The Evolution of Pointwise Statistics in Hyperbolic Equations with Random Data

We consider one-dimensional hyperbolic PDEs, linear and nonlinear, with random initial data. Our focus is the {\em pointwise statistics,} i.e., the probability measure of the solution at any fixed point in space and time. For linear hyperbolic equations, the probability density function (PDF) of these statistics satisfies the same linear PDE. For nonlinear hyperbolic PDEs, we derive a linear transport equation for the cumulative distribution function (CDF) and a nonlocal linear PDE for the PDF. Both results are valid only as long as no shocks have formed, a limitation which is inherent to the problem, as demonstrated by a counterexample. For systems of linear hyperbolic equations, we introduce the multi-point statistics and derive their evolution equations. In all of the settings we consider, the resulting PDEs for the statistics are of practical significance: they enable efficient evaluation of the random dynamics, without requiring an ensemble of solutions of the underlying PDE, and their cost is not affected by the dimension of the random parameter space. Additionally, the evolution equations for the statistics lead to a priori statistical error bounds for Monte Carlo methods (in particular, Kernel Density Estimators) when applied to hyperbolic PDEs with random data.