arXiv research

Classifies invariant measures on specific character varieties.

Dynamic risk measures follow law invariance principles over time.

The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …

New concept of partial law invariance connects decision theory and financial risk management.

The paper studies invariant measures for specific actions in algebraic groups.

We consider the family of harmonic measures on a lamination $\mathcal{L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\simeq Γ_L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\hat{\mathc…

Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.

Researchers develop a method to infer reference measures from observed functionals.

New measure defined on surface strata, invariant under scaling.

Let S be a non-exceptional oriented surface of finite type. We classify all Radon measures on the space of measured geodesic laminations for S which are invariant under the mapping class group.

It is shown that any transverse invariant measure of a foliated space can be considered as a measure on the ambient space.

Study measures invariant under horospherical subgroups for finitely generated Kleinian groups.

One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe h…

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures correspondin…

Curvature measures uniquely determined by invariance under embeddings.

Study invariant measures on measured laminations for subgroups of mapping class group.

In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property, which was introduced in [17], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on…

In this paper we analyze a dynamic recursive extension of the (static) notion of a deviation measure and its properties. We study distribution invariant deviation measures and show that the only dynamic deviation measure which is law invariant and recursive is the variance. We also solve the problem of optimal risk-sha…

The paper shows measures equidistribute on affine submanifolds with a rate.

Introduces bounded scale measure and generalizes property A.

The article discusses invariant measures outside homogeneous dynamics.

Non-ergodic measures found in horocycle flow on Abelian differentials.

The paper explores non-convex risk measures and their characterizations.

A one-to-one correspondence is drawn between law invariant risk measures and divergences, which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy a…

Extends Dirac operator results to foliations with invariant measures.

We characterize when a convex risk measure associated to a law-invariant acceptance set in $L^\infty$ can be extended to $L^p$, $1\leq p<\infty$, preserving finiteness and continuity. This problem is strongly connected to the statistical robustness of the corresponding risk measures. Special attention is paid to concre…

New tools study curvature measures of convex bodies, revealing structured spaces.

The paper characterizes risk measures with the Fatou property in function spaces.

This paper improves the robustness of risk estimation for financial positions.

We introduce the notion of pullback along a measurable cocycle and we use it to extend the Borel invariant studied by Bucher, Burger and Iozzi to the world of measurable cocycles. The Borel invariant is constant along cohomology classes and has bounded absolute value. This allows to define maximal cocycles. We conclude…

Using optimal transport we study some dynamical properties of expanding circle maps acting on measures by push-forward. Using the definition of the tangent space to the space of measures introduced by Gigli, their derivative at the unique absolutely continuous invariant measure is computed. In particular it is shown th…

A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem

DID measures similarity invariant to diffeomorphisms.

Worst-case risk measures refer to the calculation of the largest value for risk measures when only partial information of the underlying distribution is available. For the popular risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), it is now known that their worst-case counterparts can be ev…

Twisted Alexander invariants have been defined for any knot and linear representation of its group. The invariants are generalized for any periodic representation of the commutator subgroup of the knot group. Properties of the new twisted invariants are given. Under suitable hypotheses, reciprocality and bounds on the …

Valuations constitute a class of functionals on convex bodies which include the Euler-characteristic, the surface area, the Lebesgue-measure, and many more classical functionals. Curvature measures may be regarded as "localised`` versions of valuations which yield local information about the geometry of a body's bounda…

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that t…

A knot's thickness is measured by its β invariant, a new numerical invariant.

Approximates measures on curved spaces using Dirac measures.

We consider various notions of strains; quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of Patrizio Neff and co-workers , is to select a Riemannian metric on $\text{GL}_n$, and use its induced geodes…

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.

We prove several versions of Driver's integration by parts formula for the horizontal Wiener measure on a totally geodesic Riemannian foliation and prove that the horizontal Wiener measure has a quasi-invariance property with respect to flows generated by suitable tangent processes.

The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space $\mathrm{Curv}^{\mathrm{U}(n)*}$ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a…

New multivariate risk measures improve on univariate OCE methods.

Identifies conditions for multiple invariant probabilities in Markov kernels.

Permutation invariant network learns Wasserstein metrics.

This paper reviews incompatibilities of comonotonic risk measures.

Deep learning approximates geometric measures of planar curves.