Wenbo V. Li

Title: Fastest Rate of Convergence for Brownian Motion with Jump Boundary

Co-Author: Yuk J. Leung

Abstract: Consider a standard Brownian motion starting inside the open interval $(a,b)$. Once it hits one of the two boundary points $a$ or $b$, the process jump (restart) inside the interval $(a,b)$ according to the measure $\nu_a$ or $\nu_b$ depending on the hitting location, and starts the Brownian motion afresh. The same evolution is repeated independently each time the process reaches the boundary. The resulting Markov process is called Brownian motion with jump boundary (BMJ). The spectral gap of non-self-adjoint generator of BMJ, which describes the exponential rate of convergence to the invariant measure, is studied. In particular, we prove the so-called $2/3$-conjecture on the largest spectral gap (fastest rate of convergence) among all possible jump measures $\nu_a$ and $\nu_b$. The extremal jump measures are point measures $\mu_a^*=\delta_{a+2(b-a)/3}$ and $\mu_b^*=\delta_{b-2(b-a)/3}$. The method of proof is Fourier analytic.
  • Corrections:
  • Updated relevant references:
    Last modified Oct. 10, 2010 by Wenbo V. Li,