For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n\_m]$. Write $J \equiv (j\_1,\dots,j\_N) \in \prod\_m [n\_m]$ with $j\_m \in [n\_m]$. 


The composite Markov generator corresponding to running each of the $N$ processes independently has off-diagonal entries given by $Q^\otimes\_{JJ'}=\lim\_{\Delta \downarrow 0} \mathbb{P}(J \overset{\Delta}{\rightarrow} J')/\Delta$ for $J \ne J'$, where the probability of a transition from $J$ to $J'$ in a time interval of duration $\Delta$ is indicated. In this limit a.s. at most one of the component processes can execute a transition, so that the only nonzero off-diagonal terms are of the form $Q^\otimes\_{JJ'} = Q^{(M)}\_{j\_M j'\_M}$ with $j'\_m = j\_m$ for $m \ne M$. 



> 
> What can be said about the spectral
>  gap of $Q^\otimes$ or the mixing time of the underlying process?
> 
> 
> 


Of particular interest is the case $Q^{(m)} \equiv Q$.


I've looked in the most obvious place (Levin, Peres, and Wilmer) and in some less obvious places, and I haven't seen this anywhere (perhaps I'm making a silly oversight). In the literature "product chains" usually mean something rather different than the discrete-time analogue of the above, and in general I expect that any terminology is overloaded. With that in mind, specific references would be most helpful.



Spectral gap of a product of Markov processes
The spectral gap is just the smallest of the spectral gaps of the component chains. The matrix $Q$ can be simply written as the sum of $Q^{(1)}\otimes I\otimes\ldots\otimes I$, $I\otimes Q^{(2)}\otimes I\otimes\cdots\otimes I$ etc.

If the matrix $Q^{(i)}$ has eigenvalues $(\lambda^{(i)}\_j)\_{j=1}^{n\_i}$, then the matrix $Q$ has eigenvalues $\lambda^{(1)}\_{i\_1}+\lambda^{(2)}\_{i\_2}+\ldots+\lambda^{(N)}\_{i\_N}$ (with eigenvectors $e^{(1)}\_{i\_1}\otimes \cdots \otimes e^{(N)}\_{i\_N}$).




