Contraction semigroups for diffusion with drift
R. Seeley · Transactions of the American Mathematical Society · 1984
Recently Dodziuk, Karp and Li, and Strichartz have given results on existence and uniqueness of contraction semigroups generated by the Laplacian Δ \Delta on a manifold M M ; earlier, Yau gave related results for L = Δ + V L = \Delta + V for a vector field V V . The present paper considers L = Δ − V − c L = \Delta - V - c , with c c a real function, and gives conditions for (a) uniqueness of semigroups on the bounded continuous functions, (b) preservation of C 0 {C_0} (functions vanishing at ∞ \infty ) by the minimal semigroup, and (c) existence and uniqueness of contraction semigroups on L p ( μ ) , 1 ⩽ p > ∞ {L^p}(\mu ),\;1 \leqslant p > \infty , for an arbitrary smooth density μ \mu on M M . The conditions concern L ρ / ρ L\rho /\rho , where ρ \rho is a smooth function, ρ → ∞ \rho \to \infty as x → ∞ x \to \infty . They variously extend, strengthen, and complement the previous results mentioned above.