The proposed algorithm seeks to provide a novel data-driven framework for the discovery of stochastic differential equations (SDEs) by application of the Weak-formulation to stochastic SINDy. This Weak formulation of the algorithm provides a noise-robust methodology that avoids traditional noisy derivative computation using finite differences. An additional novelty is the adoption of spatial Gaussian test functions in place of temporal test functions, wherein, the use of the kernel weight $K_j(X_{t_n})$ guarantees unbiasedness in expectation and prevents the structural regression bias that is otherwise pertinent temporal test functions. The proposed framework converts the SDE identification problem into two SINDy based linear sparse identification problems. We validate the algorithm on three SDEs, for which we recover all active non-linear terms with coefficient errors below 4\%, stationary-density total-variation distances below 0.01, and autocorrelation functions that reproduce true relaxation timescales across all three benchmarks faithfully.

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