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On stochastic differential equations with random delay

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 $ε$. 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 $ε$ goes to zero is studied in detail on the example of a first-order linear differential equation.