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Search: MSC category 60 ( Probability theory and stochastic processes )

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1. CMB Online first

Lehec, Joseph
Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems
We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prékopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.

Keywords:functional inequalities, Brownian motion
Categories:39B62, 60J65

2. CMB 2012 (vol 57 pp. 113)

Madras, Neal
A Lower Bound for the End-to-End Distance of Self-Avoiding Walk
For an $N$-step self-avoiding walk on the hypercubic lattice ${\bf Z}^d$, we prove that the mean-square end-to-end distance is at least $N^{4/(3d)}$ times a constant. This implies that the associated critical exponent $\nu$ is at least $2/(3d)$, assuming that $\nu$ exists.

Keywords:self-avoiding walk, critical exponent
Categories:82B41, 60D05, 60K35

3. CMB 2011 (vol 56 pp. 13)

Alon, Gil; Kozma, Gady
Ordering the Representations of $S_n$ Using the Interchange Process
Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett, and Richthammer, we define an associated order $\prec$ on the irreducible representations of $S_n$. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the ``Aldous order'' completely is a generalized question. We show a few additional entries for this order.

Keywords:Aldous' conjecture, interchange process, symmetric group, representations
Categories:82C22, 60B15, 43A65, 20B30, 60J27, 60K35

4. CMB 2011 (vol 55 pp. 597)

Osękowski, Adam
Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
We determine the best constants $C_{p,\infty}$ and $C_{1,p}$, $1 < p < \infty$, for which the following holds. If $u$, $v$ are orthogonal harmonic functions on a Euclidean domain such that $v$ is differentially subordinate to $u$, then $$ \|v\|_p \leq C_{p,\infty} \|u\|_\infty,\quad \|v\|_1 \leq C_{1,p} \|u\|_p. $$ In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of $\mathbb R^2$. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.

Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem
Categories:31B05, 60G44, 60G40

5. CMB 2011 (vol 55 pp. 487)

Deng, Xinghua; Moody, Robert V.
Weighted Model Sets and their Higher Point-Correlations
Examples of distinct weighted model sets with equal $2,3,4, 5$-point correlations are given.

Keywords:model sets, correlations, diffraction
Categories:52C23, 51P05, 74E15, 60G55

6. CMB 2011 (vol 54 pp. 464)

Hwang, Tea-Yuan; Hu, Chin-Yuan
A Characterization of the Compound-Exponential Type Distributions
In this paper, a fixed point equation of the compound-exponential type distributions is derived, and under some regular conditions, both the existence and uniqueness of this fixed point equation are investigated. A question posed by Pitman and Yor can be partially answered by using our approach.

Keywords:fixed point equation, compound-exponential type distributions
Categories:62E10, 60G50

7. CMB 2011 (vol 54 pp. 338)

Nakazi, Takahiko
Szegö's Theorem and Uniform Algebras
We study Szegö's theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra.

Keywords:Szegö's theorem, uniform algebras, disc algebra, weighted Bergman space
Categories:32A35, 46J15, 60G25

8. CMB 2010 (vol 54 pp. 113)

Hytönen, Tuomas P.
On the Norm of the Beurling-Ahlfors Operator in Several Dimensions
The generalized Beurling-Ahlfors operator $S$ on $L^p(\mathbb{R}^n;\Lambda)$, where $\Lambda:=\Lambda(\mathbb{R}^n)$ is the exterior algebra with its natural Hilbert space norm, satisfies the estimate $$\|S\|_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*-1),\quad p^*:=\max\{p,p'\}$$ This improves on earlier results in all dimensions $n\geq 3$. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.

Categories:42B20, 60G46

9. CMB 2010 (vol 53 pp. 614)

Böröczky, Károly J.; Schneider, Rolf
The Mean Width of Circumscribed Random Polytopes
For a given convex body $K$ in ${\mathbb R}^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of $K^{(n)}$ and $K$ as $n$ tends to infinity. For a simplicial polytope $P$, a precise asymptotic formula for the difference of the mean widths of $P^{(n)}$ and $P$ is obtained.

Keywords:random polytope, mean width, approximation
Categories:52A22, 60D05, 52A27

10. CMB 2010 (vol 53 pp. 526)

Milian, Anna
On Some Stochastic Perturbations of Semilinear Evolution Equations
We consider semilinear evolution equations with some locally Lipschitz nonlinearities, perturbed by Banach space valued, continuous, and adapted stochastic process. We show that under some assumptions there exists a solution to the equation. Using the result we show that there exists a mild, continuous, global solution to a semilinear Itô equation with locally Lipschitz nonlinearites. An example of the equation is given.

Keywords:evolution equation, mild solution, non-Lipschitz drift, Ito integral
Category:60H20

11. CMB 2010 (vol 53 pp. 503)

Kurenok, V. P.
The Time Change Method and SDEs with Nonnegative Drift
Using the time change method we show how to construct a solution to the stochastic equation $dX_t=b(X_{t-})dZ_t+a(X_t)dt$ with a nonnegative drift $a$ provided there exists a solution to the auxililary equation $dL_t=[a^{-1/\alpha}b](L_{t-})d\bar Z_t+dt$ where $Z, \bar Z$ are two symmetric stable processes of the same index $\alpha\in(0,2]$. This approach allows us to prove the existence of solutions for both stochastic equations for the values $0<\alpha<1$ and only measurable coefficients $a$ and $b$ satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.

Keywords:One-dimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence
Categories:60H10, 60J60, 60J65, 60G44

12. CMB 2008 (vol 51 pp. 146)

Zhou, Xiaowen
Stepping-Stone Model with Circular Brownian Migration
In this paper we consider the stepping-stone model on a circle with circular Brownian migration. We first point out a connection between Arratia flow on the circle and the marginal distribution of this model. We then give a new representation for the stepping-stone model using Arratia flow and circular coalescing Brownian motion. Such a representation enables us to carry out some explicit computations. In particular, we find the distribution for the first time when there is only one type left across the circle.

