1-D Stochastic Wave Equation Driven By Poisson White Noise

The wave equation is one of the most important partial differential equations. It is widely used in Physics, Acoustics, Electrodynamics, Fluid Mechanics and so on. In history, many famous scientists devoted themselves to the research of wave equations and made a great contribution, such as d’Alembert, Euler, and so on. However, in real life, the random factors are everywhere. As a kind of deterministic equation, the wave equation can not perfectly describe the complicated circumstances. The study of the partial differential equation driven by random noise has been gradually focused on by more and more mathematicians and scholars in other related subjects.1-D stochastic wave equation driven by (pure) Poisson white noise is investigated in this paper. We consider the mild solution of1-D stochastic wave equation. By using Picard iteration, we prove the existence of the solution of the1-D stochastic wave equation. In addition, we prove the stochastic processes un in Picard iteration have some regularity results, that is to say, these variables are modified cadlag in t and continuous in x, as a result of which u also has these regularity results. At last, we prove the uniqueness of the solution by Gronwall inequality.