Say that we have a system evolving over discrete timesteps.

The quantity we are interested is X and is given by a distribution $P_X$. This distribution is evolving temporally, and we have a nonlinear equation $F$ that gives us this evolution.

$P_{\Delta X}(x;n) = F\big(~P_X(x;n)~,~P_X(x;n-1)~\big)~~~,~~~n \in \mathbb{N}$

where: $~~~X(x;n+1)=X(x;n)+\Delta X(x;n)$

We also know the initial condition $P_X(x;0) = \delta(x-k)$.

Unfortunately $X$ and $\Delta X$ are correlated.

My question is, what is the appropriate framework to investigate the evolution of $P_X$?


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