COMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSION PROCESSES AND SCHR¨ODINGER OPERATORS ON AN INTERVAL

The spectral gaps and thus the exponential rates of convergence to equilibrium are compared for ergodic one-dimensional diffusions on an interval. One of the results may be thought of as the diffusion analogue of a recent result for the spectral gap of one-dimensional Schr¨odinger operators. The similarities and differences between spectral gap results for diffusions and for Schr¨odinger operators are also discussed.