  A free non-relativistic particle moving in two dimensions on a half-plane can
be described by self-adjoint Hamiltonians characterized by boundary conditions
imposed on the systems. The most general boundary condition is parameterized in
terms of the elements of an infinite-dimensional matrix. We construct the
Brownian functional integral for each of these self-adjoint Hamiltonians.
Non-local boundary conditions are implemented by allowing the paths striking
the boundary to jump to other locations on the boundary. Analytic continuation
in time results in the Green's functions of the Schrodinger equation satisfying
the boundary condition characterizing the self-adjoint Hamiltonian.
