TP2=Bruhat

It is shown that the Bruhat partial order on permutations is equivalent to a certain natural partial order induced by ${\mathrm{TP}}_2$ matrices and that a positive matrix is ${\mathrm{TP}}_2$ if (and only if) it satisfies certain inequalities induced by Bruhat order.     

Ergodic properties of generalized Ornstein–Uhlenbeck processes

We investigate ergodic properties of the solution of the SDE $d{V}_t=V_{t?}d{U}_t+d{L}_t$, where $(U,L)$ is a bivariate Lévy process. This class of processes includes the generalized Ornstein–Uhlenbeck processes. We provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster–Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.