Suppose one has the an Ito process of the form:
$$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$
The infinitesimal generator $LV(x)$ is defined by:
$$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) \right]-V(x)}{t}$$
One can show that $$LV(x) = \sum_{i}b_i(x)\frac{\partial V}{\partial x_i}(x) + \frac{1}{2}\sum_{i,j}(\sigma(x)\sigma(x)^T)_{ij}\frac{\partial^2V}{\partial x_i \partial x_j}(x)$$
I'm wondering if there's an equivalent one infinitesimal generator for $$dX_t = b(X_t)dt + \sigma(X_t)d\eta(t)$$
$$d\eta(t) = \lambda\eta(t) dt + \alpha dW(t)$$
This is a stochastic process where the perturbations are from an Ornstein-Uhlenbeck process instead of a Brownian Motion. Wiki gives the infinitesimal generator of an Ornstein-Uhlenbeck process to be:
$$LV(x) = -\lambda x V'(x) + \frac{\alpha^2}{2}V''(x)$$
But I don't know if there's a way to use that fact to combine it with $LV(x)$ for the Ito process to get the infinitesimal generator for the stochastic process perturbed by an Ornstein-Uhlenbeck process