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.     

New bounds for the distance Ramsey number

In this paper we study the distance Ramsey number $R_D(s,t,d)$. The distance Ramsey number $R_D(s,t,d)$ is the minimum number $n$ such that for any graph $G$ on $n$ vertices, either $G$ contains an induced $s$-vertex subgraph isomorphic to a distance graph in ${\mathbb{R}}^d$ or $\stackrel{?}{G}$ contains an induced $t$-vertex subgraph isomorphic to the distance graph in ${\mathbb{R}}^d$. We obtain the upper and lower bounds on $R_D(s,s,d)$, which are similar to the bounds for the classical Ramsey number $R(?\frac{s}{[d/2]}?,?\frac{s}{[d/2]}?)$.