Consider a Markov process on a finite state space $S$, whose dynamic is determined by a certain infinitesimal generator $Q$ (that is a matrix in this case) and an initial distribution $m$.
1) Is there anything general that can be said on the spectrum of the matrix $-Q$?
2) Is there any bounds on the eigenvalues of $-Q$? And on their ratios?
I am interested in linear algebra results as well as more probabilistic ones, maybe using information about the structure of the state space or on the process itself.
If can be of any help, for the problem I am addressing, I usually have a quite big state space and the Q-matrix is quite sparse, but unfortunately without any symmetry or nice structure.
For discrete-time Markov chains there is an huge literature linking properties of the spectrum of the transition matrix $P$ to various mixing times, geometric properties of the state space, etc. Of course one can move from the continuous setting to the discrete one via uniformization and thus "translate" most of these results, but I was wondering the following
3) Is there any bounds on the eigenvalues developed specifically for the continuous time setting? And generally speaking, does anyone know any references for such continuous-time theory?
Best Answer
One way to approach the problem is to scale $Q$ (multiply by $1/a$) so that all entries are $< 1$. To this matrix add I and the result is a stochastic matrix. Then apply the Peron-Frobenius theorem and you get that $0$ is an eigenvalue of geometric multiplicity $1$ and all other eigenvalues have negative real part. The result is that the eigenvector whose eigenvalue is $0$ is the unique steady state if the process is irreducible