Random self-affine multifractal Sierpinski sponges in $${\mathbb{R}^d}$$

In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in \({\mathbb{R}^{d}}\).     

Geometry of \mathbb{Z}^d and the Central Limit Theorem for Weakly Dependent Random Fields

We study the asymptotic distribution of where A is a subset of , A _N = A[–N, N] ^d , v(A) = lim _N card(A _N) (2N+1) ^–d (0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, S _N(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if has a limit F(n; A) as N for each . We also study the class of sets A that satisfy this condition.     

Zero-temperature Glauber dynamics on {\mathbb{Z}^d}

We study zero-temperature Glauber dynamics on ${\mathbb{Z}^d}$ , which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define ${p_c(\mathbb{Z}^d)}$ to be the infimum over p such that the system fixates at ???+??? with probability 1. It is a folklore conjecture that ${p_c(\mathbb{Z}^d) = 1/2}$ for every ${2 \le d \in \mathbb{N}}$ . We prove that ${p_c(\mathbb{Z}^d) \to 1/2}$ as d ?? ??.     

Suitable families of boxes and kernels of staircase starshaped sets in {\mathbb{R}^d}

Let S be an orthogonal polytope in ${\mathbb{R}^d}$ . There exists a suitable family ${\mathcal{C}}$ of boxes with ${S = \cup \{C : C {\rm in} \mathcal{C}\}}$ such that the following properties hold:

The staircase kernel Ker S is a union of boxes in ${\mathcal{C}}$ . Let ${\mathcal{V}}$ be the family of vertices of boxes in ${\mathcal{C}}$ , and let ${v_o\, \epsilon \mathcal{V}}$ . Point v _o belongs to Ker S if and only if v _o sees via staircase paths in S every point w in ${\mathcal{V}}$ . Moreover, these staircase paths may be selected to consist of edges of boxes in ${\mathcal{C}}$ . Let B be a box in ${\mathcal{C}}$ with vertices of B in Ker S. Box B lies in Ker S if and only if, for some b in rel int B and for every translate H of a coordinate hyperplane at ${b, b \epsilon}$ Ker (H ∩ S). For point p in S, p belongs to Ker S if and only if, for every x in S, there exist some p ? x geodesic λ (p, x) and some corresponding ${\mathcal{C}}$ - chain D containing λ (p, x) such that D is staircase starshaped at p.

     

The Monge problem in $ {\mathbb{R}^d} $: Variations on a theme

In a recent paper, the authors proved that, under natural assumptions on the first marginal, the Monge problem in \mathbbR^d {\mathbb{R}^d} for the cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will give a proof of the result in the simpler case of a uniformly convex norm, and we will also use very recent results by Ahmad, Kim, and McCann. This allows us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set in the particular case considered in this paper is original. Bibliography: 22 titles.     

Convergence of symmetric Markov chains on {\mathbb{Z}^d}

For each n let ${Y^{(n)}_t}$ be a continuous time symmetric Markov chain with state space ${n^{-1} \mathbb{Z}^d}$ . Conditions in terms of the conductances are given for the convergence of the ${Y^{(n)}_t}$ to a symmetric Markov process Y _t on ${\mathbb{R}^d}$ . We have weak convergence of $\{{Y^{(n)}_t: t \leq t_0\}}$ for every t ₀ and every starting point. The limit process Y has a continuous part and may also have jumps.     

Dimensions of random self-affine multifractal Sierpinski sponges in {{\mathbb{R}}^d}

In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in ${{\mathbb{R}}^{d}}$ .     

Lower large deviations for the maximal flow through a domain of {\mathbb{R}^d} in first passage percolation

