Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process

We consider a discrete-time two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ on $\mathbb{Z}_{+}^{2}$ with a background process {J _n } on a finite set, where individual processes $\{L_{n}^{(1)}\}$ and $\{L_{n}^{(2)}\}$ are both skip free. We assume that the joint process $\{Y_{n}\}=\{(L_{n}^{(1)},L_{n}^{(2)},J_{n})\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ are modulated depending on the state of the background process {J _n }. This modulation is space homogeneous, but the transition probabilities in the inside of $\mathbb{Z}_{+}^{2}$ and those around the boundary faces may be different. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process, and obtain the decay rates of the stationary distribution in the coordinate directions. We also distinguish the case where the stationary distribution asymptotically decays in the exact geometric form, in the coordinate directions.     

Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures

We present three sets of results for the stationary distribution of a two-dimensional semimartingale-reflecting Brownian motion (SRBM) that lives in the non-negative quadrant. The SRBM data can equivalently be specified by three geometric objects, an ellipse and two lines, in the two-dimensional Euclidean space. First, we revisit the variational problem (VP) associated with the SRBM. Building on Avram et al. (Queueing Syst. 37: 259–289, 2001), we show that the value of the VP at a point in the quadrant is equal to the optimal value of a linear function over a convex domain. Depending on the location of the point, the convex domain is either $\mathcal{D}^{(1)}$ or $\mathcal{D}^{(2)}$ or $\mathcal{D}^{(1)}\cap \mathcal{D}^{(2)},$ where each $\mathcal{D}^{(i)},$ $i=1, 2,$ can easily be described by the three geometric objects. Our results provide a geometric interpretation for the value function of the VP and allow one to see geometrically when one edge of the quadrant has influence on the optimal path traveling from the origin to a destination point. Second, we provide a geometric condition that characterizes the existence of a product form stationary distribution. Third, we establish exact tail asymptotics of two boundary measures that are associated with the stationary distribution; a key step in our proof is to sharpen two asymptotic inversion lemmas in Dai and Miyazawa (Stoch. Syst. 1:146–208, 2011) which allow one to infer the exact tail asymptotic of a boundary measure from the singularity of its moment-generating function.     

Central limit theorems for hyperbolic spaces and Jacobi processes on [0,\infty [

We present a unified approach to a couple of central limit theorems for the radial behavior of radial random walks on hyperbolic spaces as well as for time-homogeneous Markov chains on $[0,\infty [$ whose transition probabilities are defined in terms of Jacobi convolutions. The proofs of all central limit theorems are based on corresponding limit results for the associated Jacobi functions $\varphi _{\lambda }^{(\alpha ,\beta )}$ . In particular, we consider the limit $\alpha \rightarrow \infty $ , the limit $\varphi _{i\rho -n\lambda }^{(\alpha ,\beta )}(t/n)$ for $n\rightarrow \infty $ , and the behavior of the Jacobi function $\varphi _{i\rho -\lambda }^{(\alpha ,\beta )}(t)$ for small $\lambda $ . The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the results are known, other improve known ones, and other are new.     

On the stability of \varphi -uniform domains

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n$ . In the sequel, we investigate a class of domains, so called $\varphi $ -uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\varphi $ from $[0,\infty )$ to itself. Finally, we discuss a number of stability properties of $\varphi $ -uniform domains. In particular, we show that the class of $\varphi $ -uniform domains is stable in the sense that removal of a geometric sequence of points from a $\varphi $ -uniform domain yields a $\varphi _1$ -uniform domain.     

Discrepancy, separation and Riesz energy of finite point sets on the unit sphere

For $d \geqslant 2,$ we consider asymptotically equidistributed sequences of $\mathbb S^d$ codes, with an upper bound $\operatorname{\boldsymbol{\delta}}$ on spherical cap discrepancy, and a lower bound Δ on separation. For such sequences, if 0?<?s?<?d, then the difference between the normalized Riesz s energy of each code, and the normalized s-energy double integral on the sphere is bounded above by $\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\,\Delta^{-s}\,N^{-s/d}\big),$ where N is the number of code points. For well separated sequences of spherical codes, this bound becomes $\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\big).$ We apply these bounds to minimum energy sequences, sequences of well separated spherical designs, sequences of extremal fundamental systems, and sequences of equal area points.     

