Let $\sigma: (R_1, R_2)\in \mathbb{R}\times \mathbb{R}_+^* \mapsto \beta_0 + \beta_1 R_1 + \beta_2 \sqrt{R_2}$ be a function, where $\beta_0 > 0$, $\beta_1 < 0$ and $\beta_2 \in (0, 1)$. Let, $\lambda_1, \lambda_2 > 0$ and consider the following Fokker-Plank equation : $$\lambda_1 \partial_{R_1}(R_1\psi) - \lambda_2 \partial_{R_2}\left((\sigma^2 - R_2)\psi\right) + \frac{1}{2}\lambda_1^2 \partial_{R_1,R_1}\left(\sigma^2 \psi\right) = 0$$
Does it admit an analytical solution? If not can the solutions be expressed in terms of hypergeometric functions ?
Thanks in advance,