Long time behaviour of random walks on the integer lattice

We consider an irreducible finite range random walk on the $d$-dimensional integer lattice and study asymptotic behaviour of its transition function $p(n; x)$. In particular, for simple random walk our asymptotic formula is valid as long as $n (n - |x|_1)^{-2}$ tends to zero

Pointwise ergodic theorems for some thin subsets of primes

We establish pointwise ergodic theorems for operators of Radon type along subsets of prime numbers of the form $\big\{\{ \varphi_1(n)\} < \psi(n)\big\}$. We achieve this by proving $\ell^p(\mathbb{Z})$ boundedness of $r$-variations, where $p > 1$ and $r > 2$

Heat kernel and Green function estimates on affine buildings

We obtain the optimal global upper and lower bounds for the transition density $p_n(x,y)$ of a finite range isotropic random walk on affine buildings. We present also sharp estimates for the corresponding Green function

Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvectors

We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and show that they satisfy strong non-subordinacy condition.Comment: 34 page

Cotlar's ergodic theorem along the prime numbers

The aim of this paper is to prove Cotlar's ergodic theorem modeled on the set of primes

Littlewood-Paley theory for triangle buildings

For the natural two parameter filtration $(\mathcal{F}_\lambda : \lambda \in P)$ on the boundary of a triangle building we define a maximal function and a square function and show their boundedness on $L^p(\Omega_0)$ for $p \in (1, \infty)$. At the end we consider $L^p(\Omega_0)$ boundedness of martingale transforms. If the building is of $\text{GL}(3, \mathbb{Q}_p)$ then $\Omega_0$ can be identified with $p$-adic Heisenberg group.Comment: Accepted for publication in The Journal of Geometric Analysis, 18 pages, 5 figure

Discrete maximal functions in higher dimensions and applications to ergodic theory

We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates $V_r$ on $L^p$ spaces for all $1<p<\infty$ and $r>\max\{p, p/(p-1)\}$. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree

Endpoint estimates for the maximal function over prime numbers

Given an ergodic dynamical system $(X, \mathcal{B}, \mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \log L)(X, \mu)$, the ergodic averages $$\frac{1}{\pi(N)} \sum_{p \in \mathbb{P}_N} f\big(T^p x\big),$$ converge for $\mu$-almost all $x \in X$, where $\mathbb{P}_N$ is the set of prime numbers not larger that $N$ and $\pi(N) = \# \mathbb{P}_N$.Comment: to appear in Journal of Fourier Analysis And Application

Variational estimates for discrete operators modeled on multi-dimensional polynomial subsets of primes

We prove the extensions of Birkhoff's and Cotlar's ergodic theorems to multi-dimensional polynomial subsets of prime numbers $\mathbb{P}^k$. We deduce them from $\ell^p$-boundedness of $r$-variational seminorms for the corresponding discrete operators of Radon type, where $p > 1$ and $r > 2$.Comment: To appear in Mathematische Annale

The Maximal Function and Square Function Control the Variation: An Elementary Proof

In this note we prove the following good-$\lambda$ inequality, for $r>2$, all $\lambda > 0$, $\delta \in \big(0, \frac{1}{2} \big)$ $$\nu\big\{ V_r(f) > 3 \lambda ; \mathcal{M}(f) \leq \delta \lambda\big\} \leq 4 \nu\{s(f) > \delta \lambda\} + {\delta^2 \left(1+\frac{16}{r-2}\right)^2} \cdot \nu\big\{ V_r(f) > \lambda\big\},$$ where $\mathcal{M}(f)$ is the martingale maximal function, $s(f)$ is the conditional martingale square function. This immediately proves that $V_r(f)$ is bounded on $L^p$, $1 < p <\infty$ and moreover is integrable when the maximal function is.Comment: 6 Pages. The current version implements suggestions from the referee. Accepted to the Proceedings of AM