Optimal prediction and the Klein-Gordon equation
O. H. Hald
TL;DR
This work develops the method of optimal prediction for linear systems with constrained initial data and applies it to the Klein–Gordon equation. It derives an exact mean evolution and a cheaper approximate predictor, along with rigorous probabilistic error bounds that hold with high probability, and proves almost-sure convergence of the exact averages as the trial space grows. The approach is validated in the Klein–Gordon setting, using local averages and Fourier truncation to produce finite-dimensional reduced models with quantifiable convergence as dimension increases. The results extend to related linear systems and offer a principled framework for resolving long-time statistics from incomplete initial data in dispersive Hamiltonian PDEs.
Abstract
The method of optimal prediction is applied to calculate the future means of solutions to the Klein-Gordon equation. It is shown that in an appropriate probability space, the difference between the average of all solutions that satisfy certain constraints at time t=0, and the average computed by an approximate method, is small with high probability.