Multi variable Langevin equation I need to solve the following system 
$\frac{\partial f(t)}{\partial t}=a_1 f(t)+a_1 g(t)+s_1(t) \\
\frac{\partial g(t)}{\partial t}=a_3 f(t)+a_4 g(t)+s_2(t) $
with $s_i$ being a noise, delta-correlated : $<s_i(t_1)s_j(t_2)>=2B ~~\delta_{i,j}~~ \delta(t_1-t_2)$
In particular I want the mean squares $<f^2>_{t \to \infty}$ and $<g^2>_{t \to \infty}$
Could you please help to solve this, or recommend a comprehensible document/book?

So I recognise here the Langevin equation, which I only know how to solve in 1 dimension. When I try this to solve I quickly fail :
From the second equation :
$
f(t)=\frac{1}{a_3} \left( \frac{\partial g(t)}{\partial t} - a_4 g(t) - s_2(t) \right)
$
Substituting in equation 1 :
$
\frac{1}{a_3}\left( \frac{\partial^2 g(t)}{\partial t^2}-a_4 \frac{\partial g(t)}{\partial t} - \frac{\partial}{\partial t} s_2(t)  \right)= a_1 f(t) + ...
$
Which is already problematic because of the term $\frac{\partial}{\partial t} s_2(t)$: a stochastic process can't be differentiated.
This is treated for the general n-dimensional case by Zwanzig but I find the book a bit dry and poorly detailed.
Thank you
 A: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
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 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}
 \newcommand{\fermi}{\,{\rm f}}
 \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}
 \newcommand{\half}{{1 \over 2}}
 \newcommand{\ic}{{\rm i}}
 \newcommand{\iff}{\Longleftrightarrow}
 \newcommand{\imp}{\Longrightarrow}
 \newcommand{\pars}[1]{\left(\, #1 \,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\pp}{{\cal P}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,}
 \newcommand{\sech}{\,{\rm sech}}
 \newcommand{\sgn}{\,{\rm sgn}}
 \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}
 \newcommand{\ul}[1]{\underline{#1}}
 \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$
$\ds{}$
\begin{align}
\totald{{\sf F}\pars{t}}{t}={\sf W}{\sf F}\pars{t} + {\sf s}\pars{t}\,,
\qquad\qquad
\left\{\begin{array}{rcl}
{\sf F}\pars{t} & \equiv & {\fermi\pars{t} \choose {\rm g}\pars{t}}
\\[3mm]
{\sf W} & \equiv & \pars{%
\begin{array}{cc}
a_{1} & a_{1}
\\
a_{3} & a_{4}
\end{array}}
\\[3mm]
{\rm s}\pars{t} & = & {{\rm s}_{1}\pars{t} \choose {\rm s}_{2}\pars{t}}
\end{array}\right.
\end{align}

$$
\expo{-{\sf W}t}\totald{{\sf F}\pars{t}}{t}
-\expo{-{\sf W}t}{\sf W}{\sf F}\pars{t}
=\expo{-{\sf W}t}{\sf s}\pars{t}\quad\imp\quad
\totald{\bracks{\expo{-{\sf W}t}{\sf F}\pars{t}}}{t}
=\expo{-{\sf W}t}{\sf s}\pars{t}
$$

$$
\expo{-{\sf W}t}{\sf F}\pars{t}
-
\expo{-{\sf W}t_{0}}{\sf F}\pars{t_{0}}
=\int_{t_{0}}^{t}\expo{-{\sf W}\xi}{\sf s}\pars{\xi}\,\dd \xi
$$

\begin{align}
{\sf F}\pars{t}
&=\expo{{\sf W}\pars{t - t_{0}}}{\sf F}\pars{t_{0}}
+\int_{t_{0}}^{t}\expo{-{\sf W}\pars{\xi - t}}{\sf s}\pars{\xi}\,\dd \xi
\\[3mm]&=\expo{{\sf W}\pars{t - t_{0}}}{\sf F}\pars{t_{0}}
+\int_{t_{0} - t}^{0}\expo{-{\sf W}\xi}{\sf s}\pars{\xi + t}\,\dd \xi
\\[3mm]&=\expo{{\sf W}\pars{t - t_{0}}}{\sf F}\pars{t_{0}}
-\int_{-t_{0} + t}^{0}\expo{{\sf W}\xi}{\sf s}\pars{-\xi + t}\,\dd \xi
\end{align}

$$\color{#66f}{\large%
{\sf F}\pars{t}
=\expo{{\sf W}\pars{t - t_{0}}}{\sf F}\pars{t_{0}}
+\int_{0}^{t - t_{0}}\expo{{\sf W}\xi}{\sf s}\pars{t - \xi}\,\dd \xi}
$$

In order to continue you have to know some details about $\ds{{\sf W}}$. You can go ahead as the usual Langevin $\ds{1D}$ equation.

