The complex matrix exponential of a Hermitian matrix is unitary: $e^{iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{Q} = S$?

A matrix $A$ such that $\exp(tA)$ is (right) stochastic for all $t > 0$ should be called a "generator of a semigroup of stochastic matrices" or an "infinitesimally stochastic matrix". Clearly, since $A=\lim_{t\to0} (\exp(tA)I)/t$, (i) the sum of the elements in each row of $A$ has to be 0, and (ii) all nondiagonal elements must be nonnegative. Conversely, a matrix $A$ satisfing (i) and (ii), for large enough $n$ produces a stochastic matrix $I+A/n$, hence $(I+A/n)^n$ and $\exp(A)=\lim_{n\to\infty}(I+A/n)^n$ are also stochastic (and so is $\exp(tA)$). That said, I would have a look at the results of a Google search with "infinitesimally stochastic" (I can't do it now). (Edit: as observed, the above is a stronger condition than the one you asked for; though it's a more close analog to your example.) 


Although I've voted to close because there is a trivial answer, based on your interest in analogies between quantum and statistical physics I think the following may be of interest to you. Since a comment isn't appropriate here I've CW'd this. Let $u$ be a generic unitary matrix, so that $\sum_j u_{ij}\bar u_{kj} = \delta_{ik}$. If we set $v_{ij} := u_{ij}^2$, then it is easy to show that $v$ is a doubly stochastic matrix (though not all doubly stochastic matrices are of this form [1]). Indeed such a matrix is called a unitarystochastic transition [2] or unistochastic [1] matrix. When one starts with a unitary matrix that is the propagator representing a time evolution operator associated to some Hamiltonian acting on a finitedimensional Hilbert space, then taking the squared norms yields the associated transition matrix. [1] See appendix A of Pakonski, P. et al. “Classical 1D maps, quantum graphs and ensembles of unitary matrices”. J. Phys. A 34, 9303 (2001). [2] Marshall, A. W., and Olkin, I. Inequalities: Theory of Majorization and Its Applications. Academic Press (1979). Cited in [1] and in Tanner, G. “Unitarystochastic matrix ensembles and spectral statistics”. J. Phys. A 34, 8485 (2001). 

