Given a two-dimensional Ito stochastic equations
\begin{align}
\mathrm{d} x &= \mu_1(x,y) \mathrm{d} t + \sigma_{11}(x,y) \mathrm{d} W_1 + \sigma_{12}(x,y) \mathrm{d} W_2, \\
\mathrm{d} y &= \mu_2(x,y) \mathrm{d} t + \sigma_{21}(x,y) \mathrm{d} W_1 + \sigma_{22}(x,y) \mathrm{d} W_2,
\end{align}
would it be possible to derive two Fokker-Planck equations for marginal probability density functions $p(x)$ and $p(y)$, instead of the joint probability $p(x,y)$?
Thanking you!
It's possible to derive two equations, that describe how the marginal probability distributions evolve in time; these need not be Fokker-Planck equations, however. If as the equations imply, the noise is multiplicative, they won't be. If the sigmai,j don't depend on x(t) and y(t) explicitly, the noise isn't multiplicative.
But, anyway, so what? The two equations provided describe the change of variables from the noise fields W1,2 to the fields x(t) and y(t).
The joint probability distribution can be readily obtained and, by integrating over x(t) or y(t) it's possible to obtain the marginal distributions.
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