Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators $X_{\alpha }={\left(X_{\alpha }\left(t\right)\right)}_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator ${\left(S_{\alpha }\left(t\right)\right)}_{t\ge 0}$ (where $\alpha \in (0,1)$); thus, with the property that ${\overline{\Pi }}_{\alpha }$, the tail function of the canonical measure of $X_{\alpha }$, is regularly varying of index $?\alpha \in (?1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha =0$, when ${\overline{\Pi }}_{\alpha }$ is slowly varying at 0. When $\alpha \in (0,1)$, we show that ${\left(t{\overline{\Pi }}_{\alpha }\left(X_{\alpha }\left(t\right)\right)\right)}^{?1}$ converges in distribution, as $t\downarrow 0$, to the random variable ${\left(S_{\alpha }\left(1\right)\right)}^{\alpha }$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha \downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The $\alpha =0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.     

Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes

This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process ${\left(X\left(t\right)\right)}_{t\ge 0}$ which drifts to $\infty$, namely $U\left(t\right)?{\mathrm{e}}^{?t}X({\mathrm{e}}^t?1)$. We point out that $U$ is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponential functional $\stackrel{?}{I}?{\int }_0^{\infty }exp\left({\stackrel{?}{\xi }}_s\right)\mathrm{d}s$, where $\stackrel{?}{\xi }$ is the dual of the real-valued Lévy process $\xi$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if ${\xi }_1?L^1\left(\mathbb{P}\right)$. In that case, we determine the family of Lévy processes $\xi$ for which $U$ fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process $\xi$, and properties of time-substitutions based on additive functionals.     

Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.     

On the centre of mass of a random walk

For a random walk $S_n$ on ${\mathbb{R}}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n=n^{?1}{\sum }_{i=1}^n{S}_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, $G_n$ is recurrent if $d=1$ and transient if $d\ge 2$. In the transient case we show that $G_n$ has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which $G_n$ is transient in $d=1$.     

Regularity theory for Ln-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity

We develop interior $W^{2,p,\mu }$ and $W^{2,\text{BMO}}$ regularity theories for $L^n$-viscosity solutions to fully nonlinear elliptic equations $T(D^{2}u,x)=f(x)$, where T is approximately convex at infinity. Particularly, $W^{2,\text{BMO}}$ regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of $D^{2}T(M)$ are at least $?C{6M6}^{?(1+{\sigma }_0)}$ as $M\to \infty$. $W^{2,\text{BMO}}$ regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of $W^{2,\text{BMO}}$ regularity theory is dense in the space of fully nonlinear uniformly elliptic operators.     

Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: $\left({?}^{\beta }+\frac{\nu }{2}{(?\Delta )}^{\alpha ∕2}\right)u(t,x)=I_t^{\gamma }\left[\rho \left(u(t,x)\right)\stackrel{?}{W}(t,x)\right],t>0,x\in {\mathbb{R}}^d,$where $\stackrel{?}{W}$ is the space–time white noise, $\alpha \in (0,2]$, $\beta \in (0,2)$, $\gamma \ge 0$ and $\nu >0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: $d<2\alpha +\frac{\alpha }{\beta }min(2\gamma ?1,0)$. In some cases, the initial data can be measures. When $\beta \in (0,1]$, we prove the sample path regularity of the solution.     

Mixing time of an unaligned Gibbs sampler on the square

The paper concerns a particular example of the Gibbs sampler and its mixing efficiency. Coordinates of a point are rerandomized in the unit square ${[0,1]}^2$ to approach a stationary distribution with density proportional to $exp(?A^2{(u?v)}^2)$ for $(u,v)\in {[0,1]}^2$ with some large parameter $A$.Diaconis conjectured the mixing time of this process to be $O\left(A^2\right)$ which we confirm in this paper. This improves on the currently known $O(exp\left(A^2\right))$ estimate.     

On a problem of Specker about Euclidean representations of finite graphs

Say that a graph $G$ is representable in ${\mathbb{R}}^n$ if there is a map $f$ from its vertex set into the Euclidean space ${\mathbb{R}}^n$ such that $6f\left(x\right)?f\left(x^{\prime }\right)6=6f\left(y\right)?f\left(y^{\prime }\right)6$ iff $\{x,x^{\prime }\}$ and$\{y,y^{\prime }\}$ are both edges or both non-edges in $G$. The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg in Einhorn and Schoenberg (1966): if $G$ finite is neither complete nor independent, then it is representable in ${\mathbb{R}}^{\left|G\right|?2}$. A similar result also holds in the case of finite complete edge-colored graphs.     

Harborth constants for certain classes of metacyclic groups

The Harborth constant of a finite group $G$ is the smallest integer $k\ge exp\left(G\right)$ such that any subset of $G$ of size $k$ contains $exp\left(G\right)$ distinct elements whose product is 1. Generalizing previous work on the Harborth constants of dihedral groups, we compute the Harborth constants for the metacyclic groups of the form $H_{n,m}=〈x,y\mid x^n=1,y^2=x^m,yx=x^{?1}y〉$. We also solve the “inverse” problem of characterizing all smaller subsets that do not contain $exp\left(H_{n,m}\right)$ distinct elements whose product is 1.     

Divisibility of L-polynomials for a family of Artin–Schreier curves

In this paper we consider the curves $C_k^{(p,a)}:y^p?y=x^{p^k+1}+ax$ defined over ${\mathbb{F}}_p$ and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves $C_k^{(p,a)}$. Our proof involves finding an exact formula for the number of ${\mathbb{F}}_{p^n}$-rational points on $C_k^{(p,a)}$ for all n, and uses a result we proved elsewhere about the number of rational points on supersingular curves.     

Stable-like fluctuations of Biggins’ martingales

Let ${\left(W_n\left(\theta \right)\right)}_{n\in {\mathbb{N}}_0}$ be Biggins’ martingale associated with a supercritical branching random walk, and let $W\left(\theta \right)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1\left(\theta \right)$ belongs to the domain of normal attraction of an $\alpha$-stable distribution for some $\alpha \in (1,2)$, then, as $n\to \infty$, there is weak convergence of the tail process ${(W\left(\theta \right)?W_{n?k}\left(\theta \right))}_{k\in {\mathbb{N}}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $\alpha$-stable marginals.