White noise in Lie group

Question

The matter is in the title.

Is there a means to define the white noise process in Lie group. A basic definition link text

Question: Can we replace $\mathbb{R}^n$ by a Lie group?

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In fact, I would like for studying the stochastic differential equation

\begin{align} X_t^{x} &=x + \sum_{k=1}^m \int_0^t \mathcal{X}_k (X_s^{x}) dB^k_s & ( *) \end{align}

For $H > \frac{1}{2}$,

1. $\mathcal{X}_k$'s are $C^{\infty}$-bounded vector fields on $\mathbb{R}^m$ which I would like to remplace it by a lie group $G$.

2. $B$ is a $m$-dimensional fractional Brownian motion taking its values ​​in $G$.

In Hida space $\mathcal{S}^*$, equation (*) becomes $$ d X_t^{x} =\sum_{k=1}^m \mathcal{X}_k (X_t^{x})\diamondsuit B^k_t dt \qquad (**) $$ whith $X_0^x=x$. In order to exploit the underlying Lie algebra should be established white noise on the Lie group.

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Thanks having taken bother to read this post.

Answer 1

We can define fractional Brownian motion in Lie groups when $H>1/4$ by using the Lyons' rough paths theory

Self-similarity and fractional Brownian motion on Lie groups