Keywords:stepping-stone model, circular coalescing Brownian motion, Arratia flow, duality, entrance law
Categories:60G57, 60J65

13. CMB 2008 (vol 51 pp. 26)

Belinschi, S. T.; Bercovici, H.
Hin\v cin's Theorem for Multiplicative Free Convolution
Hin\v cin proved that any limit law, associated with a triangular array of infinitesimal random variables, is infinitely divisible. The analogous result for additive free convolution was proved earlier by Bercovici and Pata. In this paper we will prove corresponding results for the multiplicative free convolution of measures definded on the unit circle and on the positive half-line.

Categories:46L53, 60E07, 60E10

14. CMB 2006 (vol 49 pp. 389)

Hiai, Fumio; Petz, Dénes; Ueda, Yoshimichi
A Free Logarithmic Sobolev Inequality on the Circle
Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

Categories:46L54, 60E15, 94A17

15. CMB 2004 (vol 47 pp. 481)

Bekjan, Turdebek N.
A New Characterization of Hardy Martingale Cotype Space
We give a new characterization of Hardy martingale cotype property of complex quasi-Banach space by using the existence of a kind of plurisubharmonic functions. We also characterize the best constants of Hardy martingale inequalities with values in the complex quasi-Banach space.

Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic function
Categories:46B20, 52A07, 60G44

16. CMB 2004 (vol 47 pp. 215)

Jaworski, Wojciech
Countable Amenable Identity Excluding Groups
A discrete group $G$ is called \emph{identity excluding\/} if the only irreducible unitary representation of $G$ which weakly contains the $1$-dimensional identity representation is the $1$-dimensional identity representation itself. Given a unitary representation $\pi$ of $G$ and a probability measure $\mu$ on $G$, let $P_\mu$ denote the $\mu$-average $\int\pi(g) \mu(dg)$. The goal of this article is twofold: (1)~to study the asymptotic behaviour of the powers $P_\mu^n$, and (2)~to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure $\mu$ on an identity excluding group and every unitary representation $\pi$ there exists and orthogonal projection $E_\mu$ onto a $\pi$-invariant subspace such that $s$-$\lim_{n\to\infty}\bigl(P_\mu^n- \pi(a)^nE_\mu\bigr)=0$ for every $a\in\supp\mu$. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of $\FC$-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

Categories:22D10, 22D40, 43A05, 47A35, 60B15, 60J50

17. CMB 2000 (vol 43 pp. 368)

Litvak, A. E.
Kahane-Khinchin's Inequality for Quasi-Norms
We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about equivalence of $L_q$-norms of logconcave random variables (Kahane-Khinchin's inequality) to the quasi-convex case. We construct examples of quasi-convex bodies $K_n \subset \R$ which demonstrate that this equivalence fails for uniformly distributed vector on $K_n$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the ``tail" volume (for convex bodies such decay was proved by M.~Gromov and V.~Milman).

Categories:46B09, 52A30, 60B11

18. CMB 1999 (vol 42 pp. 321)

Kikuchi, Masato
Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces
We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section~2 the equivalence between Shimogaki's theorem and some martingale inequalities will be established, and in Section~3 the equivalence between Boyd's theorem and martingale inequalities with change of probability measure will be established.

Keywords:martingale inequalities, rearrangement invariant function spaces
Categories:60G44, 60G46, 46E30

19. CMB 1999 (vol 42 pp. 344)

20. CMB 1999 (vol 42 pp. 221)

Liu, Peide; Saksman, Eero; Tylli, Hans-Olav
Boundedness of the $q$-Mean-Square Operator on Vector-Valued Analytic Martingales
We study boundedness properties of the $q$-mean-square operator $S^{(q)}$ on $E$-valued analytic martingales, where $E$ is a complex quasi-Banach space and $2 \leq q < \infty$. We establish that a.s. finiteness of $S^{(q)}$ for every bounded $E$-valued analytic martingale implies strong $(p,p)$-type estimates for $S^{(q)}$ and all $p\in (0,\infty)$. Our results yield new characterizations (in terms of analytic and stochastic properties of the function $S^{(q)}$) of the complex spaces $E$ that admit an equivalent $q$-uniformly PL-convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the $L^p$-boundedness of the usual square-function on scalar-valued analytic martingales.

Categories:46B20, 60G46

21. CMB 1998 (vol 41 pp. 166)

Hof, A.
Percolation on Penrose tilings
In Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely~0 or~1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of $\hbox{\Bbbvii Z}^d$, and to other percolation processes, including Bernoulli bond percolation.

Categories:60K35, 82B43

22. CMB 1997 (vol 40 pp. 67)

Knight, Frank B.
On a Brownian motion problem of T. Salisbury
Let $B$ be a Brownian motion on $R$, $B(0)=0$, and let $f(t,x)$ be continuous. T.~Salisbury conjectured that if the total variation of $f(t,B(t))$, $0\leq t\leq 1$, is finite $P$-a.s., then $f$ does not depend on $x$. Here we prove that this is true if the expected total variation is finite.

Category:60J65

23. CMB 1997 (vol 40 pp. 19)

Derbez, Eric; Slade, Gordon
Lattice trees and super-Brownian motion
This article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion called {\it integrated super-Brownian excursion\/} ($\ISE$), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is $\ISE$, in all dimensions. A connection is drawn between $\ISE$ and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point.

Categories:82B41, 60K35, 60J65

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