A Probabilistic Approach to the First Dirichlet Eigenvalue on Non-compact Riemannian Manifold

By using diffusion process with absorbing boundary, some lower bounds are obtained for the first Dirichlet eigenvalue of operator .+.h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean's estimate for .h =0. Moreover, the first Dirichlet eigen value for elliptic operators on R^d and the first mixed eigenvalue are also studied. Some examples show that our estimates can be sharp even for .h.0.