WEAK COLLOCATION REGRESSION FOR INFERRING STOCHASTIC DYNAMICS WITH LÉVY NOISE

The paper’s weak-form derivation multiplies the FP equation (2) by a Gaussian kernel, integrates by parts (drift once, diffusion twice), and uses the fractional Laplacian identity (−(−Δ)^{α/2} = ∂^α/∂x^α in 1D) to move the fractional operator from p to the test function, yielding (5). This is exactly the model’s Part (a). The paper then supplies a closed form for the fractional derivative of a Gaussian via the confluent hypergeometric 1F1 (equation (7)) and rewrites the weak identity in expectation form (8)–(9), discretizing time with the trapezoidal LMM (10) and stacking block systems (17)–(18). The model’s Parts (b)–(c) mirror these steps, including Gaussian tests, basis expansions for m and G, Monte Carlo estimators from snapshots, and a trapezoid/midpoint-type temporal differencing to build an overdetermined linear system. Minor differences are cosmetic (the model allows basis expansion also for |ξ_i|^α, whereas the paper’s main independent-noise case treats ξ_i as constants and expands it when appropriate in later examples). Overall, both arguments align closely and are methodologically indistinguishable in substance (equations (2), (3)–(5), (7), (8)–(10), (14)–(18) in the paper). Citations: equation (2) and definitions ; weak form (5) ; fractional operator definitions (3)–(4) , ; fractional derivative of Gaussian (7) ; MC and time-discretization (8)–(10) , ; linear system construction (14)–(18) , ; fractional integration-by-parts via Fourier (Appendix B) .