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I've got the following relation: For any $t_m \in (0, t_f)$,

$$I(k | t_0,t_f,x_0) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I(k_m | t_0, t_m, x_0) I(k-k_m|t_m,t_f,x_m) dk_m dx_m $$

I want to solve for $I$. I have reason to believe that a very broad class of solutions is possible, so the more general form, the better.

I have no idea where to begin. A solution would be appreciated, but absent that, advice on how to attack this problem would also be nice.


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