The Infinitesimal Generator: Bridging SDEs and PDEs for Stochastic Processes

Students often confuse the generator

\mathcal{L}

with the Kolmogorov Forward Equation (Fokker-Planck). Remember: the generator acts on test functions

f

in the 'Backward' sense, while the Fokker-Planck equation describes the evolution of the probability density

p(x,t)

itself.

Institutional Reference: Advanced Stochastic Processes