Martingale decompositions and weak differential subordination in UMD Banach spaces

Stochastic integration in UMD Banach spaces

In these lectures we shall present an introduction of the theory of stochastic integration in UMD Banach spaces and some of its applications. The Hilbert space approach to stochastic partial differential equations (SPDEs) was pioneered in the 1980s by Da Prato and Zabczyk. Under suitable Lipschitz conditions, mild solutions of semilinear SPDEs in Hilbert spaces can be obtained by solving a fixe...

Weak symplectic forms and differential calculus in Banach spaces

1Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry, second ed., Chapter 2. 2Serge Lang, Differential and Riemannian Manifolds, p. 150, Theorem 8.1; Mircea Puta, Hamiltonian Mechanical Systems and Geometric Quantization, p. 12, Theorem 1.3.1. 3Andreas Kriegl and Peter W. Michor, The Convenient Setting of Global Analysis, p. 522, §48; Peter W. Michor, Some geometric ev...

A Clark-ocone Formula in Umd Banach Spaces

Let H be a separable real Hilbert space and let F = (Ft)t∈[0,T ] be the augmented filtration generated by an H-cylindrical Brownian motion (WH(t))t∈[0,T ] on a probability space (Ω,F ,P). We prove that if E is a UMD Banach space, 1 ≤ p < ∞, and F ∈ D(Ω;E) is FT -measurable, then F = E(F ) + ∫ T 0 PF(DF ) dWH , where D is the Malliavin derivative of F and PF is the projection onto the F-adapted ...

Weak Interpolation in Banach Spaces

Weak Thresholding Greedy Algorithms in Banach Spaces

We consider weak thresholding greedy algorithms with respect to Markushevich bases in general Banach spaces. We find sufficient conditions for the equivalence of boundedness and convergence of the approximants. We also show that if there is a weak thresholding algorithm for the system which gives the best n-term approximation up to a multiplicative constant, then the system is already “greedy”....