Symbolic Weak-form Recovery of 2-D Stochastic Generators
Recovering two-dimensional Ito generators from trajectory data is difficult because drift increments have low signal-to-noise, bivariate weak designs can be ill-conditioned, and unconstrained tensor estimates need not be positive semidefinite. We study WG-SINDy estimator combining covariance-shaped spatial kernels, a ridge-stabilized local-polynomial projection, adaptive-LASSO/STLSQ selection, one in-sample per-component feasible diagonal GLS pass, and a PSD projection--Cholesky read-out with mild isotropic shrinkage. The released estimator uses a data-dependent full-cloud smoother and one in-sample per-component feasible diagonal GLS pass; accordingly, we do not claim exact finite-sample martingale cancellation or a feasible-GLS efficiency theorem for the reported implementation. We evaluate the estimator on 29 synthetic two-dimensional systems: 19 meet their declared per-system recovery contracts, eight are retained as named limits, and two remain scoped reviews. Across the 19 PASS rows, the median central-grid drift metric is 0.204 and the median tensor error is 0.0397. Among the six systems with a finite, non-degenerate off-diagonal target, the median $a_{12}$ cosine is 0.997. Positive-semidefinite validity is imposed by construction. These results are synthetic, in-sample sampled-region diagnostics and do not establish universal or real-data recovery.