Invariant density adaptive estimation for ergodic jump diffusion processes over anisotropic classes

in Journal of Statistical Planning and Inference (in press)

We consider the solution of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density μ. We assume that a continuous record of ... [more ▼]

We consider the solution of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density μ. We assume that a continuous record of observations is available. In the case without jumps, Reiss and Dalalyan [7] and Strauch [24] have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H ̈older smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d ≥ 2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldenshluger and Lepski method [11] which leads us to an adaptive non-parametric kernel estimator of the stationary density μ of the jump diffusion X. [less ▲]

The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

E-print/Working paper (in press)

We provide new tools for the calculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups : An algorithm for finding particularly useful fundamental domains, and an analysis ... [more ▼]

We provide new tools for the calculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups : An algorithm for finding particularly useful fundamental domains, and an analysis of the equivariant spectral sequence combined with torsion subcomplex reduction. [less ▲]

Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows

in Analysis and PDE (in press)

Let M be a differentiable manifold endowed with a family of complete Riemannian metrics g(t) evolving under a geometric flow over the time interval [0,T[. In this article, we give a probabilistic ... [more ▼]

Let M be a differentiable manifold endowed with a family of complete Riemannian metrics g(t) evolving under a geometric flow over the time interval [0,T[. In this article, we give a probabilistic representation for the derivative of the corresponding conjugate semigroup on M which is generated by a Schrödinger type operator. With the help of this derivative formula, we derive fundamental Harnack type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows. [less ▲]

Invariance in a class of operations related to weighted quasi-geometric means

in Fuzzy Sets and Systems (in press)

Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left ... [more ▼]

Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \cdot g\left( y\right) \right) \text{,} \end{equation*} where $f,g$ are bijections of $I$ are considered. Their connections with generalized weighted quasi-geometric means is presented. It is shown that invariance\ question within the class of this operations leads to means of iterative type and to a problem on a composite functional equation. An application of the invariance identity to determine effectively the limit of the sequence of iterates of some generalized quasi-geometric mean-type mapping, and the form of all continuous functions which are invariant with respect to this mapping are given. The equality of two considered operations is also discussed. [less ▲]

Multiple Sets Exponential Concentration and Higher Order Eigenvalues

in Potential Analysis (in press)

On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the ... [more ▼]

On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the metric Laplacian gives exponential improved concentration with k sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of Chung, Grigor’yan & Yau, Upper bounds for eigenvalues of the discrete and continuous Laplace operators. Adv. Math. 117(2), 165–178 (1996). [less ▲]

Characterization of field homomorphisms through Pexiderized functional equations

in Journal of Difference Equations and Applications (in press)

Vector-valued statistics of binomial processes: Berry-Esseen bounds in the convex distance

in Annals of Applied Probability (in press)

Visual characterization of associative quasitrivial nondecreasing functions on finite chains

in Fuzzy Sets and Systems (in press)

Maximal and Typical topology of real polynomial singularities

in Annales de l'Institut Fourier (in press)

On deformation quantization of quadratic Poisson structures

in Communications in Mathematical Physics (in press)

We study the deformation complex of the dg wheeled properad of Z-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application ... [more ▼]

We study the deformation complex of the dg wheeled properad of Z-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmüller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of Z-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps. [less ▲]