On a class of hypoelliptic operators with unbounded coefficients in \(\mathbb R^N\) (Q1016655)

The authors consider operators in \(\mathbb{R}^N\) of Ornstein-Uhlenbeck type \[ Au(x)= \sum^p_{i,j=1} q_{ji}(x) D_{ji}u+ \sum^N_{i,j=1} b_{ji} x_j D_i u+\sum^N_{j=1} F_j(x) D_j u \] with \(p< N\), satisfying the Hörmander's hypoellipticity condition. For the coefficients \(q_{ji}\) a quadratic growth is allowed, whereas \(F_j\) has at most linear growth. Under general assumptions, the authors prove the existence of a semigroup \(T(t)\) of bounded linear operators, associated to \(A\), with applications to Hölder estimates for the related elliptic and parabolic problems. For a stochastic approach to similar equations we address to \textit{E. Priela} [J. Evol. Equ. 6, No. 4, 577--600 (2006; Zbl 1116.35075)].