Consider an iterative algorithm with incremental updates \begin{align} x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}], \end{align} where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$, $\alpha_t>0$ is the stepsize, $h \colon \mathcal{H} \rightarrow \mathcal{H}$ is a Lipschitz continuous mapping, and $\{ M_{t+1} \}$ is a Martingale difference sequence.

In classical stochastic approximation theory (e.g.,[1]), if $\mathcal{H}$ is the $n$-dimensional Euclidean space $\mathbb{R}^n$ equipped with the Euclidean norm, it is well known that $\{ x_t\}_{t\geq 0}$ converges almost surely to the stable equilibria of an ordinary differential equation (ODE) \begin{align} \dot{x} = h(x). \end{align}

I was wondering if the theory in classical ODE approximation theory can be readily extended to RKHS by replacing the Euclidean norm in $\mathbb{R}^n$ by the RKHS norm $\| \cdot \|_{\mathcal{H} } $ in $\mathcal{H}$. Any pointer to relevant works of literature would be appreciated.

[1]. Stochastic Approximation: A Dynamical Systems Viewpoint. The basic theory for stochastic approximation in $\mathbb{R}^n$ is in Chapter 2.