Spectral gap and rate of convergence to equilibrium for a class of conditioned Brownian motions

If a Brownian motion is physically constrained to the interval [0, y] by reflecting it at the endpoints, one obtains an ergodic process whose exponential rate of convergence to equilibrium is pi(2)/2y(2). On the other hand, if Brownian motion is conditioned to remain in (0, y) tip to time t, then in the limit as t -> infinity one obtains an ergodic process whose exponential rate of convergence to equilibrium is 3 pi(2)/2y(2). A recent paper [Grigorescu and Kang, J. Theoret. Probab. 15 (2002) 817-844] considered a different kind of physical coil straint-when the Brownian motion reaches an endpoint, it is catapulted to the point py, where p is an element of (0, 1/2], and then continues until it again hits an endpoint at which time it is catapulted again to py, etc. The resulting process-Brownian motion physically returned to the point py-is ergodic and the exponential rate of convergence to equilibrium is independent of p and equals 2 pi(2)/y(2). In this paper we define a conditioning analog of the process physically returned to the point py and study its rate of convergence to equilibrium. (c) 2005 Elsevier B.V. All rights reserved.