$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

Abstract

In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay $$\tau >0$$ : $$x'(t)=ax(t)+bx(t-\tau )$$ , $$t\ge 0$$ , with initial condition $$x(t)=g(t)$$ , $$-\tau \le t\le 0$$ . The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using $$\mathrm {L}^p$$ -calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an $$\mathrm {L}^p$$ -solution too. An analysis of $$\mathrm {L}^p$$ -convergence when the delay $$\tau $$ tends to 0 is also performed in detail.