  We prove existence of optimal control for the deterministic and stochastic
shallow lake problem without any restrictions on the parameter space and we
establish a generalization of the Arrhenius Law in the case of noise-dependent
potentials, which naturally arise in control theory problems. We also prove a
result about convergence of the derivatives in the viscosity approximation of
the value function and use this result to derive the Arrhenius Law for the
shallow lake problem.
