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!



Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space
The solution to the linear PDE is just the PDF corresponding to $\mathcal{N}(\mu\_t, \Sigma\_t)$ where $$
\mu\_t = \exp(A t) x \;, \quad \text{and} \quad \Sigma\_t = 2 \int\_0^t \exp(A (t-s) \begin{bmatrix} 1/4 & 0 \\ 0 & 1 \end{bmatrix} \exp(A^T (t-s)) ds
$$ where $A= -\begin{bmatrix} 1/4 & -4 \\ 4 & 1 \end{bmatrix}$.


This solution can be easily obtained by using the correspondence between SDE/PDE. In particular, the corresponding SDE is $$
d X\_t = A X\_t + \sqrt{2} \begin{bmatrix} 1/2 & 0 \\ 0 & 1 \end{bmatrix} d B\_t
$$ where $B\_t$ is a standard $2$-dimensional Brownian motion. This SDE is a non-symmetric Ornstein-Uhlenbeck process due to the presence of a skew term in the drift, but since the SDE is linear, it can be explicitly solved to obtain the transition distribution $\mathcal{N}(\mu\_t, \Sigma\_t)$. It is also straightforward to verify that $\mathcal{N}(\mu\_t, \Sigma\_t) \to \mathcal{N}(0,I\_{2\times2})$ as $t \to \infty$.


Standard references for this are:


*Metafune, G.; Pallara, D.; Priola, E.*, [**Spectrum of Ornstein-Uhlenbeck operators in $L ^{p}$ spaces with respect to invariant measures**](http://dx.doi.org/10.1006/jfan.2002.3978), J. Funct. Anal. 196, No. 1, 40-60 (2002). [ZBL1027.47036](https://zbmath.org/?q=an:1027.47036).


*Pavliotis, Grigorios A.*, [**Stochastic processes and applications. Diffusion processes, the Fokker-Planck and Langevin equations**](http://dx.doi.org/10.1007/978-1-4939-1323-7), Texts in Applied Mathematics 60. New York, NY: Springer (ISBN 978-1-4939-1322-0/hbk; 978-1-4939-1323-7/ebook). xiii, 339 p. (2014). [ZBL1318.60003](https://zbmath.org/?q=an:1318.60003).


