There is a Langevin equation: $$ \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -a\frac{\mathrm{d}x}{\mathrm{d}t} + \xi(t), $$ where $\xi(t)$ is a Gaussian white noise. I want to know how to prove that $x$ satisfies a Gaussian distribution without using the Fokker-Planck equation.
Additionally, is there a general method for solving the distribution of a variable in any linear stochastic differential equation?
Thanks for any help.
$$\begin{align} &\iff e^{at}\left(x''(t) +ax'(t) \right)=e^{at}\xi(t)\\ &\iff \left(e^{at}\cdot x'(t)\right)'=e^{at}\xi(t)\\ &\iff e^{at}\cdot x'(t)-x'(0)=\int_0^te^{as}\xi(s)ds\\ &\iff x'(t)=e^{-at}\left(x'(0)+\int_0^te^{as}\xi(s)ds\right)\\ &\iff x(t) = x(0) +\int_0^t \left(e^{-au}\left(x'(0)+\int_0^ue^{as}\xi(s)ds\right)\right)du \\ &= x(0) +x'(0)\frac{1-e^{-at}}{a}+ \iint_ {u\le s\le u\le t}e^{a(s-u)}\xi(s)dsdu\\ &=x(0) +x'(0)\frac{1-e^{-at}}{a}+ \int_0^t \left(\int_s^te^{a(s-u)}\xi(s)du\right)ds\\ &=x(0) +x'(0)\frac{1-e^{-at}}{a}+ \int_0^t \frac{1-e^{-a(t-s)}}{a}\xi(s)ds\\ \end{align}$$
The general method does not exist. Perhaps you could try to guess a particular solution, denoted by $x_1(t)$, after that, you solve the linear differential equation $x''(t)+ax'(t) = 0$ to get $x_2(t)$. And the solution of the linear stochastic equation will be $x_1(t)+x_2(t)$. However, gussing $x_1(t)$ is not easy in practice.