Sparse recovery of Diffusion Dynamics: Handling High-Dimensionality in Repeated Short Trajectories

Viewed as systems of interacting particles, high-dimensional stochastic differential equations encode complex interaction structures within their drift component. We propose a novel approach to estimate this drift from independent high-frequency trajectory data observed over a short time horizon. Each trajectory is modelled as the solution of a Brownian-driven stochastic differential equation, while the number of time points within each path tends to infinity. We further assume that the drift function governing the dynamics can be expressed as a linear combination of a growing number of Lipschitz basis functions. To promote accurate recovery of the underlying dynamics under sparsity constraints, we propose a Lasso-regularised likelihood criterion. Under suitable regularity conditions, we establish convergence rates for the resulting estimator and emphasise how they depend on the dimensional parameters of the problem, in particular on the number of observed trajectories. We assess the performance of the estimator on synthetic datasets, both from an estimation and a generative perspective. Finally, we illustrate the practical relevance of the approach on a real-world climate dataset, highlighting its ability to perform variable selection.