The Langevin Diffusion in Continuous Time

Every progressive process with finite probability-time L2 energy has an Ito integral that is an adapted continuous martingale, is characterized at each deterministic time by the restricted terminal L2 completion, and satisfies the Ito isometry.

At every deterministic time, the second moment of the Ito integral equals the probability-time L2 energy accumulated by the integrand up to that time.

The expected square of an elementary stochastic integral expands to the sum of its diagonal increment terms because distinct adapted Brownian increments are orthogonal.

The second moment of the elementary Ito integral equals the expected time integral of the squared elementary integrand.

Localization permits progressive integrands whose squared time integral is finite almost surely, without requiring its expectation to be finite.

The squared L2 norm of an integrand on the product probability-time space is the expectation of its squared time integral, and admissible integrands have finite value.

An elementary adapted process is a finite sum of bounded left-endpoint measurable coefficients on half-open time intervals.

The Ito integral of an elementary process is defined by the finite sum of its adapted coefficients times Brownian increments stopped at the terminal time.

Informally, a curve is absolutely continuous when its finite metric derivative exists almost everywhere.

A functional is alpha-geodesically convex when its value along each geodesic lies below endpoint interpolation minus the alpha quadratic distance correction.

Geodesic alpha-convexity implies the first-order lower bound at the initial endpoint, with the derivative pairing and alpha times squared distance correction.

For a stationary reversible generator, the two negative-generator pairings equal each other and the integrated carre du champ.

The negative generator of a reversible Markov semigroup has a nonnegative quadratic form.

The dual optimal transport value is the supremum of the two potential integrals over feasible integrable potential pairs.

The dual value is the supremum of integral f dmu plus integral g dnu over dual-feasible potentials.

A Markov process satisfies a Poincare inequality when variance is bounded by a constant times its generator Dirichlet energy.

A Markov process satisfies an LSI when density entropy is bounded by C/2 times the Dirichlet form of the density and its logarithm.

Standard Brownian motion starts at zero, has independent centered Gaussian increments with covariance proportional to elapsed time, and has almost surely continuous paths.

A localizing sequence is an increasing stopping-time sequence whose stopped integrands have finite L2 norm and which converges almost surely to the terminal time.

A local martingale is adapted and admits increasing stopping times tending almost surely to infinity for which every centered stopped process is a martingale.

The Markov operator sends an observable to its conditional expectation after elapsed time t, given the initial state.

The 2-Wasserstein distance is the positive square root of the optimal quadratic coupling cost.

The square of W2 equals the infimum of integrated squared distance over all couplings.

A Markov semigroup records how the law or observables evolve with time.

Identity and Chapman-Kolmogorov transition-kernel laws induce the zero-time, composition, and commutation laws of Markov operators.

The infinitesimal generator is the right derivative at zero of the semigroup orbit on its convergence domain.

The right derivative of P_t f is P_t Lf, and P_t f remains in the generator domain with generator P_t Lf.

The generator is the derivative at time zero of the Markov semigroup.

A martingale is an adapted integrable process whose conditional expectation at an earlier time equals its earlier value.

A stopping time is a random time whose occurrence by time t is measurable using the information available at time t.

A Markov semigroup is reversible when every time operator is symmetric in the L2 inner product of its stationary law.

A Markov semigroup satisfies the pointwise Jensen inequality: the square of P_t f is bounded by P_t applied to the square of f.

The carre du champ is the bilinear defect between applying the generator after multiplication and multiplying after applying the generator.

The carre du champ of a Markov generator is pointwise nonnegative on the diagonal.

For the Langevin differential operator, the carre du champ is the gradient inner product, and on the diagonal it is the squared gradient norm.

The iterated carre du champ applies the generator to Gamma and subtracts the two mixed generator terms.

The Bakry-Emery curvature-dimension condition requires positive alpha and the pointwise inequality Gamma_2(f) at least alpha Gamma(f).

A differentiable scalar curve satisfying g'(t) at most c times g(t) is bounded by g(0) exp(ct) on the same finite interval.

For overdamped Langevin diffusion, Itô's formula displays the formal generator as a Laplacian minus a score-directional derivative.

A weighted integration-by-parts calculation makes the Langevin generator formally symmetric under its Gibbs weight.

The Gibbs measure is the stationary law suggested by the weighted generator identity.

Variance is bounded by an energy involving the gradient.

Entropy is controlled by a Fisher-information or Dirichlet-form term.

P2,ac consists of Euclidean probability laws with finite second moment that are absolutely continuous with respect to Lebesgue measure.

The Wasserstein geodesic between two P2,ac laws is the law of the affine interpolation of an optimally coupled endpoint pair; it is also called displacement or McCann interpolation.

The Kantorovich transport cost is the infimum of expected cost over all joint probability laws with the prescribed marginals.

The transport value expands as the infimum of the coupling lintegrals of the cost.

A coupling is a joint probability law with two prescribed marginals.

The pathwise local-square-integrability condition admits an increasing canonical sequence of stopping times approaching the horizon such that each stopped integrand has finite global L2 energy.

Each canonical energy truncation is globally square-integrable, so its Ito integral is an adapted continuous martingale with the deterministic-time restriction representation used in the localization proof.

The Ito integral of a progressive locally square-integrable integrand has a continuous version that is a local martingale.