The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Doléans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenou...

A classical theorem of S. Bochner states that a function f : R → C is the Fourier transform of a finite Borel measure if and only if f is positive definite. In 1938, I. Schoenberg found a beautiful converse to Bochner’s theorem. We present a non-technical derivation of of Schoenberg’s theorem that relies chiefly on the law of large numbers of classical probability theory.

For an arbitrary Dirac-harmonic map (φ, ψ) between compact oriented Riemannian surfaces, we shall study the zeros of |ψ|. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of |ψ| and the genus of M and N . On the basis, we could clarify all of nontrivial Dirac-harmonic maps from S to S.

On a Riemannian manifold (M, g) endowed with flow, we study in this paper the curvature term Bochner–Weitzenböck formula of basic Laplacian. We prove that splits into two parts; first part depends on operator M and second can be expressed terms O’Neill tensor flow. After getting lower bound for depending each these parts, establish an eigenvalue estimate then discuss limiting case latter that, ...

In this paper, we proved the Normal Scalar Curvature Conjecture and the Böttcher–Wenzel Conjecture. We developed a new Bochner formula and it becomes useful with the first conjecture we proved. Using the results, we established some new pinching theorems for minimal submanifolds in spheres. Published by Elsevier Inc.

We give simple and unified proofs of weak holomorhpic Morse inequalities on complete manifolds, $q$-convex pseudoconvex domains, weakly $1$-complete manifolds covering manifolds. This paper is essentially based the asymptotic Bergman kernel functions Bochner-Kodaira-Nakano formulas.