Multidimensional Fokker-Planck equation with exact solution

Question

In my research I consider the following diffusion processes obeying an Ito stochastic differential equation: $$ d X_t = - \nabla \Psi(X_t) dt + \sqrt{2 \beta^{-1}} d W(t) \quad (1) $$ Here $\Psi(x) : \mathbb{R}^n \rightarrow \mathbb{R}_+$ - potential, $\beta$ is constant and $W(t)$ is standard n-dimensional Wiener process. Actually, the marginal densities $\rho_t$ of the process $X_t$ in form $(1)$ governed by Fokker-Planck equation: $$ \frac{\partial \rho_t(x)}{\partial t} = \text{div}\left(\nabla \Psi(x) \rho_t\right) + \beta^{-1} \Delta \rho_t \quad (2) $$ I have the following question: Is there a multidimensional nonlinear diffusion process which satisfies Fokker-Planck equation $(2)$ and has close-form solution. I know about Ornstein–Uhlenbeck processes (with quadratic potential) but they model linear diffusion. Thank you in advance for your ansvers! This will help me a lot.