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...

This paper is devoted to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both themapping and the constraint set are perturbed by different parameters. By using the properties of recession cones, several equivalent characterizations are given for the set-valued vector equilibrium problems to ha...

We give a comparison inequality that allows one to estimate the tail probabilities of sums of independent Banach space valued random variables in terms of those of independent identically distributed random variables. More precisely, let X1, . . . , Xn be independent Banach-valued random variables. Let I be a random variable independent of X1, . . . , Xn and uniformly distributed over {1, . . ....

In this paper we prove a central limit theorem for Borel measurable nonseparably valued random elements in the case of Banach space valued fuzzy random variables.

Motivated by multi-task machine learning with Banach spaces, we propose the notion of vectorvalued reproducing kernel Banach spaces (RKBS). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RKBS. The theory is then applied to multi-task machine learning. Especially, the...