I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$: $$ \partial_t f = {div} \left [\left( \begin{array}{rr} 1/4 & 0 \\ 0 & 1 \\ \end{array}\right) \nabla f + \left( \begin{array}{rr} 1/4 & -4 \\ 4 & 1 \\ \end{array}\right)xf \right ], $$ with $x = (x_1,x_2) \in \mathbb{R}^2,$ $t >0$ and $f=f(t,x_1,x_2).$ For this equation it known that there is sufficient regular and positive, unique solution. The problem is similar to the diffusion equation, for which, as is well known, there is a fundamental solution on the whole space. The equation can also be written as: $$ \partial_t f = \frac{1}{4} \partial_{x_1 x_1}^2f + \partial_{x_2 x_2}^2f + \left(\frac{1}{4}x_1 - 4x_2 \right) \partial_{x_1} f + \left(4x_1 + x_2 \right) \partial_{x_2} f + \frac{5}{4}f.$$ The (normalized) steady-state $f_{\infty}$, solution of the corresponding elliptic equation $$ 0 = {div} \left [\left( \begin{array}{rr} 1/4 & 0 \\ 0 & 1 \\ \end{array}\right) \nabla f + \left( \begin{array}{rr} 1/4 & -4 \\ 4 & 1 \\ \end{array}\right)xf \right ]$$ is known, it would be $$ f_{\infty}(x_1,x_2) = \frac{1}{2 \pi} {exp}\left(-\frac{1}{2}(x_1^2+x_2^2)\right).$$ Unfortunately I am not that very well versed in PDEs. So are there any results on explicit solutions of such equations on the whole space in 2D? I strongly assume the solution should be very similar to one of the heat equation.

I would be very grateful for any help!

I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$: $$ \partial_t f = {div} \left [\left( \begin{array}{rr} 1/4 & 0 \\ 0 & 1 \\ \end{array}\right) \nabla f + \left( \begin{array}{rr} 1/4 & -4 \\ 4 & 1 \\ \end{array}\right)xf \right ], $$ with $x = (x_1,x_2) \in \mathbb{R}^2,$ $t >0$ and $f=f(t,x_1,x_2).$ For this equation it is known that there is sufficient regular and positive, unique solution if we impose a positive initial state $f_0 \in L^1(\mathbb{R}^2).$ The problem is similar to the diffusion equation, for which, as is well known, there is a fundamental solution on the whole space. The equation can also be written as: $$ \partial_t f = \frac{1}{4} \partial_{x_1 x_1}^2f + \partial_{x_2 x_2}^2f + \left(\frac{1}{4}x_1 - 4x_2 \right) \partial_{x_1} f + \left(4x_1 + x_2 \right) \partial_{x_2} f + \frac{5}{4}f.$$ The (normalized) steady-state $f_{\infty}$, solution of the corresponding elliptic equation $$ 0 = {div} \left [\left( \begin{array}{rr} 1/4 & 0 \\ 0 & 1 \\ \end{array}\right) \nabla f + \left( \begin{array}{rr} 1/4 & -4 \\ 4 & 1 \\ \end{array}\right)xf \right ]$$ is known, it would be $$ f_{\infty}(x_1,x_2) = \frac{1}{2 \pi} {exp}\left(-\frac{1}{2}(x_1^2+x_2^2)\right).$$ Unfortunately I am not that very well versed in PDEs. So are there any results on explicit solutions of such equations on the whole space in 2D? I strongly assume the solution should be very similar to one of the heat equation.

I would be very grateful for any help!