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1. CJM 2014 (vol 66 pp. 1358)

Osėkowski, Adam
Sharp Localized Inequalities for Fourier Multipliers
In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator .

Keywords:Fourier multiplier, martingale, laminate
Categories:42B15, 60G44, 42B20

2. CJM 2013 (vol 66 pp. 1050)

Holmes, Mark; Salisbury, Thomas S.
Random Walks in Degenerate Random Environments
We study the asymptotic behaviour of random walks in i.i.d. random environments on $\mathbb{Z}^d$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience, and in 2-dimensions the existence of a deterministic limiting velocity.

Keywords:random walk, non-elliptic random environment, zero-one law, coupling
Category:60K37

3. CJM 2012 (vol 65 pp. 600)

Kroó, A.; Lubinsky, D. S.
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials
We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain - in particular this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex, under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.

Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions
Categories:42C05, 42C99, 42B05, 60B20

4. CJM 2011 (vol 64 pp. 1201)

Aistleitner, Christoph; Elsholtz, Christian
The Central Limit Theorem for Subsequences in Probabilistic Number Theory
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying \begin{equation} \tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty. \end{equation} If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of \begin{equation} \tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}} \end{equation} converges to a Gaussian distribution. In the case $$ 1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty $$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$ \frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}} $$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem
Categories:60F05, 42A55, 11D04, 05C55, 11K06

5. CJM 2011 (vol 64 pp. 1075)

Raja, Chandiraraj Robinson Edward
A Stochastic Difference Equation with Stationary Noise on Groups
We consider the stochastic difference equation $$\eta _k = \xi _k \phi (\eta _{k-1}), \quad k \in \mathbb Z $$ on a locally compact group $G$ where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$-valued random variables and $\eta _k$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on one-dimensional torus. We consider the case when $\xi _k$ have a common law $\mu$ and prove that if $G$ is a distal group and $\phi$ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu$ is supported on $Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

Keywords:dissipating, distal automorphisms, probability measures, pointwise distal groups, shifted convolution powers
Categories:60B15, 60G20

6. CJM 2011 (vol 64 pp. 961)

Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim
Densities of Short Uniform Random Walks
We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

Keywords:random walks, hypergeometric functions, Mahler measure
Categories:60G50, 33C20, 34M25, 44A10

7. CJM 2011 (vol 64 pp. 805)

Chapon, François; Defosseux, Manon
Quantum Random Walks and Minors of Hermitian Brownian Motion
Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.

Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process
Categories:46L53, 60B20, 14L24

8. CJM 2011 (vol 64 pp. 869)

Hu, Ze-Chun; Sun, Wei
Balayage of Semi-Dirichlet Forms
In this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayaged measures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting.

Keywords:balayage, semi-Dirichlet form, potential theory
Categories:31C25, 60J45

9. CJM 2010 (vol 63 pp. 153)

Hambly, B. M.
Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet
We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function, and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$ and $\phi$ a periodic function. We also discuss similar results for the heat content of affine nested fractals.

Categories:35K05, 28A80, 35B40, 60J65

10. CJM 2010 (vol 63 pp. 222)

Wang, Jiun-Chau
Limit Theorems for Additive Conditionally Free Convolution
In this paper we determine the limiting distributional behavior for sums of infinitesimal conditionally free random variables. We show that the weak convergence of classical convolution and that of conditionally free convolution are equivalent for measures in an infinitesimal triangular array, where the measures may have unbounded support. Moreover, we use these limit theorems to study the conditionally free infinite divisibility. These results are obtained by complex analytic methods without reference to the combinatorics of c-free convolution.

Keywords:additive c-free convolution, limit theorems, infinitesimal arrays
Categories:46L53, 60F05

11. CJM 2010 (vol 63 pp. 104)

Feng, Shui; Schmuland, Byron; Vaillancourt, Jean; Zhou, Xiaowen
Reversibility of Interacting Fleming-Viot Processes with Mutation, Selection, and Recombination
Reversibility of the Fleming--Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming--Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.

