2020/02/11 by Marc Nualart, Nualart, Marc
Economics, Econometrics and Finance · Engineering · Mathematics · #35A08 #35D30 #35L05 #35Q91 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2002.04249
openalex publication_date 2020/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work presents results on solutions of the one-dimensional damped wave equation, also called telegrapher's equation, when the initial conditions are general distributions, not only functions. We make a complete deduction of its fundamental solutions, both for positive and negative times. To obtain them we use only self-similarity arguments and distributional calculus, making no use of Fourier or Laplace transforms. We next use these fundamental solutions to prove both the existence and the uniqueness of solutions to the distributional initial value problem. As applications we recover the semigroup property for initial data in classical function spaces and also the probability distribution function for a certain financial model of evolution of prices.