This question is regarding the convolution theorem. So far, I have only used it as a tool to solve some recurrence relations. A nice example that I am familiar with is the renewal formula for finding first passage probabilities $-$ consider a continuous time Markov process on a discrete state space $\mathcal{S}$. We denote by $P(C,t|C_0)$ (where $C,C_0\in\mathcal{S}$) the probability density that the process is in state $C$ at time $t$, given that initially it was in state $C_0$. Then, we have $$P(C,t|C_0) = \int_0^t dt'~F(C,t'|C_0)P(C,t-t'|C)$$
where $F(C,t|C_0)$ denotes the probability density that, starting from $C_0$, the first visit to configuration $C$ happens at time $t$. We can define Laplace transforms of $P$ and $F$, and using convolution theorem, we get $$F(C,s|C_0)=\frac{P(C,s|C_0)}{P(C,s|C)}.$$ where $s$ is the Laplace variable.
Thinking more along these lines, consider the following equation $$P(C,t|C_0) = \sum_{C'\in\mathcal{S}} P(C',t^*|C_0)P(C,t-t^*|C')$$ where $t^*$ is some intermediate time between $0$ and $t$. Could the above equation be classified as a convolution in some abstract configuration space? If yes, could a convolution theorem like result be used to simplify the equation (bring it into a product form in the abstract space)? I understand that if we were considering a $1D$ lattice, then going from $C_0=x_0$ to $C'=x'$ would be equivalent to a displacement of $x'-x_0$. And similarly going from $C'$ to $C=x$ is equivalent to covering up the rest of the displacement $x-x'$ needed to finally reach $x$. This allows us to view the problem as a conventional convolution in space. But what if the state space isn't ordered like a lattice, and instead has an arbitrary graph structure?
I would appreciate any comments/suggestions that you may have regarding the question.