  PDDSparse is a new hybrid parallelisation scheme for solving large-scale
elliptic boundary value problems on supercomputers, which can be described as a
Feynman-Kac formula for domain decomposition. At its core lies a stochastic
linear, sparse system for the solutions on the interfaces, whose entries are
generated via Monte Carlo simulations. Assuming small statistical errors, we
show that the random system matrix ${\tilde G}(\omega)$ is near a nonsingular
M-matrix $G$, i.e. ${\tilde G}(\omega)+E=G$ where $||E||/||G||$ is small. Using
nonstandard arguments, we bound $||G^{-1}||$ and the condition number of $G$,
showing that both of them grow moderately with the degrees of freedom of the
discretisation. Moreover, the truncated Neumann series of $G^{-1}$ -- which is
straightforward to calculate -- is the basis for an excellent preconditioner
for ${\tilde G}(\omega)$. These findings are supported by numerical evidence.
