I'm going to ask a very vague question, and then give specifics for the version I particularly care about. I'm interested in answers at all levels of vagueness.
At the most vague version, I am in the following situation. I have some particular (smooth) function $A(t_0,q_0,t_1,q_1)$, where $t_0,t_1$ are real variables and $q_0,q_1$ vary over some manifold $M$, i.e. $A: \mathbb R \times M \times \mathbb R \times M \to \mathbb R$. I have another (smooth) function $Q(t_0,q_0,t_1,q_1;t) : \mathbb R \times M \times \mathbb R \times M \times \mathbb R \to M$. I am interested in (smooth) solutions $F(t_0,q_0,t_1,q_1)$ to the "composition" formula: $$ A(t_0,q_0,t_1,q_1)\,F(t_0,q_0,t,Q(t_0,q_0,t_1,q_1;t))\,F(t,Q(t_0,q_0,t_1,q_1;t),t_1,q_1) = F(t_0,q_0,t_1,q_1)$$ I know one such (nonzero) solution explicitly. I am hoping that, presumably with slightly more data (some sort of "initial data"), I can conclude that my solution is uniquely determined. So my question is:
What extra data for $F$, and what restrictions on $A$, are needed for a functional equation of the above type to have a unique solution?
For those interested, I can give you more precise information. I have some "physics" encoded by a function $S(t_0,q_0,t_1,q_1)$, which I am thinking of as a "Hamilton-Jacobi function". This function defines $Q$ via: $$ \left. \frac{\partial}{\partial q} \bigl[ S(t_0,q_0,t,q) + S(t,q,t_1,q_1) \bigr] \right]_{q = Q(t_0,q_0,t_1,q_1;t)} = 0 $$ and satisfies: $$ S(t_0,q_0,t,q) + S(t,q,t_1,q_1) \bigr|_{q = Q(t_0,q_0,t_1,q_1;t)} = S(t_0,q_0,t_1,q_1)$$ The function $A$ is given by $$ \left| \det \left[ \frac{\partial^2}{\partial q^2} \bigl[ S(t_0,q_0,t,q) + S(t,q,t_1,q_1) \bigr] \right]_{q = Q(t_0,q_0,t_1,q_1;t)} \right|^{-1/2} $$ Then the function $F$ is: $$ F(t_0,q_0,t_1,q_1) = \left| \det \frac{\partial^2}{\partial q_0\partial q_1} \bigl[ S(t_0,q_0,t_1,q_1) \bigr] \right|^{1/2}$$ which as defined makes sense only on $\mathbb R^n$. Actually, $A$ makes sense as a volume form, and $F$ as a half density in each variable, and then everything transforms correctly under changes of coordinates.
Anyway, I would really like to conclude that this $F$ is the only solution, but I don't know if I can.
$a + f_0 + f_1 = f_{01}$, to make it even more clear that I do not expect there to be a unique solution --- multiply the original$F(t_0,q_0,t_1,q_1)$by$e^{\lambda(t_1 - t_0)}$for any $\lambda$, for example. But I do expect that there's a "small" family, like is the case for differential equations? $\endgroup$