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,γ] by reflecting it at the endpoints, one obtains an ergodic process whose exponential rate of convergence to equilibrium is π2/2γ2. On the other hand, if Brownian motion is conditioned to remain in (0,γ) up to time t, then in the limit as t→∞ one obtains an ergodic process whose exponential rate of convergence to equilibrium is 3π2/2γ2. A recent paper [Grigorescu and Kang, J. Theoret. Probab. 15 (2002) 817-844] considered a different kind of physical constraint-when the Brownian motion reaches an endpoint, it is catapulted to the point pγ, where p∈(0,12], and then continues until it again hits an endpoint at which time it is catapulted again to pγ, etc. The resulting process-Brownian motion physically returned to the pointpγ-is ergodic and the exponential rate of convergence to equilibrium is independent of p and equals 2π2/γ2. In this paper we define a conditioning analog of the process physically returned to the point pγ and study its rate of convergence to equilibrium.