# Statistical Spatially Inhomogeneous Diffusion Inference

Ren, Yinuo, Lu, Yiping, Ying, Lexing, Rotskoff, Grant M.

December 2023
### Abstract

Inferring a diffusion equation from discretely-observed measurements is a statistical challenge of significant importance in a variety of fields, from single-molecule tracking in biophysical systems to modeling financial instruments. Assuming that the underlying dynamical process obeys a \$d\$-dimensional stochastic differential equation of the form \$\$\textbackslash mathrm\{d\}\textbackslash boldsymbol\{x\}\_t=\textbackslash boldsymbol\{b\}(\textbackslash boldsymbol\{x\}\_t)\textbackslash mathrm\{d\} t+\textbackslash Sigma(\textbackslash boldsymbol\{x\}\_t)\textbackslash mathrm\{d\}\textbackslash boldsymbol\{w\}\_t,\$\$ we propose neural network-based estimators of both the drift \$\textbackslash boldsymbol\{b\}\$ and the spatially-inhomogeneous diffusion tensor \$D = \textbackslash Sigma\textbackslash Sigma\^\{T\}\$ and provide statistical convergence guarantees when \$\textbackslash boldsymbol\{b\}\$ and \$D\$ are \$s\$-H\textbackslash “older continuous. Notably, our bound aligns with the minimax optimal rate \$N\^\{-\textbackslash frac\{2s\}\{2s+d\}\}\$ for nonparametric function estimation even in the presence of correlation within observational data, which necessitates careful handling when establishing fast-rate generalization bounds. Our theoretical results are bolstered by numerical experiments demonstrating accurate inference of spatially-inhomogeneous diffusion tensors.