Categories:60J60, 60J70

12. CJM 2009 (vol 61 pp. 1279)

Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval
Tail Bounds for the Stable Marriage of Poisson and Lebesgue
Let $\Xi$ be a discrete set in $\rd$. Call the elements of $\Xi$ {\em centers}. The well-known Voronoi tessellation partitions $\rd$ into polyhedral regions (of varying volumes) by allocating each site of $\rd$ to the closest center. Here we study allocations of $\rd$ to $\Xi$ in which each center attempts to claim a region of equal volume $\alpha$. We focus on the case where $\Xi$ arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is {\em stable} in the sense of the Gale--Shapley marriage problem. We study the distance $X$ from a typical site to its allocated center in the stable allocation. The model exhibits a phase transition in the appetite $\alpha$. In the critical case $\alpha=1$ we prove a power law upper bound on $X$ in dimension $d=1$. (Power law lower bounds were proved earlier for all $d$). In the non-critical cases $\alpha<1$ and $\alpha>1$ we prove exponential upper bounds on $X$.

Keywords:stable marriage, point process, phase transition
Category:60D05

13. CJM 2009 (vol 61 pp. 534)

Chen, Chuan-Zhong; Sun, Wei
Girsanov Transformations for Non-Symmetric Diffusions
Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form $(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(E;m)$. For $u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process $Y$ of $X$. First, let $\widehat{X}$ be the dual process of $X$ and $\widehat{Y}$ the Girsanov transformed process of $\widehat{X}$. We give a necessary and sufficient condition for $(Y,\widehat{Y})$ to be in duality with respect to the measure $e^{2u}m$. We also construct a counterexample, which shows that this condition may not be satisfied and hence $(Y,\widehat{Y})$ may not be dual processes. Then we present a sufficient condition under which $Y$ is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Keywords:Diffusion, non-symmetric Dirichlet form, Girsanov transformation, $h$-transformation, perturbation of Dirichlet form, generalized Feynman-Kac semigroup
Categories:60J45, 31C25, 60J57

14. CJM 2008 (vol 60 pp. 822)

Kuwae, Kazuhiro
Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms
Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Keywords:positivity preserving form, semi-Dirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity condition
Categories:31C25, 35B50, 60J45, 35J, 53C, 58

15. CJM 2008 (vol 60 pp. 334)

Curry, Eva
Low-Pass Filters and Scaling Functions for Multivariable Wavelets
We show that a characterization of scaling functions for multiresolution analyses given by Hern\'{a}ndez and Weiss and that a characterization of low-pass filters given by Gundy both hold for multivariable multiresolution analyses.

Keywords:multivariable multiresolution analysis, low-pass filter, scaling function
Categories:42C40, 60G35

16. CJM 2008 (vol 60 pp. 457)

Teplyaev, Alexander
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18

17. CJM 2008 (vol 60 pp. 313)

Choi, Yong-Kab; o, Miklós Csörg\H
Asymptotic Properties for Increments of $l^{\infty}$-Valued Gaussian Random Fields
This paper establishes general theorems which contain both moduli of continuity and large incremental results for $l^\infty$-valued Gaussian random fields indexed by a multidimensional parameter under explicit conditions.

Keywords:$l^\infty$-valued Gaussian random field, modulus of continuity, regularly varying function, large deviation probability
Categories:60F15, 60G15, 60G60

18. CJM 2007 (vol 59 pp. 828)

Ortner, Ronald; Woess, Wolfgang
Non-Backtracking Random Walks and Cogrowth of Graphs
Let $X$ be a locally finite, connected graph without vertices of degree $1$. Non-backtracking random walk moves at each step with equal probability to one of the ``forward'' neighbours of the actual state, \emph{i.e.,} it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of old results concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but \emph{small cycles are dense in} $X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen.

