The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function

This paper deals with the damped pendulum random differential equation: X¨(t)+2ω0ξX˙(t) + ω 2 0 X(t) = Y(t), t ∈ [0, T ], with initial conditions X(0) = X0 and X˙(0) = X1. The forcing term Y(t) is a stochastic process and X0 and X1 are random variables in a common underlying complete probability space (Ω, F, P). The term X(t) is a stochastic process that solves the random differential equation in both the sample path and in the Lp senses. To understand the probabilistic behavior of X(t), we need its joint finite-dimensional distributions. We establish mild conditions under which X(t) is an absolutely continuous random variable, for each t, and we find its probability density function fX(t)(x). Thus, we obtain the first finite-dimensional distributions. In practice, we deal with two types of forcing term: Y(t) is a Gaussian process, which occurs with the damped pendulum stochastic differential equation of Itô type; and Y(t) can be approximated by a sequence {YN (t)} ∞ N=1 in L2 ([0, T ] × Ω), which occurs with Karhunen–Loève expansions and some random power series. Finally, we provide numerical examples in which we choose specific random variables X0 and X1 and a specific stochastic process Y(t), and then, we find the probability density function of X(t)