We consider the standard first passage percolation model in the rescaled graph ${\mathbb{Z}^d/n}$ for d??? 2, and a domain ?? of boundary ?? in ${\mathbb{R}^d}$ . Let ??¹ and ??² be two disjoint open subsets of ??, representing the parts of ?? through which some water can enter and escape from ??. We investigate the asymptotic behaviour of the flow ${\phi_n}$ through a discrete version ?? _n of ?? between the corresponding discrete sets ${\Gamma^{1}_{n}}$ and ${\Gamma^{2}_{n}}$ . We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of ${\phi_n/ n^{d-1}}$ below a certain constant are of surface order.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      13. Meteor process on {mathbb Z}^d      Krzysztof Burdzy 《Probability Theory and Related Fields》2015,163(3-4):667-711       14. Sampling and reconstruction in shift-invariant spaces on mathbb {R}^d      A. Antony Selvan  R. Radha 《Annali di Matematica Pura ed Applicata》2015,194(6):1683-1706       15. On Weak*-Convergence in \boldsymbol{H^1_L(\boldsymbol{\mathbb R}^d)}      Luong Dang Ky 《Potential Analysis》2013,39(4):355-368 Let L?=???Δ?+?V be a Schrödinger operator on $\mathbb R^d$ , d?≥?3, where V is a nonnegative function, $V\ne 0$ , and belongs to the reverse Hölder class RH d/2. In this paper, we prove a version of the classical theorem of Jones and Journé on weak*-convergence in the Hardy space $H^1_L(\mathbb R^d)$ .      16. On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}      Maria Luiza Leite  Jaime Ripoll 《Results in Mathematics》2011,60(1-4):351-360 Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M n having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$       17. Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions      Qun Chen  Qing Cui 《Results in Mathematics》2010,57(3-4):319-333 In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M n , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .      18. Discrete fractal dimensions of the ranges of random walks in {{\mathbb Z}^d} associate with random conductances      Yimin Xiao  Xinghua Zheng 《Probability Theory and Related Fields》2013,156(1-2):1-26 Let ${X= \{X_t, t \ge 0\}}$ be a continuous time random walk in an environment of i.i.d. random conductances ${\{\mu_e \in [1,\infty), e \in E_d\}}$ , where E d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice ${\mathbb{Z}^d}$ and d ≥ 3. Let ${{\rm R} = \{x \in \mathbb{Z}^d: X_t = x {\rm \,for\, some}\,t \ge 0\}}$ be the range of X. It is proved that, for almost every realization of the environment, dimH R = dimP R = 2 almost surely, where dimH and dimP denote, respectively, the discrete Hausdorff and packing dimension. Furthermore, given any set ${A \subseteq \mathbb{Z}^d}$ , a criterion for A to be hit by X t for arbitrarily large t > 0 is given in terms of dimH A. Similar results for Bouchoud’s trap model in ${\mathbb{Z}^d}$ (d ≥ 3) are also proven.      19. Multibubble analysis on N-Laplace equation in {\mathbb{R}^N}      Adimurthi  Yunyan Yang 《Calculus of Variations and Partial Differential Equations》2011,40(1-2):1-14 In this note, we describe the asymptotic behavior of sequences of solutions to N-Laplace equations with critical exponential growth in smooth bounded domain in ${\mathbb{R}^N}$ . Precisely we prove multibubble phenomena and obtain an energy inequality for those concentrating solutions. In fact we partly extend the corresponding two-dimensional results of Adimurthi and Struwe (J Funct Anal 175:125?C167, 2000) and Druet (Duke Math J 132:217?C269, 2006) to high dimensional case.      20. A Version of the {\cal G} \rm{-conditional Bipolar Theorem in} {L^0 (\mathbb{R}^{d}_{+};\Omega}, {\cal F},\mathbb{P})      B. Bouchard 《Journal of Theoretical Probability》2005,18(2):439-467 Motivated by applications in financial mathematics, Ref. 3 showed that, although fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of : if we place this space in polarity with itself, the bipolar of a set of non-negative random variables is equal to its closed (in probability), solid, convex hull. This result was extended by Ref. 1 in the multidimensional case, replacing by a closed convex cone K of [0, )d, and by Ref. 12 who provided a conditional version in the unidimensional case. In this paper, we show that the conditional bipolar theorem of Ref. 12 can be extended to the multidimensional case. Using a decomposition result obtained in Ref. 3 and Ref. 1, we also remove the boundedness assumption of Ref. 12 in the one dimensional case and provide less restrictive assumptions in the multidimensional case. These assumptions are completely removed in the case of polyhedral cones K.

On Weak*-Convergence in \boldsymbol{H^1_L(\boldsymbol{\mathbb R}^d)}

Let L?=???Δ?+?V be a Schrödinger operator on $\mathbb R^d$ , d?≥?3, where V is a nonnegative function, $V\ne 0$ , and belongs to the reverse Hölder class RH _d/2. In this paper, we prove a version of the classical theorem of Jones and Journé on weak*-convergence in the Hardy space $H^1_L(\mathbb R^d)$ .     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

Discrete fractal dimensions of the ranges of random walks in {{\mathbb Z}^d} associate with random conductances

Let ${X= \{X_t, t \ge 0\}}$ be a continuous time random walk in an environment of i.i.d. random conductances ${\{\mu_e \in [1,\infty), e \in E_d\}}$ , where E _d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice ${\mathbb{Z}^d}$ and d ≥ 3. Let ${{\rm R} = \{x \in \mathbb{Z}^d: X_t = x {\rm \,for\, some}\,t \ge 0\}}$ be the range of X. It is proved that, for almost every realization of the environment, dim_H R = dim_P R = 2 almost surely, where dim_H and dim_P denote, respectively, the discrete Hausdorff and packing dimension. Furthermore, given any set ${A \subseteq \mathbb{Z}^d}$ , a criterion for A to be hit by X _t for arbitrarily large t > 0 is given in terms of dim_H A. Similar results for Bouchoud’s trap model in ${\mathbb{Z}^d}$ (d ≥ 3) are also proven.     

Multibubble analysis on N-Laplace equation in {\mathbb{R}^N}

In this note, we describe the asymptotic behavior of sequences of solutions to N-Laplace equations with critical exponential growth in smooth bounded domain in ${\mathbb{R}^N}$ . Precisely we prove multibubble phenomena and obtain an energy inequality for those concentrating solutions. In fact we partly extend the corresponding two-dimensional results of Adimurthi and Struwe (J Funct Anal 175:125?C167, 2000) and Druet (Duke Math J 132:217?C269, 2006) to high dimensional case.     

A Version of the {\cal G} \rm{-conditional Bipolar Theorem in} {L^0 (\mathbb{R}^{d}_{+};\Omega}, {\cal F},\mathbb{P})

Motivated by applications in financial mathematics, Ref. 3 showed that, although fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of : if we place this space in polarity with itself, the bipolar of a set of non-negative random variables is equal to its closed (in probability), solid, convex hull. This result was extended by Ref. 1 in the multidimensional case, replacing by a closed convex cone K of [0, )^d, and by Ref. 12 who provided a conditional version in the unidimensional case. In this paper, we show that the conditional bipolar theorem of Ref. 12 can be extended to the multidimensional case. Using a decomposition result obtained in Ref. 3 and Ref. 1, we also remove the boundedness assumption of Ref. 12 in the one dimensional case and provide less restrictive assumptions in the multidimensional case. These assumptions are completely removed in the case of polyhedral cones K.