Consider a family of probability measures {mu_y : y on the boundary of D}
on a bounded open domain D in R^d with smooth boundary. For any starting point x in D, we run a
a standard d-dimensional Brownian motion B(t) in R^d until it first exits D at time tau,
at which time it jumps to a point in the domain D according to the measure mu_{B(tau)} at the exit time,
and starts the Brownian motion afresh. The same evolution is repeated independently each time the process reaches the boundary.
The resulting diffusion 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.
The main analytic tool is Fourier transforms with only real zeros.
This is a joint work with Yuk Leung and Rakesh.
