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Transactions of the London Mathematical Society
Number of Followers: 1

This is an Open Access Journal

Open Access journal
ISSN (Online) 2052-4986
Published by Oxford University Press

[419 journals]

On the Ф-variation of stochastic processes with exponential moments
- Authors: Basse-O'Connor A; Weber M.
  Pages: 1 - 27
  Abstract: AbstractWe obtain sharp sufficient conditions for exponentially integrable stochastic processes $X=\{X(t): t\in [0,1]\}$, to have sample paths with bounded $\Phi $-variation. When $X$ is moreover Gaussian, we also provide a bound of the expectation of the associated $\Phi $-variation norm of $X$. For a Hermite process $X$ of order $m\in {\mathbb N}$ and of Hurst index $H\in (1/2,1)$, we show that $X$ is of bounded $\Phi $-variation where $\Phi (x)=x^{1/H}(\log (\log 1/x))^{-m/(2H)}$, and that this $\Phi $ is optimal. This shows that in terms of $\Phi $-variation, the Rosenblatt process (corresponding to $m=2$) has more rough sample paths than the fractional Brownian motion (corresponding to $m=1$).
  PubDate: Mon, 27 Jun 2016 00:00:00 GMT
  DOI: 10.1112/tlms/tlw001
  Issue No: Vol. 3, No. 1 (2016)
The R∞ property for nilpotent quotients of surface groups
- Authors: Dekimpe K; Gonçalves DL.
  Pages: 28 - 46
  Abstract: AbstractA group $G$ is said to have the $R_\infty $ property if, for any automorphism $\varphi $ of $G$, the number $R(\varphi )$ of twisted conjugacy classes (or Reidemeister classes) is infinite. It is well known that when $G$ is the fundamental group of a closed surface of negative Euler characteristic, it has the $R_{\infty }$ property. In this work, we compute the least integer $c$, called the $R_{\infty }$-nilpotency degree of $G$, such that the group $G/ \gamma _{c+1}(G)$ has the $R_{\infty }$ property, where $\gamma _r(G)$ is the $r$th term of the lower central series of $G$. We show that $c=4$ for $G$ the fundamental group of any orientable closed surface $S_g$ of genus $g>1$. For the fundamental group of the non-orientable surface $N_g$ (the connected sum of $g$ projective planes) this number is $2(g-1)$ (when $g>2$). A similar concept is introduced using the derived series $G^{(r)}$ of a group $G$. Namely, the $R_{\infty }$-solvability degree of $G$, which is the least integer $c$ such that the group $G/G^{(c)}$ has the $R_{\infty }$ property. We show that the fundamental group of an orientable closed surface $S_g$ has $R_{\infty }$-solvability degree 2. As a by-product of our research, we find a lot of new examples of nilmanifolds on which every self-homotopy equivalence can be deformed into a fixed point free map.
  PubDate: Thu, 29 Sep 2016 00:00:00 GMT
  DOI: 10.1112/tlms/tlw002
  Issue No: Vol. 3, No. 1 (2016)

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