Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {_Zs≥ζ(s),s∈S}$\{Z_s\ge \zeta \left(s\right),s\in S\}$, where Z$Z$ is a continuous Gaussian process with stationary increments, ζ$\zeta$ is a function that belongs to the reproducing kernel Hilbert space R$R$ of process Z$Z$, and S⊂R$S\subset \mathbb{R}$ is compact. The main problem considered in this paper is identifying the function β^∗∈R${\beta }^{\ast }\in R$ satisfying β^∗(s)≥ζ(s)${\beta }^{\ast }\left(s\right)\ge \zeta \left(s\right)$ on S$S$ and having minimal R$R$-norm. The smoothness (mean square differentiability) of Z$Z$ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s$\zeta \left(s\right)=s$ for s∈[0,1]$s\in [0,1]$ and Z$Z$ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.     

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X$X$ in a bounded κ$\kappa$-fat open set; if u$u$ is a positive harmonic function with respect to X$X$ in a bounded κ$\kappa$-fat open set D$D$ and h$h$ is a positive harmonic function in D$D$ vanishing on D^c$D^c$, then the non-tangential limit of u/h$u/h$ exists almost everywhere with respect to the Martin-representing measure of h$h$.     

Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

Uniqueness of the Cheeger set of a convex body

We prove that if C⊂R^N$C\subset {\mathbb{R}}^N$ is an open bounded convex set, then there is only one Cheeger set inside C$C$ and it is convex. A Cheeger set of C$C$ is a set which minimizes the ratio perimeter over volume among all subsets of C$C$.     

Disjoint cycles with different length in 4-arc-dominated digraphs

A d$d$-arc-dominated digraph is a digraph D$D$ of minimum out-degree d$d$ such that for every arc (x,y)$(x,y)$ of D$D$, there exists a vertex u$u$ of D$D$ of out-degree d$d$ such that (u,x)$(u,x)$ and (u,y)$(u,y)$ are arcs of D$D$. Henning and Yeo [Vertex disjoint cycles of different length in digraphs, SIAM J. Discrete Math. 26 (2012) 687–694] conjectured that a digraph with minimum out-degree at least four contains two vertex-disjoint cycles of different length. In this paper, we verify this conjecture for 4-arc-dominated digraphs.     

Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic p$p$-Laplace equation, p∈(1,2)$p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1)$r\in (0,1)$ on a bounded open domain Λ⊂R^d$\Lambda \subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p$p$ and r$r$ respectively (in the Hilbert spaces L²(Λ)$L^2\left(\Lambda \right)$, H⁻¹(Λ)$H^{-1}\left(\Lambda \right)$ respectively). The highly singular limit case p=1$p=1$ is treated with the help of stochastic evolution variational inequalities, where P$\mathbb{P}$-a.s. convergence, uniformly in time, is established.     

Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence

We give a functional limit theorem for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion and α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence process involving sub-fractional Brownian motion. We also give an analogous result for the system without branching and d<α$d<\alpha$, which involves fractional Brownian motion. We use a space–time random field approach.     

A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail

We study the probability distribution F(u)$F\left(u\right)$ of the maximum of smooth Gaussian fields defined on compact subsets of R^d${\mathbb{R}}^d$ having some geometric regularity.     

An upper bound for a valence of a face in a parallelohedral tiling

Consider a face-to-face parallelohedral tiling of R^d${\mathbb{R}}^d$ and a (d−k)$(d-k)$-dimensional face F$F$ of the tiling. We prove that the valence of F$F$ (i.e. the number of tiles containing F$F$ as a face) is not greater than 2^k$2^k$. If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay k$k$-cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.     

Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions

We give functional limit theorems for the fluctuations of the rescaled occupation time process of a critical branching particle system in R^d${\mathbb{R}}^d$ with symmetric α$\alpha$-stable motion in the cases of critical and large dimensions, d=2α$d=2\alpha$ and d>2α$d>2\alpha$. In a previous paper [T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., this issue.] we treated the case of intermediate dimensions, α<d<2α$\alpha <d<2\alpha$, which leads to a long-range dependence limit process. In contrast, in the present cases the limits are generalized Wiener processes. We use the same space–time random field method of the previous paper, the main difference being that now the tightness requires a new approach and the proofs are more difficult. We also give analogous results for the system without branching in the cases d=α$d=\alpha$ and d>α$d>\alpha$.     

Monotonicity of the reflected Bessel transition density on the diagonal

For α∈R$\alpha \in \mathbb{R}$, let p^R(t,x,x)$p^R(t,x,x)$ denote the diagonal of the transition density of the α$\alpha$-Bessel process in (0,1]$(0,1]$, killed at 0 and reflected at 1. As a function of x$x$, if either α≥3$\alpha \ge 3$ or α=1$\alpha =1$, then for t>0$t>0$, the diagonal is nondecreasing. This monotonicity property fails if 1≠α<3$1\ne \alpha <3$.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Weak convergence of censored and reflected stable processes

It is shown that if a sequence of open n$n$-sets _Dk$D_k$ increases to an open n$n$-set D$D$ then reflected stable processes in _Dk$D_k$ converge weakly to the reflected stable process in D$D$ for every starting point x$x$ in D$D$. The same result holds for censored α$\alpha$-stable processes for every x$x$ in D$D$ if D$D$ and _Dk$D_k$ satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains.