First assume $\zeta_i=0$, which conditions guarantee that $w^*$ is a stable fixed point? If it's not a stable fixed point for noise-free case, then the process will not be ergodic with additive noise.
Consider
\begin{equation} \Sigma=\left( \begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix} \right) \end{equation}
One condition on step size $\eta=\alpha$ which guarantees convergence
$$\frac{2}{\alpha}<\text{Tr}(\Sigma)+2\|H\|$$
Which gives that $$\alpha<\frac{2}{7}$$
However, this condition is too strict. The necessary and sufficient condition is that
$$\alpha<\frac {4} {9 + \sqrt {17}}$$
This note derives necessary and sufficient condition above for a general Gaussian. I haven't seen anyone else derives this, please correct me if it occurs in literature.
Background for non-Gaussian case:
It can be shown that in the case of 1 dimension and deterministic $x$, the following condition on $\alpha$ is necessary and sufficient for convergence \begin{equation}\label{supersimple} \alpha x^4 < 2 x^2 \end{equation}
Since we have $h=x^2$ for Hessian $h$, this reduces to the well known bound on convergent learning rate: $\alpha < 2/h$
In the case of stochastic x, the following is necessary and sufficient
\begin{equation}\label{eq:0} \alpha E[x^4] < 2 E[x^2] \end{equation}
For the case of $x$ being distributed as standard normal, this gives $2/(3h)$ for the largest learning rate, three times smaller than what's allowed in deterministic case
For the case of $d$ dimensions, the following is a sufficient condition, with $\prec$ indicating Loewner order\footnote{assumption A.6 in Bach paper
\begin{equation}\label{eq:1} \alpha E[xx'xx'] \prec E[xx'] \end{equation}
The right-hand side can be tightened to \begin{equation}\label{eq:1x} \alpha E[xx'xx'] \prec 2 E[xx'] \end{equation}
Defossez, Bach showed that the following optimization over symmetric matrices gives sufficient condition for convergence, and conjectured it to also be necessary (Lemma 1 of \citep{Defossez2015-fr})
\begin{equation}\label{eq:2} \frac{1}{\alpha} < \sup_{A\in \mathcal{S}(R^d)} \frac{E[(x'Ax)^2]}{2 E[x'A^2 x] } \end{equation}
We can show this to be equivalent to the following positive semi-definite constraint \begin{equation}\label{eq:3} \alpha E[xx' \otimes xx'] \prec E[xx'\otimes I] + E[I\otimes xx'] \end{equation}
Most recently, Jain generalized last Eq to batch sizes beyond 1 and formally showed it to be a necessary condition for convergence.