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As mentioned in the comments, you are interested in solving a boundary value problem and not an initial value problem. Some of the most used methods are: Finite Difference Method; Finite Element Method. For this particular problem the Ritz method might be a good choice, but in general you are better using the Finite Element Method.


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Here is a commonly used alternative: Let's assume that your probability density $f(x,y,v_x,v_y)$ lives in a domain $\Omega$ that is bounded and that you can subdivide into "cells" $\Omega_i$. In one of the comments, you mention that $\Omega$ is simply a 4-dimensional box, and then the $\Omega_i$ could simply be subdivision into a regular mesh ...


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Simply redoing my earlier comment with a bit more space ... Depending on the particulars, a simple acceptance-rejection method might be a good place to start. Suppose you know $f$ never gets bigger than $f_{\text{max}}$. Generate $X$ uniformly between $x_{\text{min}}$ and $x_{\text{max}}$, then similarly $Y$ from between $y_{\text{min}}$ and $y_{\text{max}}$ ...


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