Quasilinear asymptotically periodic Schrödinger equations with critical growth

It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in ${\mathbb{R}^N}$ with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in ${H^1(\mathbb{R}^N)}$ and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration–Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.     

Laplace Equation on a Domain With a Cuspidal Point in Little Hölder Spaces

In this paper, we give new results about existence, uniqueness and regularity properties for solutions of Laplace equation $$\Delta u = h \quad {\rm in} \, \Omega$$ where Ω is a cusp domain. We impose nonhomogeneous Dirichlet conditions on some part of ?Ω. The second member h will be taken in the little Hölder space ${h^{2 \sigma}(\bar{\Omega})}$ with ${\sigma \, \in \, ]0, \, 1/2[}$ . Our approach is based essentially on the study of an abstract elliptic differential equation set in an unbounded domain. We will use the continuous interpolation spaces and the generalized analytic semigroup theory.     

Sharp geometric rigidity of isometries on Heisenberg groups

We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\varepsilon )$ -quasi-isometry on a John domain of the Heisenberg group $\mathbb H ^n, n>1,$ is close to some isometry up to proximity order $\sqrt{\varepsilon }+\varepsilon $ in the uniform norm, and up to proximity order $\varepsilon $ in the $L_p^1$ -norm. We give examples showing the asymptotic sharpness of our results.     

Proof of the BMV conjecture

We prove the BMV (Bessis, Moussa, Villani, [1]) conjecture, which states that the function ${t \mapsto \mathop{\rm Tr}\exp(A-tB)}$ , ${t \geqslant 0}$ , is the Laplace transform of a positive measure on [0,∞) if A and B are ${n \times n}$ Hermitian matrices and B is positive semidefinite. A semi-explicit representation for this measure is given.     

Universal geometric cluster algebras

We consider, for each exchange matrix $B$ , a category of geometric cluster algebras over $B$ and coefficient specializations between the cluster algebras. The category also depends on an underlying ring $R$ , usually $\mathbb {Z},\,\mathbb {Q}$ , or $\mathbb {R}$ . We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over $B$ with universal geometric coefficients, or the universal geometric cluster algebra over $B$ . Constructing universal geometric coefficients is equivalent to finding an $R$ -basis for $B$ (a “mutation-linear” analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan ${\mathcal {F}}_B$ , which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between ${\mathcal {F}}_B$ and $\mathbf{g}$ -vectors. We construct universal geometric coefficients in rank $2$ and in finite type and discuss the construction in affine type.     

On the signed star domination number of regular multigraphs

Let $G$ be a graph with the vertex set $V(G)$ and the edge set $E(G)$ . A function $f: E(G)\longrightarrow \{-1, 1\}$ is said to be a signed star dominating function of $G$ if $\sum _{e \in E_G(v)}f (e)\ge 1 $ , for every $v \in V(G)$ , where $E_G(v) = \{uv\in E(G)\,|\,u \in V (G)\}$ . The minimum values of $\sum _{e \in E_G(v)}f (e)$ , taken over all signed star dominating functions $f$ on $G$ , is called the signed star domination number of $G$ and denoted by $\gamma _{SS}(G)$ . In this paper we determine the signed star domination number of regular multigraphs.     

Strongly dominating sets of reals

We analyze the structure of strongly dominating sets of reals introduced in Goldstern et al. (Proc Am Math Soc 123(5):1573–1581, 1995). We prove that for every ${\kappa < \mathfrak{b}}$ κ < b a ${\kappa}$ κ -Suslin set ${A\subseteq{}^\omega\omega}$ A ? ω ω is strongly dominating if and only if A has a Laver perfect subset. We also investigate the structure of the class l of Baire sets for the Laver category base and compare the σ-ideal of sets which are not strongly dominating with the Laver ideal l ⁰.     

