Suppose we have a Ito diffusion process on $\mathbb{R}^d$ given by $$ dx_t = F_0(x_t,\alpha)\, dt + \sum_{i=1}^n F_i(x_t)\,dW_t^i$$ where the drift vector field depends smoothly on a parameter $\alpha\in\mathbb{R}$. Assume that the Hormander condition is satisfied for each $\alpha$ so that we have smooth Markov transition density functions which we could denote $p_t(x_0,x_1;\alpha)$ and are obtained by solving the corresponding Fokker-Planck PDE.
Are there any results which allow us to conclude that these transition density functions vary smoothly with $\alpha$?
I know I'm not being very precise about what I mean by smoothness here but I'm just looking for some results in this direction. I cannot find anything.