A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in ${\mathbb{Z}}^d$. Denote by $Z_n\left(z\right)$ the number of particles of generation $n$ located at site $z\in {\mathbb{Z}}^d$. We give the second order asymptotic expansion for $Z_n\left(z\right)$. The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on ${\mathbb{Z}}^d$, which is used in the proof of the main theorem and is of independent interest.     

Analysis of random walks in dynamic random environments via L2-perturbations

We consider random walks in dynamic random environments given by Markovian dynamics on ${\mathbb{Z}}^d$. We assume that the environment has a stationary distribution $\mu$ and satisfies the Poincaré inequality w.r.t. $\mu$. The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $\mu$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of $L^2$- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.     

Fermat-type configurations of lines in P3 and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals I in $\mathbb{K}[x,y,z,w]$ for which the containment $I^{(3)}?I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb{P}}^3$, which resemble Fermat configurations of points in ${\mathbb{P}}^2$, see [14]. All examples exhibiting the failure of the containment $I^{(3)}?I^2$ constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.     

A regularity theory for quasi-linear Stochastic PDEs in weighted Sobolev spaces

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$-domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain $L_p$ and Hölder estimates of both the solution and its gradient.     

Proof of the Hénon–Lane–Emden conjecture in R3

We study the Hénon–Lane–Emden conjecture, which states that there is no non-trivial non-negative solution for the Hénon–Lane–Emden elliptic system whenever the pair of exponents is subcritical. By scale invariance of the solutions and Sobolev embedding on $S^{N?1}$, we prove this conjecture is true for space dimension $N=3$; which also implies the single elliptic equation has no positive classical solutions in ${\mathbb{R}}^3$ when the exponent lies below the Hardy–Sobolev exponent, this covers the conjecture of Phan–Souplet [22] for ${\mathbb{R}}^3$.     

Hod up to ADR+Θ is measurable

Suppose M is a transitive class size model of $A{D}_{\mathbb{R}}+\text{“Θ is regular”}$. M is a minimal model of $A{D}_{\mathbb{R}}+\text{“Θ is measurable”}$ if (i) $\mathbb{R},Ord?M$ (ii) there is $\mu \in M$ such that $M?\text{“}\mathit{\mu }\text{ is a normal }\mathbb{R}\text{-complete measure on Θ”}$ and (iii) for any transitive class size $N?M$ such that $\mathbb{R}?N$, $N?\text{“there is no }\mathbb{R}\text{-complete measure on Θ”}$. Continuing Trang's work in [8], we compute HOD of a minimal model of $A{D}_{\mathbb{R}}+\text{“Θ is measurable”}$.     

Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

On weak uniqueness and distributional properties of a solution to an SDE with α-stable noise

For an SDE driven by a rotationally invariant $\alpha$-stable noise we prove weak uniqueness of the solution under the balance condition $\alpha +\gamma >1$, where $\gamma$ denotes the Hölder index of the drift coefficient. We prove the existence and continuity of the transition probability density of the corresponding Markov process and give a representation of this density with an explicitly given “principal part”, and a “residual part” which possesses an upper bound. Similar representation is also provided for the derivative of the transition probability density w.r.t. the time variable.     

Quasineutral limit of the two-fluid Euler–Poisson system in a bounded domain of R3

The quasineutral limit of the two-fluid Euler–Poisson system (one for ions and another for electrons) in a bounded domain of ${\mathbb{R}}^3$ is rigorously proved by investigating the existence and the stability of boundary layers. The non-penetration boundary condition for velocities and Dirichlet boundary condition for electric potential are considered.     

Hydrodynamic limit for the Ginzburg–Landau ∇ϕ interface model with non-convex potential

The hydrodynamic limit of the Ginzburg–Landau $??$ interface model was derived in Funaki and Spohn (1997) and Nishikawa (2003) for strictly convex potentials. This paper deals with non-convex potentials under suitable assumptions on the free energy and identification of the extremal Gibbs measures which have been recently established at sufficiently high temperature in Cotar and Deuschel (2012). Because of the non-convexity, many difficulties arise, especially, on the identification of equilibrium states. We show the equivalence between the stationarity and the Gibbs property under quite general settings, and we complete the identification of equilibrium states. We also establish some uniform estimates for variances of extremal Gibbs measures.     

Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in RN

In this paper, we study the existence of least energy sign-changing solutions for a Kirchhoff-type problem involving the fractional Laplacian operator. By using the constraint variation method and quantitative deformation lemma, we obtain a least energy nodal solution u_b for the given problem. Moreover, we show that the energy of u_b is strictly larger than twice the ground state energy. We also give a convergence property of u_b as b ↘ 0, where b is regarded as a positive parameter.     

Uniformly bounded representations of SL(2,R)

We compute the “norm” of irreducible uniformly bounded representations of $\mathrm{SL}(2,\mathbb{R})$. We show that the Kunze–Stein version of the uniformly bounded representations has minimal norm in its similarity class of uniformly bounded representations.