Danyu Yang

G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

The present article is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the viewpoint of rough path theory. As the starting point, by using techniques in rough path theory, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by G-Brownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of developing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this article is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. In particular, we establish the generating nonlinear heat equation for such G-Brownian motion on a Riemannian manifold. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest.

The partial sum process of orthogonal expansions as geometric rough process with Fourier series as an example—An improvement of Menshov–Rademacher theorem

Abstract The partial sum process of orthogonal expansion ∑ n ⩾ 0 c n u n is a geometric 2-rough process, for any orthonormal system { u n } n ⩾ 0 in L 2 and any sequence of numbers { c n } satisfying ∑ n ⩾ 0 ( log 2 ( n + 1 ) ) 2 | c n | 2 ∞ . Since being a geometric 2-rough process implies the existence of a limit function up to a null set, our theorem could be treated as an improvement of Menshov–Rademacher theorem. For Fourier series, the condition can be strengthened to ∑ n ⩾ 0 log 2 ( n + 1 ) | c n | 2 ∞ , which is equivalent to ∫ − π π ∫ − π π | f ( u ) − f ( v ) | 2 | sin u − v 2 | d u d v ∞ (with f the limit function).

Séminaire de Probabilités XLVI

This volume provides a broad insight on current, high level researches in probability theory.