Liouville's equations for random systems

• Published in: Stochastic analysis and applications Vol. 40; no. 6; pp. 1026 - 1047
• Main Author: Jornet, Marc
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02.11.2022; Taylor & Francis Ltd
• Subjects: Fields (mathematics); Liouville's equation; Mathematical analysis; ordinary and fractional differential equation; Partial differential equations; Probability density function; Probability density functions; random partial; Stochastic processes
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1080/07362994.2021.1980015

Given a random system, a Liouville's equation is an exact partial differential equation that describes the evolution of the probability density function of the solution. In this article, we derive Liouville's equations for the first-order homogeneous semilinear random partial differential equation. This is done for all finite-dimensional distributions of the random field solution, starting with dimension one, then dimension two, and finally generalizing to any dimension. Several examples, including the linear advection equation with random coefficients, are treated. As a corollary, we deduce Liouville's equations for path-wise stochastic integrals and nonlinear random ordinary differential equations.