Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

Long-time existence for signed solutions of the heat equation with a noise term

Summary.   Let ? be the circle [0,J] with the ends identified. We prove long-time existence for the following equation.

Virtual Eigenvalues of the High Order Schrödinger Operator I

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space $$L_2 (\mathbb{R}^d ),$$ where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and $$\lim _{|{\mathbf{x}}| \to \infty } V({\mathbf{x}}) = 0.$$ We obtain an asymptotic expansion as $$\gamma \uparrow 0$$of the bottom negative eigenvalue of H_γ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H₀. Furthermore, we extract a finite-rank portion Φ(λ) from the Birman- Schwinger operator $$X_V (\lambda ) = V^{\frac{1} {2}} R_\lambda (H_0 )V^{\frac{1}{2}} ,$$ which yields the leading terms for the desired asymptotic expansion.     

Asymptotic behavior of the critical probability for ρ-percolation in high dimensions

We consider oriented bond or site percolation on ℤ ^d ₊. In the case of bond percolation we denote by P _p the probability measure on configurations of open and closed bonds which makes all bonds of ℤ ^d ₊ independent, and for which P _p {e is open} = 1 −P _p e {is closed} = p for each fixed edge e of ℤ ^d ₊. We take X(e) = 1 (0) if e is open (respectively, closed). We say that ρ-percolation occurs for some given 0 < ρ≤ 1, if there exists an oriented infinite path v ₀ = 0, v ₁, v ₂, …, starting at the origin, such that lim inf _{n →∞} (1/n) ∑ _i=1 ⁿ X(e _i ) ≥ρ, where e _i is the edge {v _i−1 , v _i }. [MZ92] showed that there exists a critical probability p _c = p _c (ρ, d) = p _c (ρ, d, bond) such that there is a.s. no ρ-percolation for p < p _c and that P _p {ρ-percolation occurs} > 0 for p > p _c . Here we find lim _{d →∞} d ^1/ρ p _c (ρd, bond) = D ₁ , say. We also find the limit for the analogous quantity for site percolation, that is D ₂ = lim _{d →∞} d ^1/ρ p _c (ρ, d, site). It turns out that for ρ < 1, D ₁ < D ₂ , and neither of these limits equals the analogous limit for the regular d-ary trees. Received: 7 January 1999 / Published online: 14 June 2000     

Left-inverses of fractional Laplacian and sparse stochastic processes

The fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} commutes with the primary coordination transformations in the Euclidean space ℝ ^d : dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I _γ which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I _γ to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/p), there exists a Schwartz function f such that I _γ f is not p-integrable. We then introduce the new unique left-inverse I _{γ, p} of the fractional Laplacian (-\triangle)^g/2(-\triangle)^{\gamma/2} with the property that I _{γ, p} is dilation-invariant (but not translation-invariant) and that I _{γ, p} f is p-integrable for any Schwartz function f. We finally apply that linear operator I _{γ, p} with p = 1 to solve the stochastic partial differential equation (-\triangle)^g/2 F = w(-\triangle)^{\gamma/2} \Phi=w with white Poisson noise as its driving term w.     

Heat Equations with Fractional White Noise Potentials

This paper is concerned with the following stochastic heat equations: where w ^H is a time independent fractional white noise with Hurst parameter H=(h ₁ , h ₂ ,..., h _d ) , or a time dependent fractional white noise with Hurst parameter H=(h ₀ , h ₁ ,..., h _d ) . Denote |H|=h ₁ +h ₂ +...+h _d . When the noise is time independent, it is shown that if ? <h _i <1 for i=1, 2,..., d and if |H|>d-1 , then the solution is in L ₂ and the L ₂ -Lyapunov exponent of the solution is estimated. When the noise is time dependent, it is shown that if ? <h _i <1 for i=0, 1,..., d and if |H|>d- 2 /( 2h ₀ -1 ) , the solution is in L ₂ and the L ₂ -Lyapunov exponent of the solution is also estimated. A family of distribution spaces S _ρ , ρ∈ \RR , is introduced so that every chaos of an element in S _ρ is in L ₂ . The Lyapunov exponents in S _ρ of the solution are also estimated. Accepted 10 October 2000. Online publication 19 February 2001.     

Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group

Denoting Δ_? the Laplacian operator on the (2N+1)-dimensional Heisenberg group ? ^N , we prove some nonexistence results for solutions of inequalities of the three types

A Faber-Krahn inequality for the Laplacian with Generalised Wentzell boundary conditions

We give an extension of the Faber-Krahn inequality to the Laplacian Δ on bounded Lipschitz domains , with generalised Wentzell boundary conditions on ∂Ω, where β, γ are nonzero real constants. We prove that when β, γ > 0, the ball B minimises the first eigenvalue with respect to all Lipschitz domains Ω of the same volume as B, and that B is the unique minimiser amongst C ²-domains. We also consider β, γ not both positive, and slightly extend what is known about the associated Wentzell operator and its resolvent in addition to considering an analogue of the Faber-Krahn inequality. This is based on the recent extension of the Faber-Krahn inequality to the Robin Laplacian. We also give a version of Cheeger’s inequality for the Wentzell Laplacian when β, γ > 0.      

Heat kernel estimates for jump processes of mixed types on metric measure spaces

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity where ν is a probability measure on , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c ₀(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ₁ and γ₂, where either γ₂ ≥ γ₁ > 0 or γ₁ = γ₂ = 0. This example contains mixed symmetric stable processes on as well as mixed relativistic symmetric stable processes on . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes. Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday. The research of Zhen-Qing Chen is supported in part by NSF Grants DMS-0303310 and DMS-06000206. The research of Takashi Kumagai is supported in part by the Grant-in-Aid for Scientific Research (B) 18340027.     

Large solutions to the p-Laplacian for large p

In this work we consider the behaviour for large values of p of the unique positive weak solution u _p to Δ _p u = u ^q in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u _p as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit.     

Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on ℤ ^d at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap of the generator of the dyamics in a box of side L centered at the origin scales like L ⁻². Such an estimate is then used to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and α = ? for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition when β crosses the critical value β _c of the pure system. Received: 10 April 2000 / Revised version: 23 October 2000 / Published online: 5 June 2001     

A stochastic approximation algorithm with multiplicative step size modification

An algorithm of searching a zero of an unknown function ϕ: ℝ → ℝ is considered: x _t = x _t−1 − γ _t−1 y _t , t = 1, 2, ..., where y _t = ϕ(x _t−1) + ξ _t is the value of ϕ measured at x _t−1 and ξ _t is the measurement error. The step sizes γ _t > 0 are modified in the course of the algorithm according to the rule: γ _t = min{uγ _t−1, } if y _t−1 y _t > 0, and γ _t = dγ _t−1, otherwise, where 0 < d < 1 < u, > 0. That is, at each iteration γ _t is multiplied either by u or by d, provided that the resulting value does not exceed the predetermined value . The function ϕ may have one or several zeros; the random values ξ _t are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on ϕ, ξ _t , and , the conditions on u and d guaranteeing a.s. convergence of the sequence {x _t }, as well as a.s. divergence, are determined. In particular, if P(ξ ₁ > 0) = P (ξ ₁ < 0) = 1/2 and P(ξ ₁ = x) = 0 for any x ∈ ℝ, one has convergence for ud < 1 and divergence for ud > 1. Due to the multiplicative updating rule for γ _t , the sequence {x _t } converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximate one of the zeros of ϕ. By adjusting the parameters u and d, one can reach arbitrarily high precision of the approximation; higher accuracy is obtained at the expense of lower convergence rate.      

Density Results in Sobolev Spaces Whose Elements Vanish on a Part of the Boundary

This paper is devoted to the study of the subspace ofW ^m,r of functions that vanish on a part γ ₀ of the boundary. The author gives a crucial estimate of the Poincaré constant in balls centered on the boundary of γ ₀. Then, the convolution-translation method, a variant of the standard mollifier technique, can be used to prove the density of smooth functions that vanish in a neighborhood of γ ₀, in this subspace. The result is first proved for m = 1, then generalized to the case where m ≥ 1, in any dimension, in the framework of Lipschitz-continuous domain. However, as may be expected, it is needed to make additional assumptions on the boundary of γ ₀, namely that it is locally the graph of some Lipschitz-continuous function.     

