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

  • J. Calatayud
  • J.-C. Cortés
  • M. JornetEmail author


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.


Random autonomous linear differential equation with discrete delay \(\mathrm {L}^p\) random calculus method of steps uncertainty quantification 

Mathematics Subject Classification

34F05 34K50 60H10 65C30 



This work has been supported by the Spanish Ministerio de Economía y Competitividad Grant MTM2017–89664–P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.

Compliance with Ethical Standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.


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Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Multidisciplinar Building 8G, access C, 2nd floorUniversitat Politècnica de ValènciaValenciaSpain

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