On fractional smoothness and $L_p$-approximation on the Gaussian space

TL;DR

This paper explores the relationship between fractional smoothness and regularity of functionals on Gaussian spaces, applying these insights to improve $L_p$-approximation of stochastic integrals in Brownian motion.

Contribution

It introduces new connections between fractional smoothness in Gaussian Besov spaces and heat extension regularity, with applications to stochastic integral approximation.

Findings

Established links between fractional smoothness and heat extension regularity.

Applied theoretical results to $L_p$-approximation of stochastic integrals.

Enhanced understanding of approximation errors in Gaussian space settings.

Abstract

We consider Gaussian Besov spaces obtained by real interpolation and Riemann-Liouville operators of fractional integration on the Gaussian space and relate the fractional smoothness of a functional to the regularity of its heat extension. The results are applied to study an approximation problem in $L_p$ for $2\le p<\infty$ for stochastic integrals with respect to the $d$-dimensional (geometric) Brownian motion.