Maximal dissipativity for degenerate Kolmogorov operators (Q2433276)

The author considers the second-order differential operator \(N_0 \phi := \frac {1}{2} {\text{trace}} (\sigma\sigma^* \nabla^2\phi) + \langle b,\nabla\phi\rangle\) defined on \(C_b^2(\mathbb R^d)\), where \(b = b(x) \in \mathbb R^d\) and \(\sigma=\sigma(x)\) is a \(d\times d\)-matrix-valued function. Let \(\mu\) be an infinitesimally invariant measure, i.e., it holds that \(\int N_0\phi \,d\mu = 0\); sufficient conditions for its existence are due to \textit{R.\,Z.\,Khas'minski} [Theor.\ Probab.\ Appl.~5, 179--196 (1969; Zbl 0106.12001)]. The principal aim of the paper under review is to find conditions on \(b\), \(\sigma\) so that \(N_0\), resp.\ its closure \(N\) is dissipative, resp.\ maximal dissipative as an operator on \(L^p(\mathbb R^d,\mu)\). The main result states that \(N\) is maximal dissipative, generates an \(L^p\)-semigroup and that \(C_b^2\) is an operator core if the vector field \(b\) and the matrix function \(\sigma\) are of class \(C^2\) and satisfy (together with their derivatives) certain polynomial growth restrictions. Note that \(\sigma\sigma^*\) may still be degenerate. An essential ingredient in the proofs is the stochastic interpretation of \(N\) as infinitesimal generator of the semigroup, resp.\ stochastic process, which arises as solution of the following stochastic differential equation driven by Brownian motion: \[ dX_t^x = b(X_t^x)\,dt + \sigma(X_t^x)\,dB_t,\quad X_0^x = x. \]