractivity in rough path analysis.

Abstract

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with Meyer's inequality and a Kolmogorov-type criterion for the r-variation of cadlag rough paths, due to Chevyrev et al. (2022). Since martingale techniques do not apply, we obtain the convergence of the finite-dimensional distributions through a bespoke version of Slutsky's lemma: First, we overcome the lack of hypercontractivity by an iterated integration-by-parts scheme which reduces the remaining analysis to finite Wiener chaos; crucially, this argument relies on Malliavin differentiability of the nonlinearity but not on chaos decay and, as a consequence, encompasses the centred absolute value function. Second, in the spirit of the law of large numbers, we show that the diagonal of the second-order process converges to an explicit symmetric correction term. Finally, we compute all the moments of the remaining process and, through a fine combinatorial analysis, show that they converge to those of the Stratonovich Brownian rough path perturbed by an antisymmetric area correction, as computed by a suitable amendment of Fawcett's theorem. All of these steps benefit from a major combinatorial reduction that is implied by the original argument of Breuer and Major (1983).