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Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as

$$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$

where $X_0,X_1,...,X_k$ are vector fields on $M$ and $k<n$.

A main topic of interest are the ergodic properties of the diffusion process on $M$ whose infinitesimal generator is given by $\mathcal{A}$.

Suppose that we know that there is a Riemannian metric $g$ on $M$ such that $X_i=\text{grad}\,\psi_i$ for some smooth functions $\psi_i$ for all $i\geq 1$ and a possibly different Riemannian metric $g_0$ such that $X_0=-\text{grad}_0\,\psi_0$. Does this help us in studying the associated process under some assumptions on $\psi_i$ (e.g. geodesic convexity)?

I'm interested in whether this specific question has been studied and if yes, I'd be interested in any known results that specifically exploit the gradient structure of the vector fields.

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