Consider a family of probability measures $\{\mu_y : y \in
\partial D\}$ on a bounded open domain $D\subset R^d$ 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.