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?

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

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

$B$ 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.

Thanks having taken bother to read this post.