Search

We compute the Wassertein-1 (or Kolmogorov-Rubinstein) distance between a random walk in $R^d$ and the Brownian motion. The proof is based on a new estimate of the Lipschitz modulus of the solution of the Stein's equation. As an application, we can evaluate the rate of convergence towards the local time at 0 of the Brownian motion.

We define in this work a notion of Young differential inclusion $$ dz_t \in F(z_t)dx_t, $$ for an $\alpha$-Holder control $x$, with $\alpha>1/2$, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, $\gamma$-H\"older continuous set-valued map on the interval $[0,1]$ has a selection with finite $p$-variation, for $p>1/\gamma$. We also give a notion of solution to the rough differential inclusion $$ dz_t \in F(z_t)dt + G(z_t)d{\bf X}_t, $$ for an $\alpha$-Holder rough path $\bf X$ with $\alpha\in \left(\frac{1}{3},\frac{1}{2}\right]$, a set-valued map $F$ and a single-valued one form $G$. Then, we prove the existence of a solution to the inclusion when $F$ is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values., Comment: v.3: 22 pages. Final version

The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this process has stationary increments and satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar property is global are characterized. Finally, we prove an integration by parts formula on the path group space and deduce the existence of a density.