Affine Dunkl processes of type $\widetilde{\mathrm {A}}_{1}$

François Chapon

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Displaying similar documents to “Affine Dunkl processes of type ${\tilde{A}}_{1}$ ”

The natural operators of general affine connections into general affine connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We reduce the problem of describing all $ℳ f_{m}$ -natural operators transforming general affine connections on $m$ -manifolds into general affine ones to the known description of all $G L (𝐑^{m})$ -invariant maps $𝐑^{m *} \otimes 𝐑^{m} \to \otimes^{k} 𝐑^{m *} \otimes \otimes^{k} 𝐑^{m}$ for $k = 1, 3$ .

A Remark on a Paper of Crachiola and Makar-Limanov

Robert Dryło (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line $¹_{k}$ , then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.

On the Hausdorff dimension of certain self-affine sets

Abercrombie Alex G.., Nair R. (2002)

Studia Mathematica

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A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $Φ (S) = ⋃_{1 \leq j \leq t} ϕ_{j} (S)$ . If Φ(S) ⊂ S let $E_{Φ} (S)$ denote $⋂_{k \geq 0} Φ^{k} (S)$ . For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{Φ} (S)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also...

Sufficient conditions for the spectrality of self-affine measures with prime determinant

Jian-Lin Li (2014)

Studia Mathematica

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Let $μ_{M, D}$ be a self-affine measure associated with an expanding matrix M and a finite digit set D. We study the spectrality of $μ_{M, D}$ when |det(M)| = |D| = p is a prime. We obtain several new sufficient conditions on M and D for $μ_{M, D}$ to be a spectral measure with lattice spectrum. As an application, we present some properties of the digit sets of integral self-affine tiles, which are connected with a conjecture of Lagarias and Wang.

Uniqueness of Cartesian Products of Compact Convex Sets

Zbigniew Lipecki, Viktor Losert, Jiří Spurný (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $X_{i}$ , i∈ I, and $Y_{j}$ , j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of $\prod_{i \in I} X_{i}$ onto $\prod_{j \in J} Y_{j}$ . We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_{i}$ onto $Y_{b (i)}$ , i∈ I.

Centroaffine differential geometry and its relations to horizontal submanifolds

Luc Vrancken (2002)

Banach Center Publications

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We relate centroaffine immersions $f : M ⁿ \to ℝ^{n + 1}$ to horizontal immersions g of Mⁿ into $S_{n + 1}^{2 n + 1} (1)$ or $H_{n}^{2 n + 1} (- 1)$ . We also show that f is an equiaffine sphere, i.e. the centroaffine normal is a constant multiple of the Blaschke normal, if and only if g is minimal.

On Paszkiewicz-type criterion for a.e. continuity of processes in $L^{p}$ -spaces

Jakub Olejnik (2010)

Banach Center Publications

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In this paper we consider processes Xₜ with values in $L^{p}$ , p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type...

Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras

Peng Shan (2011)

Annales scientifiques de l'École Normale Supérieure

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We define the $i$ -restriction and $i$ -induction functors on the category $𝒪$ of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

Distances to spaces of affine Baire-one functions

Jiří Spurný (2010)

Studia Mathematica

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Let E be a Banach space and let $ℬ ₁ (B_{E *})$ and $₁ (B_{E *})$ denote the space of all Baire-one and affine Baire-one functions on the dual unit ball $B_{E *}$ , respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between $d i s t (f, ℬ ₁ (B_{E *}))$ and $d i s t (f, ₁ (B_{E *}))$ , where f is an affine function on $B_{E *}$ . If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.

Integral representations for solutions of exponential Gauß-Manin systems

Marco Hien, Céline Roucairol (2008)

Bulletin de la Société Mathématique de France

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Let $f, g : U \to 𝔸^{1}$ be two regular functions from the smooth affine complex variety $U$ to the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system $𝒪_{U} e^{g}$ with respect to $f$ . We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.

A remarkable $σ$ -finite measure unifying supremum penalisations for a stable Lévy process

Yuko Yano (2013)

Annales de l'I.H.P. Probabilités et statistiques

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The $σ$ -finite measure $𝒫_{sup}$ which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s $h$ -transform processes with respect to these functions are utilized for the construction of $𝒫_{sup}$ .

Self-affine measures and vector-valued representations

Qi-Rong Deng, Xing-Gang He, Ka-Sing Lau (2008)

Studia Mathematica

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Let A be a d × d integral expanding matrix and let $S_{j} (x) = A^{- 1} (x + d_{j})$ for some $d_{j} \in ℤ^{d}$ , j = 1,...,m. The iterated function system (IFS) ${S_{j}}_{j = 1}^{m}$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof...

Generalizing the Johnson-Lindenstrauss lemma to k-dimensional affine subspaces

Alon Dmitriyuk, Yehoram Gordon (2009)

Studia Mathematica

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Let ε > 0 and 1 ≤ k ≤ n and let ${W_{l}}_{l = 1}^{p}$ be affine subspaces of ℝⁿ, each of dimension at most k. Let $m = O (ε^{- 2} (k + l o g p))$ if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map $H : ℝ ⁿ \to ℝ^{m}$ such that for all 1 ≤ l ≤ p and $x, y \in W_{l}$ we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces...

Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Kevin G. Hare, Nikita Sidorov (2015)

Studia Mathematica

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Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if $| d e t M | \geq 2^{- 1 / d}$ , then the attractor $A_{M}$ has non-empty interior. We also consider the set $_{M}$ of points in $A_{M}$ which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of $_{M}$ is positive. For this special class the full description of $_{M}$ is given as well. This paper continues our...

On some ergodic properties for continuous and affine functions

Charles J. K. Batty (1978)

Annales de l'institut Fourier

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Two problems posed by Choquet and Foias are solved: (i) Let $T$ be a positive linear operator on the space $C (X)$ of continuous real-valued functions on a compact Hausdorff space $X$ . It is shown that if $n^{- 1} \sum_{r = 0}^{n - 1} T^{r} 1$ converges pointwise to a continuous limit, then the convergence is uniform on $X$ . (ii) An example is given of a Choquet simplex $K$ and a positive linear operator $T$ on the space $A (K)$ of continuous affine real-valued functions on $K$ , such that $\inf {(T^{n} 1) (x) : n \geq} < 1$ for each...

Regularity of stopping times of diffusion processes in Besov spaces

Xicheng Zhang (2002)

Studia Mathematica

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We prove that the exit times of diffusion processes from a bounded open set Ω almost surely belong to the Besov space $B_{p, q}^{α} (Ω)$ provided that pα < 1 and 1 ≤ q < ∞.

The tame automorphism group of an affine quadric threefold acting on a square complex

Cinzia Bisi, Jean-Philippe Furter, Stéphane Lamy (2014)

Journal de l’École polytechnique — Mathématiques

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We study the group $Tame ({SL}_{2})$ of tame automorphisms of a smooth affine $3$ -dimensional quadric, which we can view as the underlying variety of ${SL}_{2} (ℂ)$ . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $CAT (0)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $Tame ({SL}_{2})$ is linearizable, and that $Tame ({SL}_{2})$ satisfies the Tits alternative.

Theorem-proving systems

Ewa Orłowska

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CONTENTSIntroduction.................................................................................................................... 6Chapter I. Theorem-proving system§ 1. Theory...................................................................................................................... 7§ 2. Fundamental theory $T_{ƒ}$ ................................................................................ 8§ 3. Theorem-proving system.......................................................................................

Two stationarity conditions in the $G I / r_{0} / M / r_{1}$ dam

M. Jankiewicz (1988)

Applicationes Mathematicae

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The theory of reproducing systems on locally compact abelian groups

Gitta Kutyniok, Demetrio Labate (2006)

Colloquium Mathematicae

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A reproducing system is a countable collection of functions $ϕ_{j} : j \in$ such that a general function f can be decomposed as $f = \sum_{j \in} c_{j} (f) ϕ_{j}$ , with some control on the analyzing coefficients $c_{j} (f)$ . Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint in the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems...

Lévy processes conditioned on having a large height process

Mathieu Richard (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In the present work, we consider spectrally positive Lévy processes $(X_{t}, t \geq 0)$ not drifting to $+ \infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$ ) before hitting $0$ . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law $ℙ_{x}^{☆}$ of this conditioned process (starting at $x g t; 0$ ) is defined as a Doob $h$ -transform via a martingale. For Lévy processes with infinite variation paths,...

Self-affine measures that are $L^{p}$ -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

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A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{q}$ to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^{p}$ -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^{p}$ -improving.

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, R. L. Schilling (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$ . The state space of $(Z, S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z, S)$ is the product of the uniform probability measure and a Gaussian distribution.

Digit sets of integral self-affine tiles with prime determinant

Jian-Lin Li (2006)

Studia Mathematica

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Let M ∈ Mₙ(ℤ) be expanding such that |det(M)| = p is a prime and pℤⁿ ⊈ M²(ℤⁿ). Let D ⊂ ℤⁿ be a finite set with |D| = |det(M)|. Suppose the attractor T(M,D) of the iterated function system ${ϕ_{d} (x) = M^{- 1} (x + d)}_{d \in D}$ has positive Lebesgue measure. We prove that (i) if D ⊈ M(ℤⁿ), then D is a complete set of coset representatives of ℤⁿ/M(ℤⁿ); (ii) if D ⊆ M(ℤⁿ), then there exists a positive integer γ such that $D = M^{γ} D ₀$ , where D₀ is a complete set of coset representatives of ℤⁿ/M(ℤⁿ). This improves the corresponding results...

A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $∥ Y ∥_{W} \leq (2 (α + 1) ²) / (2 α + 1) ∥ X ∥_{L^{\infty}}$ . Here W is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

Fabio Nicola (2010)

Studia Mathematica

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We study Fourier integral operators of Hörmander’s type acting on the spaces $ℱ L^{p} {(ℝ^{d})}_{c o m p}$ , 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in $L^{p}$ . We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on $ℱ L^{p} {(ℝ^{d})}_{c o m p}$ if the mapping $x \mapsto \nabla_{x} Φ (x, η)$ is constant on the fibres, of codimension r,...

Displaying similar documents to “Affine Dunkl processes of type A ˜ 1 ”

Displaying similar documents to “Affine Dunkl processes of type ${\tilde{A}}_{1}$ ”