Related papers: Small deviations in p-variation for stable process…

This paper establishes small ball probabilities for a class of time-changed processes $X\circ E$, where $X$ is a self-similar process and $E$ is an independent continuous process, each with a certain small ball probability. In particular,…

In this paper we define the closure under weak convergence of the class of p-tempered {\alpha}-stable distributions. We give necessary and sufficient conditions for convergence of sequences in this class. Moreover, we show that any element…

In this paper, we establish a $C^{1,\alpha}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy…

In this paper, we broaden Shiu's Brun-Titchmarsh theorem to allow for functions that are larger and/or smooth-supported. In particular, let $f$ be a nonnegative multiplicative function. We prove that if there exists a $\beta<1$ such that…

We prove several results which imply the following consequences. For any $\varepsilon>0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv…

Let $u_k$ be a solution of the Helmholtz equation with the wave number $k$, $\Delta u_k+k^2 u_k=0$, on a small ball in either $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For a fixed point $p$, we define $M_{u_k}(r)=\max_{d(x,p)\le…

We introduce a parameter $p$, called partial survival, in the persistence of stochastic processes and show that for smooth processes the persistence exponent $\theta(p)$ changes continuously with $p$, $\theta(0)$ being the usual persistence…

We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for \Gamma_0(2). Numerical evidence indicates that a sharper bound holds for the…

We study asymptotic properties of the following Markov system of $N \geq 3$ points in~$[0,1]$. At each time step, the point farthest from the current centre of mass, multiplied by a constant $p>0$, is removed and replaced by an independent…

The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < \alpha_*(p) \leq…

We show that, for every $1 \leq p < +\infty$ and for every Borel probability measure $\mathbb{P}$ over $\mathbb{R}$, every element of $L^{p}(\mathbb{R}, \mathscr{B}_{\mathbb{R}}, \mathbb{P})$ is the $L^{p}$-limit of some sequence of bounded…

A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the…

For the following Neumann problem in a ball $$\begin{cases} -\Delta_p u+u^{p-1}=u^{q-1}\quad&\text{in }B,\\ u>0,\,u\text{ radial}\quad&\text{in }B,\\ \frac{\partial u}{\partial \nu}=0\quad&\text{on }\partial B, \end{cases}$$ with…

We analyze Jim Propp's P-machine, a simple deterministic process that simulates a random walk on $Z^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful…

In this article, the small ball probability is obtained for the collision local time of two independent symmetric $\alpha-$stable processes with parameters $\alpha_1,\alpha_2\in(0,2]$ satisfying $\max\{\alpha_1,\alpha_2\}>1$. The proof is…

We prove non-trivial lower bounds for sums of type $\sum_{p\sim P}g(\gamma\Log p)$, where $g$ is a non-negative $2\pi$-periodical function and $\gamma$ is a given parameter. As an application we prove that $\zeta(1+it)^{\pm1}\ll\Log\Log…

Let G=SL_n. Let K=Z/pZ, p a prime. Let A\subset G(K) generate G(K). Suppose that |A|<p^{n+1-\delta}, delta>0. Then |A A A|>>|A|^{1+\epsilon}, where epsilon>0 and the implied constant depend only on n and delta.

For each 1 < p < infinity, there exists a positive constant c_p, depending only on p, such that the following holds. Let (d_k), (e_k) be real-valued martingale difference sequences. If for for all bounded nonnegative predictable sequences…

We consider upper exponential bounds for the probability of the event that an absolute deviation of sample mean from mathematical expectation p is bigger comparing with some ordered level epsilon. These bounds include 2 coefficients {alpha,…

For positive integers $n, k, q, p$, let $A_k(n; q, p)$ be the largest integer $N$ such that there exists an edge coloring of $K_N^{(k)}$ with $q$ colors that does not contain a tight monotone path of length $n$ that consists of at most $p$…