  This paper is concerned with solutions to a one dimensional linear diffusion
equation and their relation to some problems in stochastic control theory. A
stochastic variational formula is obtained for the logarithm of the solution to
the diffusion equation, with terminal data which is the characteristic function
of a set. In this case the terminal data for the control problem is singular,
and hence standard theory does not apply. The variational formula is used to
prove convergence in the zero noise limit of the cost function for the
stochastic control problem and its first derivatives, to the corresponding
quantities for a classical control problem.
