Stochastic Analysis

Definition of a Gaussian probability space, reducibility — Hermite polynomials on ℝ — Hermite polynomials on ℝ^N — Numerical model of a Gaussian probability space — Intrinsic geometry on a Gaussian probability space — The Ornstein-Uhlenbeck semigroup, chaos decomposition — The Cameron-Martin representation — Abstract Wiener space.

Continuity of the Cameron-Martin representation over L^∞−0 — The space D ₁ ^∞ of differentiable vectors of the Cameron-Martin representation — Gradient operator — Generalized polynomials — Cauchy operator; Krée-Meyer inequality for the gradient — The spaces D ₁ ^p — Gradient of Hilbert-valued functionals — Recursive approach to higher derivatives and to Krée-Meyer inequalities of higher order — The space D_∞ of smooth functionals, its approximation by C^∞ cylindrical functionals — Divergence as the adjoint of the gradient — Divergence of smooth vector fields — Shigekawa acyclicity of the complex of differential forms — Appendix: Proof of the L ^p inequality for the Hilbert transform.

Divergence of differentiable flow — Divergence as the adjoint operator of a derivation — Non-degenerate maps and their covariance matrices — Lifting up vector fields through a non-degenerate map — Pushing down divergences — Hölder regularity under D ₂ ^∞−0 hypothesis — Smoothness under D_∞ hypothesis — Lifting up and pushing down through a non-degenerate D_∞ map — Inverse image of a distribution — Absolute continuity of scalar functionals under D ₁ ¹ hypothesis — Appendix: Computation of derivatives by means of divergence operators — Law of a weakly non degenerated map.

Capacities on a numerical model — Tigtness of capacities — Quasicontinuous functions — Tchebycheff inequalities — Redefinitions — Positive generalized functionals, their representation by a Borel measure — Equilibrium potentials — Continuity of capacities — Capacitability of Borel stes — Equilibrium measures — Measures of finite energy — Charge and capacity — Slim sets are null sets for all measures of finite energy — Invariance of capacities under a change of numerical model, precise Gaussian probability space — Quasi-sure analysis on an abstract Wiener space.

Regular disintegration corresponding to a non-degenerate functional; principle of descent — Partial function associated to a splitting — Finite codimensional projection of a slim set is slim — Implicit function theorem in finite codimension — Differential forms of degree p — Non-normalized conditional expectation of differential forms — Currents; currents along a fibre of a non-degenerate map — Submanifolds; their defining functions — Gauss map of a submanifold, its approximate continuity — Hausdorff area measure of a submanifold — Functional characterization of the area measure — Coarea formula along a non-degenerate map — The operator δ; its commutation with the non-normalized conditional expectation — Oriented submanifold — Stokes’ theorem.

Gaussian probability space over an L² space — Construction of Seg (L²(Г)) by Wiener-Itô multiple integrals — Fock space, naturality of Wiener — Itô multiple integrals — Stroock-Taylor formula — Nualart-Pardoux construction of the Skorokhod stochastic integral — The Gaveau-Trauber theorem identifying the Skorokhod integral with a divergence — Energy estimate for the Skorokhod integral — Existence of traces and of Stratonovich integrals.

The probability space of Brownian motion and its filtration — Energy identity for stochastic integral of an adapted process — Itô’s stochastic integral of an adapted process — Chaos expansion in terms of iterated Itô stochastic integrals — Itô representation of a martingale by a stochastic integral — Clark-Bismut-Ocone representation of a martingale in D ₁ ^p — Itô calculus on semi-martingales — Covariance under C^∞-maps of the Stratonovich representation of semi-martingales — Change of variables formula — Appendix: Estimates for Brownian martingales.

Stratonovich SDE — Intrinsic stochastic integral — Itô SDE, uniqueness of the Cauchy problem — The Stroock-Varadhan piecewise linear approximation — Algebraic analysis on the group of C^∞-diffeomorphisms: the exponential map, the adjoint action — Reduced variation of the Stroock-Varadhan approximation sequence of a smooth SDE, its a priori bound; Bootstrap and proof of the limit theorem — Cauchy problem for SDE with Lipschitz coefficients — Critique of our approach.

Variation of an ODE: linearized ODE determining the Jacobian ma­trix — The control map, its Jacobian — Jacobian of a stochastic flow — Higher derivatives of a stochastic flow — Itô functionals, their differentiability — Malliavin matrix of an Itô functional — Reduced variation of an Itô functional — Reduced vari­ation and regularity in the backward variables — Bismut’s identity — Hörmander’s hypoellipticity theorem.

The Ornstein-Uhlenbeck (OU) flow on a finite-dimensional Gaussian space — Lifting the OU flow to an abstract Wiener space, its axiomatic definition — Representation of the OU flow on the probability space of Brownian motion — Itô calculus of variations along the OU flow — Equilibrium processes and regularity of law.

Stochastic calculus of variation on a Lie group: Reduced variation and adjoint representation — Path groups: left infinitesimal quasi-invariance of Wiener measure — Path group on a compact Lie group — Orthonormal frame bundles over a Riemannian manifold: Levi-Civita parallelism, structure equations — Lifting to the frame bundle of the Riemannian diffusion: the stochastic parallel displacement — The horizontal stochastic flow, determination of its Jacobian on a Riemannian diffusion — Invariance of Brownian motion by orthogonal transformations — Tangent processes — Stochastic analysis on the path space of a Riemannian manifold: Twisted differential, Clark-Bismut-Ocone representation — Harnack estimates via the reduced variation — Loop spaces.