For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is C (and C on every curve)...

In this paper we prove the differentiability of Lipschitz maps X → V , where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direct...

We study a BSDE with random terminal time that appears in the modeling of counterparty risk in finance. We proceed by reduction of the original BSDE into a simpler BSDE posed with respect to a smaller filtration and a changed probability measure. This is done under a relaxation of the classical immersion hypothesis, stated in terms of the changed probability measure, of which we characterize th...

Last decade, at rst the variational problems in Sobolev spaces, later the Radon-Nikodym problem in vector integration, and some other problems leads us to the new function characteristics, both scalar and convex compact, which are closely connected to a certain decomposition of the space under consideration. That decomposition is generated by the Banach subspaces spanned by the all absolutely c...

Erratum to " The Clark-Ocone formula for vector valued random variables in abstract Wiener space " , Jour. In this paper we considered the extension of the Clark-Ocone formula for a random variable defined on an abstract Wiener space (W, H, µ) and taking values in a Banach space (denoted there either B or Y). The main result appears in Theorem 3.4. Unfortunately, as first pointed out to us by J...

We first establish the following Lemma which relates to the results in [1]. Lemma 1. Consider random variables X ∈ R and Y ∈ R. Let f Y |X be the Radon-Nikodym derivative of the probability measure P θ Y |X with respect to arbitrary measures QY provided that P θ Y |X QY . θ ∈ R is a parameter. f θ Y is the Radon-Nikodym derivative of probability measure P θ Y with respect to QY provided that P ...

Semilinear multivalued equations are considered, in separable Banach spaces with the Radon-Nikodym property. An effective criterion for the existence of solutions to the associated Floquet boundary value problem is showed. Its proof is obtained combining a continuation principle with a Liapunov-like technique and a Scorza-Dragoni type theorem. A strictly localized transversality condition is as...

We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting bi-Lipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximati...

Anomalous emanation of radon (222Rn) was observed preceding large earthquakes and is considered to be linked to preseismic electromagnetic phenomena (e.g. great changes of atmospheric electric field and ionospheric disturbances). Here we analyze atmospheric radon concentration and estimate changes of electrical conditions in atmosphere due to preseismic radon anomaly. The increase of radon eman...