  We consider stochastic dynamical systems defined by differential equations
with a uniform random time delay. The latter equations are shown to be
equivalent to deterministic higher-order differential equations: for an $n$-th
order equation with random delay, the corresponding deterministic equation has
order $n+1$. We analyze various examples of dynamical systems of this kind, and
find a number of unusual behaviors. For instance, for the harmonic oscillator
with random delay, the energy grows as $\exp((3/2)\,t^{2/3})$ in reduced units.
We then investigate the effect of introducing a discrete time step $\epsilon$.
At variance with the continuous situation, the discrete random recursion
relations thus obtained have intrinsic fluctuations. The crossover between the
fluctuating discrete problem and the deterministic continuous one as $\epsilon$
goes to zero is studied in detail on the example of a first-order linear
differential equation.