On the Zeros of Laplace Transforms

Suppose that f is a positive, nondecreasing, and integrable function in the interval $(0,1)$ . Then, by Pólya's theorem, all the zeros of the Laplace transform $$F(z) = \int_0^1 {e^{zt} f(t)dt} $$ lie in the left-hand half-plane $\operatorname{Re} z \leqslant 0$ . In this paper, we assume that the additional condition of logarithmic convexity of f in a left-hand neighborhood of the point 1 is satisfied. We obtain the form of the left curvilinear half-plane and also, under the condition $f( + 0) >0$ , the form of the curvilinear strip containing all the zeros of $f(z)$ .     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

On the Stretch Factor of Randomly Embedded Random Graphs

We consider a random graph $\mathcal{G}(n,p)$ whose vertex set $V,$ of cardinality $n,$ has been randomly embedded in the unit square and whose edges, which occur independently with probability $p,$ are given weight equal to the geometric distance between their end vertices. Then each pair $\{u,v\}$ of vertices has a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all $\{u,v\}\subseteq V.$ We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for $p$ not too close to 0 or 1, these bounds are the best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if $n(1-p)$ tends to 0, answering a question of O’Rourke.     

The canonical shrinking soliton associated to a Ricci flow

To every Ricci flow on a manifold ${\mathcal{M}}$ over a time interval ${I\subset\mathbb{R}_-}$ , we associate a shrinking Ricci soliton on the space–time ${\mathcal{M}\times I}$ . We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author Topping (J Reine Angew Math 636:93–122, 2009), and McCann and the second author (Am J Math 132:711–730, 2010); we briefly survey the link between these subjects.     

Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane

A random walk in $\mathbf{Z}_+^2$ spatially homogeneous in the interior, absorbed at the axes, starting from an arbitrary point $(i_0,j_0)$ and with step probabilities drawn on Fig. 1 is considered. The trivariate generating function of probabilities that the random walk hits a given point $(i,j) \in \mathbf{Z}_+^2 $ at a given time $k\ge 0$ is made explicit. Probabilities of absorption at a given time $k$ and at a given axis are found, and their precise asymptotic is derived as the time $k\rightarrow \infty $ . The equivalence of two typical ways of conditioning this random walk to never reach the axes is established. The results are also applied to the analysis of the voter model with two candidates and initially, in the population $\mathbf{Z}$ , four connected blocks of same opinions. Then, a citizen changes his mind at a rate proportional to the number of his neighbors that disagree with him. Namely, the passage from four to two blocks of opinions is studied.     

A minimum principle for the problem of St-Venant in {\mathbb{R}^N, N \geq 2}

This paper deals mainly with the St-Venant problem in a convex domain ?? of ${\mathbb{R}^N, N \geq 2}$ . A minimum principle for a combination of the stress function ${\psi}$ and ${|\nabla \psi|}$ is derived. Some possible applications are indicated.     

Totally geodesic discs in strongly convex domains

We prove that every isometry from the unit disk Δ in ${\mathbb{C}}$ , endowed with the Poincaré distance, to a strongly convex bounded domain Ω of class ${\mathcal{C}^3}$ in ${\mathbb{C}^n}$ , endowed with the Kobayashi distance, is the composition of a complex geodesic of Ω with either a conformal or an anti-conformal automorphism of Δ. As a corollary we obtain that every isometry for the Kobayashi distance, from a strongly convex bounded domain of class ${\mathcal{C}^3}$ in ${\mathbb{C}^n}$ to a strongly convex bounded domain of class ${\mathcal{C}^3}$ in ${\mathbb{C}^m}$ , is either holomorphic or anti-holomorphic.