  The complex Helmholtz equation $(\Delta + k^2)u=f$ (where $k\in{\mathbb
R},u(\cdot),f(\cdot)\in{\mathbb C}$) is a mainstay of computational wave
simulation. Despite its apparent simplicity, efficient numerical methods are
challenging to design and, in some applications, regarded as an open problem.
Two sources of difficulty are the large number of degrees of freedom and the
indefiniteness of the matrices arising after discretisation. Seeking to meet
them within the novel framework of probabilistic domain decomposition, we set
out to rewrite the Helmholtz equation into a form amenable to the Feynman-Kac
formula for elliptic boundary value problems. We consider two typical
scenarios, the scattering of a plane wave and the propagation inside a cavity,
and recast them as a sequence of Poisson equations. By means of stochastic
arguments, we find a sufficient and simulatable condition for the convergence
of the iterations. Upon discretisation a necessary condition for convergence
can be derived by adding up the iterates using the harmonic series for the
matrix inverse -- we illustrate the procedure in the case of finite
differences.
  From a practical point of view, our results are ultimately of limited scope.
Nonetheless, this unexpected -- even paradoxical -- new direction of attack on
the Helmholtz equation proposed by this work offers a fresh perspective on this
classical and difficult problem. Our results show that there indeed exists a
predictable range $k<k_{max}$ in which this new ansatz works with $k_{max}$
being far below the challenging situation.