Keywords:graph, oriented line grap, covering tree, random walk, cogrowth, amenability
Categories:05C75, 60G50, 20F69

19. CJM 2007 (vol 59 pp. 795)

Jaworski, Wojciech; Neufang, Matthias
The Choquet--Deny Equation in a Banach Space
Let $G$ be a locally compact group and $\pi$ a representation of $G$ by weakly$^*$ continuous isometries acting in a dual Banach space $E$. Given a probability measure $\mu$ on $G$, we study the Choquet--Deny equation $\pi(\mu)x=x$, $x\in E$. We prove that the solutions of this equation form the range of a projection of norm $1$ and can be represented by means of a ``Poisson formula'' on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law $\mu$. The relation between the space of solutions of the Choquet--Deny equation in $E$ and the space of bounded harmonic functions can be understood in terms of a construction resembling the $W^*$-crossed product and coinciding precisely with the crossed product in the special case of the Choquet--Deny equation in the space $E=B(L^2(G))$ of bounded linear operators on $L^2(G)$. Other general properties of the Choquet--Deny equation in a Banach space are also discussed.

Categories:22D12, 22D20, 43A05, 60B15, 60J50

20. CJM 2006 (vol 58 pp. 1026)

Handelman, David
Karamata Renewed and Local Limit Results
Connections between behaviour of real analytic functions (with no negative Maclaurin series coefficients and radius of convergence one) on the open unit interval, and to a lesser extent on arcs of the unit circle, are explored, beginning with Karamata's approach. We develop conditions under which the asymptotics of the coefficients are related to the values of the function near $1$; specifically, $a(n)\sim f(1-1/n)/ \alpha n$ (for some positive constant $\alpha$), where $f(t)=\sum a(n)t^n$. In particular, if $F=\sum c(n) t^n$ where $c(n) \geq 0$ and $\sum c(n)=1$, then $f$ defined as $(1-F)^{-1}$ (the renewal or Green's function for $F$) satisfies this condition if $F'$ does (and a minor additional condition is satisfied). In come cases, we can show that the absolute sum of the differences of consecutive Maclaurin coefficients converges. We also investigate situations in which less precise asymptotics are available.

Categories:30B10, 30E15, 41A60, 60J35, 05A16

21. CJM 2006 (vol 58 pp. 691)

Bendikov, A.; Saloff-Coste, L.
Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups
On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(\mu_t)_{t>0}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if $(\mu_t)_{t>0} $ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto \mu_t(x)$, $t>0$, such that $\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega$ can be represented by a continuous function in $\Omega$. Examples are discussed.

Categories:60B15, 43A77, 35H10, 46F25, 60J45, 60J60

22. CJM 2005 (vol 57 pp. 506)

Gross, Leonard; Grothaus, Martin
Reverse Hypercontractivity for Subharmonic Functions
Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, $e^{-tA}$, can be bounded {\it below} from $L^p$ to $L^q$ when $p,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions.

Keywords:Reverse hypercontractivity, subharmonic
Categories:58J35, 47D03, 47D07, 32Q99, 60J35

23. CJM 2005 (vol 57 pp. 204)

Xiong, Jie; Zhou, Xiaowen
On the Duality between Coalescing Brownian Motions
A duality formula is found for coalescing Brownian motions on the real line. It is shown that the joint distribution of a coalescing Brownian motion can be determined by another coalescing Brownian motion running backward. This duality is used to study a measure-valued process arising as the high density limit of the empirical measures of coalescing Brownian motions.

Keywords:coalescing Brownian motions, duality, martingale problem,, measure-valued processes
Categories:60J65, 60G57

24. CJM 2004 (vol 56 pp. 963)

Ishiwata, Satoshi
A Berry-Esseen Type Theorem on Nilpotent Covering Graphs
We prove an estimate for the speed of convergence of the transition probability for a symmetric random walk on a nilpotent covering graph. To obtain this estimate, we give a complete proof of the Gaussian bound for the gradient of the Markov kernel.

Categories:22E25, 60J15, 58G32

25. CJM 2004 (vol 56 pp. 431)

Rosenblatt, Joseph; Taylor, Michael
Group Actions and Singular Martingales II, The Recognition Problem
We continue our investigation in [RST] of a martingale formed by picking a measurable set $A$ in a compact group $G$, taking random rotates of $A$, and considering measures of the resulting intersections, suitably normalized. Here we concentrate on the inverse problem of recognizing $A$ from a small amount of data from this martingale. This leads to problems in harmonic analysis on $G$, including an analysis of integrals of products of Gegenbauer polynomials.

Categories:43A77, 60B15, 60G42, 42C10
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