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Kolmogorov Operator
Differential operator appearing in the Fokker-Planck equation, consisting of a drift term (first order) and a diffusion term (second order) governing the dynamics of the process.
Forward Probability Density
Evolution of the probability distribution in forward time, described by the Fokker-Planck equation, progressively transforming data into Gaussian noise.
Continuous Diffusion Process
Continuous-time limit of discrete diffusion models, where noise addition becomes a differentiable stochastic process governed by the Fokker-Planck equation.
Drift Term
Deterministic part of the Fokker-Planck equation representing the average tendency of the process to evolve in a particular direction of the state space.
Diffusion Matrix
Symmetric positive definite matrix in the Fokker-Planck equation quantifying the intensity and correlation of noise added in different directions of the state space.
Score Potential
Gradient of the log of the probability density, closely related to the Fokker-Planck equation and used to guide the reverse generation process in diffusion models.
Kolmogorov Forward Equation
Alternative name for the Fokker-Planck equation, describing the forward evolution of the probability density as opposed to the backward equation for expectations.
Hörmander's Theorem
Mathematical condition guaranteeing the hypoellipticity of the Fokker-Planck operator, ensuring that solutions become infinitely differentiable even with degenerate diffusion.
Stationary State
Probability distribution invariant under the flow of the Fokker-Planck equation, often a centered Gaussian towards which the diffusion process converges asymptotically.
Probability Flux
Divergence term in the Fokker-Planck equation representing the net flow of probability through state space boundaries, crucial for mass conservation.
Langevin Process
Stochastic differential equation whose probability density satisfies the Fokker-Planck equation, providing a microscopic description of the diffusion process.
Smoluchowski Equation
Special case of the Fokker-Planck equation for overdamped diffusion where inertial terms are neglected, commonly used in molecular modeling.
Reflective Boundary Condition
Boundary condition for the Fokker-Planck equation where the probability flux is zero at the boundaries, ensuring probability conservation in a bounded domain.
Doob Decomposition
Decomposition of a martingale into a quadratic variation part and a pure martingale part, used to analyze solutions of the Fokker-Planck equation.
Gibbs Invariant Measure
Specific stationary distribution of the Fokker-Planck equation taking the form of a Gibbs measure, fundamental for applications in generative learning.
First Passage Time
Analysis based on the Fokker-Planck equation to determine the average time required for the process to reach a certain state or region of space.