Diffusion in Lagrangian Grid-Based Predictors

This paper focuses on state prediction for stochastic dynamic models with linear dynamics, emphasizing a recently proposed efficient and robust Lagrangian approach for solving the Chapman-Kolmogorov equation. In contrast to the standard Eulerian perspective, the Lagrangian method separates the solution into two sequential steps: advection and diffusion. Advection is handled by moving a carefully designed grid, while diffusion is addressed using the convolution theorem. This approach significantly reduces computational complexity while preserving the same accuracy. In this paper, we propose formulating diffusion as a continuous-time process, leading to a partial differential equation (PDE). Various methods for solving this PDE are presented and compared within a unified framework, along with evaluations of their properties and example implementations. We demonstrate that the continuous formulation can yield substantial reductions in computational complexity with only marginal loss in accuracy.