The divergence of fluctuations for shape in first passage percolation

We consider the first passage percolation model on Z ^d for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R ^d and It is well known that if p < p _c , there exists a compact shape B _d ⊂ R ^d such that for all > 0, t B _d (1 − ) ⊂ B(t) ⊂ tB _d (1 + ) and G(t)(1 − ) ⊂ B(t) ⊂ G(t)(1 + ) eventually w.p.1. We denote the fluctuations of B(t) from tB _d and G(t) by In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB _d ) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting. Research supported by NSF grant DMS-0405150     

Conditions for the existence of a graph with given diameter, connectivity, and ball diversity vector

Given some arbitrary integers d ≥ 2, ? ? 1 and an integer vector $ \bar \tau Given some arbitrary integers d ≥ 2, ϰ ⩾ 1 and an integer vector = (τ ₀, τ ₁, …, τ _d ) with τ ₀ ⩾ τ ₁ ⩾ … ⩾ τ _d = 1 and τ _{d − 1} ⩾ d ²ϰ + 3, the existence is proved of a graph of diameter d and connectivity ϰ whose ball diversity vector is . Moreover, the nonexistence is proved of a graph of diameter d with connectivity ϰ and ball diversity vector (τ ₀, τ ₁, …, τ _d ), where τ ₀ < (d − 1)ϰ + 2. Original Russian Text ? K.L. Rychkov, 2007, published in Diskretnyi Analiz i Issledovanie Operatsii, Ser. 1, 2007, Vol. 14, No. 4, pp. 43–56.     

A central limit theorem for “critical” first-passage percolation in two dimensions

Summary. Consider (independent) first-passage percolation on the edges of ℤ ² . Denote the passage time of the edge e in ℤ ² by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C ₀ } = 0 for some constant C ₀ >0 and that E[t ^δ (e)]<∞ for some δ>4. Denote by b _0,n the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T( 0 ,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0<C ₁ , C ₂ <∞ and γ _n such that C ₁ ( log n) ^1/2 ≦γ _n ≦ C ₂ ( log n) ^1/2 and such that γ _n ⁻¹ [b _0,n −Eb _0,n ] and (√ 2γ _n ) ⁻¹ [T( 0 ,nu) − ET( 0 ,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed). A similar result holds for the site version of first-passage percolation on ℤ ² , when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C ₀ } = p _c (ℤ ² , site ) := critical probability of site percolation on ℤ ² , and E[t ^δ (u)]<∞ for some δ>4. Received: 6 February 1996 / In revised form: 17 July 1996     

On States of Total Weighted Occupation Times for Superdiffusions

Suppose X is a superdiffusion in R ^d with general branching mechanism ψ, and Y _{r D} denotes the total weighted occupation time of X in a bounded smooth domain D. We discuss the conditions on ψ to guarantee that Y _{r D} has absolutely continuous states. And for particular ψ₍ z) = z ^1+β, 0 < β≤ 1, we prove that, in the case d < 2 + 2/β, Y _{r D} is absolutely continuous with respect to the Lebesgue measure in , whereas in the case d > 2 + 2/β, it is singular. As we know the absolute continuity and singularity of Y _{r D} have not been discussed before. Received February 1, 1999, Revised February 25, 2000, Accepted March 9, 2000     

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d≥ 3 under diffusive rescaling starting from the Bernoulli product measure with density 0 < α < 1. We prove that the density fluctuation field Y ^N _t converges to a generalized Ornstein–Uhlenbeck process, which is formally the solution of the stochastic differential equatin dY _t = ?Y _t dt + dB ^∇ _t , where ? is a second order differential operator and B ^∇ _t is a mean zero Gaussian field with known covariances. Received: 31 May 1999 / Revised version: 15 June 2000 / Published online: 24 January 2001     

A nucleation-and-growth model

Summary. We consider the following simple nucleation-and-growth model. On the lattice ℤ ^d , starting with all sites unoccupied, a site becomes occupied at rate e ^−ℬΓ if it has no occupied neighbors, at rate ɛ= e ^−βγ if it has 1 occupied neighbor, and at rate 1 if it has 2 or more occupied neighbors. Occupied sites remain occupied forever. The parameters Γ≧γ are fixed, and we are interested in the behavior of the system as β→∞. We show that the relaxation time of this system scales as e ^βκc , where κ _c = max {γ,( Γ + γ)/(d+1)}. Received: 20 February 1996 / In revised form: 15 June 1996     

Computing the genus of the 2-amalgamations of graphs

The above authors [2] and S. Stahl [3] have shown that if a graphG is the 2-amalgamation of subgraphsG ₁ andG ₂ (namely ifG=G ₁∪G ₂ andG ₁∩G ₂={x, y}, two distinct points) then the orientable genus ofG,γ(G), is given byγ(G)=γ(G ₁)+γ(G ₂)+ε, whereε=0,1 or −1. In this paper we sharpen that result by giving a means by whichε may be computed exactly. This result is then used to give two irreducible graphs for each orientable surface.