Contraction semigroups for diffusion with drift

Abstract:

Recently Dodziuk, Karp and Li, and Strichartz have given results on existence and uniqueness of contraction semigroups generated by the Laplacian $\Delta$ on a manifold $M$; earlier, Yau gave related results for $L = \Delta + V$ for a vector field $V$. The present paper considers $L = \Delta - V - c$, with $c$ a real function, and gives conditions for (a) uniqueness of semigroups on the bounded continuous functions, (b) preservation of ${C_0}$ (functions vanishing at $\infty$) by the minimal semigroup, and (c) existence and uniqueness of contraction semigroups on ${L^p}(\mu ),\;1 \leqslant p < \infty$, for an arbitrary smooth density $\mu$ on $M$. The conditions concern $L\rho /\rho$, where $\rho$ is a smooth function, $\rho \to \infty$ as $x \to \infty$. They variously extend, strengthen, and complement the previous results mentioned above.