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

Abstract

[EN] This paper deals with the damped pendulum random differential equation: (X) over double dot(t)+2 omega(0)xi(X) over dot(t) + omega X-2(0)(t) = Y(t), t is an element of [0, T], with initial conditions X(0) = X-0 and (X) over dot(0) = X-1. The forcing term Y(t) is a stochastic process and X-0 and X-1 are random variables in a common underlying complete probability space (Omega, F, P). The term X(t) is a stochastic process that solves the random differential equation in both the sample path and in the L-P 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 f(X(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 Ito type; and Y(t) can be approximated by a sequence {Y-N(t)}(N-1)(infinity) in L-2([0, T] x Omega), which occurs with Karhunen-Loeve expansions and some random power series. Finally, we provide numerical examples in which we choose specific random variables X-0 and X-1 and a specific stochastic process Y(t), and then, we find the probability density function of X(t). (C) 2018 Elsevier B.V. All rights reserved.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewers that have improved the final version of the paper.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A Statistical Mechanics and its Applications. 512:261-279. https://doi.org/10.1016/j.physa.2018.08.024S26127951