Almost everywhere convergence of entangled ergodic averages

We study pointwise convergence of entangled averages of the form $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f,$$ where $f\in L^2(X,\mu)$, $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$, and the $T_i$ are ergodic measure preserving transformations on the standard probability space $(X,\mu)$. We show that under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, almost everywhere convergence holds for all $f\in L^2$. We also present results for the general $L^p$ case ($1\leq p<\infty$) and for polynomial powers, in addition to continuous versions concerning ergodic flows.Comment: 16 pages, to appear in Integral Equations and Operator Theor

On the pointwise entangled ergodic theorem

We present some twisted compactness conditions for almost everywhere convergence of one-parameter entangled ergodic averages of Dunford-Schwartz operators $T_0,\ldots, T_a$ on a Borel probability space of the form $$\sum_{n=1}^N T_a^n A_{a-1}T_{a-1}^nA_{a-1}\cdot \ldots \cdot A_0 T_0^nf$$ for $f\in L^p(X,\mu)$, $p\geq 1$. We also discuss examples and present a continuous version of the result.Comment: 18 pages, minor changes of the revised version incorporating referee's suggestions, where the results are formulated and proved for Dunford-Schwartz operators instead of Koopman operator

Pointwise entangled ergodic theorems for Dunford–Schwartz operators

We investigate pointwise convergence of entangled ergodic averages of Dunford-Schwartz operators $T_0,T_1,\ldots, T_m$ on a Borel probability space. These averages take the form $$\frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f,$$ where $f\in L^p(X,\mu)$ for some $1\leq p<\infty$, and $\alpha:\left\{1,\ldots,m\right\}\to\left\{1,\ldots,k\right\}$ encodes the entanglement. We prove that under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, almost everywhere convergence holds for all $f\in L^p$. We also present an extension to polynomial powers in the case $p=2$, in addition to a continuous version concerning Dunford-Schwartz $C_0$-semigroups.Comment: 16 page

The cut norm and Sampling Lemmas for unbounded kernels

Generalizing the bounded kernel results of Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi (2008), we prove two Sampling Lemmas for unbounded kernels with respect to the cut norm. On the one hand, we show that given a (symmetric) kernel $U\in L^p([0,1]^2)$ for some $3<p<\infty$, the cut norm of a random $k$-sample of $U$ is with high probability within $O(k^{-\frac14+\frac{1}{4p}})$ of the cut norm of $U$. The cut norm of the sample has a strong bias to being larger than the original, allowing us to actually obtain a stronger high probability bound of order $O(k^{-\frac 12+\frac1p+\varepsilon})$ for how much smaller it can be (for any $p>2$ here). These results are then partially extended to the case of vector valued kernels. On the other hand, we show that with high probability, the $k$-samples are also close to $U$ in the cut metric, albeit with a weaker bound of order $O((\ln k)^{-\frac12+\frac1{2p}})$ (for any appropriate $p>2$). As a corollary, we obtain that whenever $U\in L^p$ with $p>4$, the $k$-samples converge almost surely to $U$ in the cut metric as $k\to\infty$

A second-order Magnus-type integrator for evolution equations with delay

We rewrite abstract delay equations as nonautonomous abstract Cauchy problems allowing us to introduce a Magnus-type integrator for the former. We prove the second-order convergence of the obtained Magnus-type integrator. We also show that if the differential operators involved admit a common invariant set for their generated semigroups, then the Magnus-type integrator will respect this invariant set as well, allowing for much weaker assumptions to obtain the desired convergence. As an illustrative example we consider a space-dependent epidemic model with latent period and diffusion

ON THE DENSE PREFERENTIAL ATTACHMENT GRAPH MODELS AND THEIR GRAPHON INDUCED COUNTERPART

Letting $\mathcal{M}$ denote the space of finite measures on $\mathbb{N}$, and $\mu_\lambda\in\mathcal{M}$ denote the Poisson distribution with parameter $\lambda$, the function $W:[0,1]^2\to\mathcal{M}$ given by $W(x,y)=\mu_{c\log x\log y}$ is called the PAG graphon with density $c$. It is known that this is the limit, in the multigraph homomorphism sense, of the dense Preferential Attachment Graph (PAG) model with edge density $c$. This graphon can then in turn be used to generate the so-called W-random graphs in a natural way, and similar constructions also work in the slightly more general context of the so-called $\mathrm{PAG}_{\kappa}$ models.\\ The aim of this paper is to compare these dense $\mathrm{PAG}_{\kappa}$ models with the W-random graph models obtained from the corresponding graphons. Motivated by the multigraph limit theory, we investigate the expected jumble norm distance of the two models in terms of the number of vertices $n$. We present a coupling for which the expectation can be bounded from above by $O(\log^{3/2} n\cdot n^{-1/2})$, and provide a universal lower bound that is coupling-independent, but without the logarithmic term

Mean-field approximation of counting processes from a differential equation perspective

Deterministic limit of a class of continuous time Markov chains is considered based purely on differential equation techniques. Starting from the linear system of master equations, ordinary differential equations for the moments and a partial differential equation, called Fokker–Planck equation, for the distribution is derived. Introducing closures at the level of the second and third moments, mean-field approximations are introduced. The accuracy of the mean-field approximations and the Fokker–Planck equation is investigated by using two differential equation-based and an operator semigroup-based approach