Consider Markov chain $\{X\_t\}\_{t\in N}\subseteq R^{n\times n}$ defined by $X\_{t} = X\_0 G\_1 \dots G\_t$ where $G\_i$'s are iid Gaussian matrices $G\_1,\dots,G\_t\sim N(0,1/n)^{n\times n}$, and $X\_0$ is some deterministic matrix with full rank fixed scale, $\|X\_0\|\_F^2=n$ and $\operatorname{rank}(X\_0)=n$. Is this chain ergodic? Put in other words, assuming that initial matrices $X\_0$ and $Y\_0$ share the some "nice" properties (eg. similar scale and full rank), can we find a coupling(\*) of two copies of this chain $\{X\_t\}\_{t\in N}$ and $\{Y\_t\}\_{t\in N}$ such that $P(X\_t\neq Y\_t) \le \alpha^t$ for some $\alpha<1$?


Intuitively this coupling should be plausible: Since Gaussian matrices $G\_i$'s are rotation invariant, we can couple the $X\_t$ and $Y\_t$ by optimising for rotation. But I come short of making these intuitions any more rigorous. I've looked up literature on random matrices but can't find anything relevant.


(\*) More elaborately, can we design $\{(G\_t,W\_t)\}\_{t\in N}$ where both $G\_1,\dots, $ and $W\_1,\dots$ are independent, both jointly are coupled to lead to mixing of the two chains?



Is this Markov chain of Gaussian matrix products $G_1 G_2 \dots G_m$ ergodic?
You are dealing here with the products of invertible matrices, and the resulting Markov chain is known as a **random walk** on the corresponding group $GL(n,\mathbb R)$. A qualitative asymptotic "boundary" theory of such products was created by Furstenberg in the early 60's (see his 1963 papers "A Poisson formula for semi-simple Lie groups" and "Noncommuting random products"). He mostly talks about $SL(n,\mathbb R)$ and more general semi-simple Lie groups, but it doesn't make much difference.


I will return to Furstenberg's theory in a moment, but let me first say a couple of words about the "coupling" conditions you mention, in the "rawest" form. When talking about a general Markov chain one should in principle distinguish its mixing and ergodicity. **Mixing** is equivalent to the triviality of the tail $\sigma$-algebra of the chain or to the asymptotic independence of the time $n$ distributions of initial states:
$$
\| (\delta\_x - \delta\_y) P^n \| \to 0 \qquad \forall x,y \;,
$$
where $\|\cdot\|$ is the total variation, and $P$ is the transition operator of the Markov chain. In a similar way, **ergodicity** is equivalent to the triviality of the shift invariant $\sigma$-algebra in the path space (i.e., to the absence of non-constant bounded harmonic functions), or to the independence of the Cesaro averages of time $n$ distributions of initial states:
$$
\frac1n \left\| \sum\_{k=1}^n (\delta\_x - \delta\_y) P^k) \right\| \to 0 \qquad \forall x,y \;.
$$


For the random walks that you consider mixing and ergodicity are actually equivalent. It follows from so-called **0-2 laws** that also provide further information about the above differences.


Now, one of the consequences of Furstenbrg's theory is that random walks on matrix groups can only be ergodic ($\equiv$ mixing) in very special cases. Gaussian random walks on the linear group are not among them. In particular, in your setup
$$
\lim\_n \| (\delta\_x - \delta\_y) P^n \| \neq 0
$$
for any two initial matrices $x,y\in GL(n,\mathbb R)$ unless $x$ and $y$ are proportional.


