Expand all Collapse all | Results 1 - 25 of 40 |
1. CJM Online first
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
LÃ©vy
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 Online first
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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 |