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1.
Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤd, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant,
and ξ = {ξ(x): x∈ℤ
d
} is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate,
then the solution u is asymptotically intermittent.
In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the
vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e
s
/θ] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result
is that, for fixed x, y∈ℤ
d
and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w
ρ∥−2
ℓ2Σz ∈ℤd
w
ρ(x+z)w
ρ(y+z). In this expression, ρ = θ/κ while w
ρ:ℤd→ℝ+ is given by w
ρ = (v
ρ)⊗
d
with v
ρ: ℤ→ℝ+ the unique centered ground state (i.e., the solution in ℓ2(ℤ) with minimal l
2-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞).
empty
It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation
coefficient of u(x, t) and u(y, t) converges to δ
x, y
(resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation
structure.
Received: 5 March 1997 / Revised version: 21 September 1998 相似文献
2.
Let Zjt, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential
equations
where the matrix A(x)=(Aij(x))1≤ i, j ≤ d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem
method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let
λ2 > λ1 > 0. We show that for any two vectors a, b∈ ℝd with |a|, |b| ∈ (λ1, λ2) and p sufficiently large, the Lp-norm of the operator whose Fourier multiplier is (|u · a|α - |u · b|α) / ∑j=1d |ui|α is bounded by a constant multiple of |a−b|θ for some θ > 0, where u=(u1 , . . . , ud) ∈ ℝd. We deduce from this the Lp-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|α / ∑j=1d |ui|α, where u=(u1,...,ud) and a is a continuous function on ℝd with |a(x)|∈ (λ1, λ2) for all x∈ ℝd.
Research partially supported by NSF grant DMS-0244737.
Research partially supported by NSF grant DMS-0303310. 相似文献
3.
Hyun Jae Yoo 《Mathematische Zeitschrift》2006,252(1):27-48
We consider fermion (or determinantal) random point fields on Euclidean space ℝd. Given a bounded, translation invariant, and positive definite integral operator J on L2(ℝd), we introduce a determinantal interaction for a system of particles moving on ℝd as follows: the n points located at x1,· · ·,xn ∈ ℝd have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)−1 is a Gibbs measure admitted to the specification. 相似文献
4.
Summary. This is a continuation of our previous work [6] on the investigation of intermittency for the parabolic equation (∂/∂t)u=Hu on ℝ+×ℤ
d
associated with the Anderson Hamiltonian H=κΔ+ξ(·) for i.i.d. random potentials ξ(·). For the Cauchy problem with nonnegative
homogeneous initial condition we study the second order asymptotics of the statistical moments <u(t,0)
p
> and the almost sure growth of u(t,0) as t→∞. We point out the crucial role of double exponential tails of ξ(0) for the formation of high intermittent peaks of the
solution u(t,·) with asymptotically finite size. The challenging motivation is to achieve a better understanding of the geometric structure
of such high exceedances which in one or another sense provide the essential contribution to the solution.
Received: 10 December 1996 / In revised form: 30 September 1997 相似文献
5.
Sandra Cerrai 《Probability Theory and Related Fields》1999,113(1):85-114
In the present paper we consider the transition semigroup P
t
related to some stochastic reaction-diffusion equations with the nonlinear term f having polynomial growth and satisfying some dissipativity conditions. We are proving that it has a regularizing effect in
the Banach space of continuous functions , where ⊂ℝ
d
is a bounded open set. In L
2() the only result proved is the strong Feller property, for d=1. Here we are able to prove that if f∈C
∞(ℝ) and d≤3, then for any and t>0. An important application is to the study of the ergodic properties of the system. These results are also of interest for
some problem in stochastic control.
Received: 20 August 1997 / Revised version: 27 May 1998 相似文献
6.
Do Yong Kwon 《Acta Mathematica Hungarica》2011,131(3):285-294
Let f(x)=a
d
x
d
+a
d−1
x
d−1+⋅⋅⋅+a
0∈ℝ[x] be a reciprocal polynomial of degree d. We prove that if the coefficient vector (a
d
,a
d−1,…,a
0) or (a
d−1,a
d−2,…,a
1) is close enough, in the l
1-distance, to the constant vector (b,b,…,b)∈ℝ
d+1 or ℝ
d−1, then all of its zeros have moduli 1. 相似文献
7.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
8.
We consider the parametric programming problem (Q
p
) of minimizing the quadratic function f(x,p):=x
T
Ax+b
T
x subject to the constraint Cx≤d, where x∈ℝ
n
, A∈ℝ
n×n
, b∈ℝ
n
, C∈ℝ
m×n
, d∈ℝ
m
, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q
p
) are denoted by M(p) and M
loc
(p), respectively. It is proved that if the point-to-set mapping M
loc
(·) is lower semicontinuous at p then M
loc
(p) is a nonempty set which consists of at most ?
m,n
points, where ?
m,n
= is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.
Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000 相似文献
9.
M. S. Bichegkuev 《Differential Equations》2012,48(6):769-778
We consider the linear difference operator (Dx)(t) = x(t) − B(t)x(t − h), t ∈ ℝ, where h ∈ ℝ is a nonzero number and B is a continuous bounded operator-valued function on ℝ ranging in the algebra of linear bounded operators on a Banach space. We obtain necessary and sufficient conditions for the continuous invertibility of D in the space of continuous bounded vector functions and present a formula for the inverse operator. 相似文献
10.
Random projection methods give distributions over k×d matrices such that if a matrix Ψ (chosen according to the distribution) is applied to a finite set of vectors x
i
∈ℝ
d
the resulting vectors Ψx
i
∈ℝ
k
approximately preserve the original metric with constant probability. First, we show that any matrix (composed with a random
±1 diagonal matrix) is a good random projector for a subset of vectors in ℝ
d
. Second, we describe a family of tensor product matrices which we term Lean Walsh. We show that using Lean Walsh matrices as random projections outperforms, in terms of running time, the best known current
result (due to Matousek) under comparable assumptions. 相似文献
11.
Hydrodynamic large scale limit for the Ginzburg-Landau ∇φ interface model was established in [6]. As its next stage this
paper studies the corresponding large deviation problem. The dynamic rate functional is given by
for h=h(t,θ),t∈[0,T],θ∈?
d
, where σ=σ(u) is the surface tension for mean tilt u∈ℝ
d
. Our main tool is H
−1-method expoited by Landim and Yau [9]. The relationship to the rate functional obtained under the static situation by Deuschel
et al. [3] is also discussed.
Received: 22 February 2000 / Revised version: 19 October 2000 / Published online: 5 June 2001 相似文献
12.
In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic
partial differential equation u(t,x)=1+∫0tκΔxu (s,x) ds+∫0t W(ds,x) u (s,x), when the spatial parameter x is continuous, specifically x∈R, and W is a Gaussian field on R+×R that is Brownian in time, but whose spatial distribution is widely unrestricted. We give a partial existence result of the
Lyapunov exponent defined as limt→∞t−1 log u(t,x). Furthermore, we find upper and lower bounds for lim supt→∞t−1 log u(t,x) and lim inft→∞t−1 log u(t,x) respectively, as functions of the diffusion constant κ which depend on the regularity of W in x. Our bounds are sharper, work for a wider range of regularity scales, and are significantly easier to prove than all previously
known results. When the uniform modulus of continuity of the process W is in the logarithmic scale, our bounds are optimal.
This author's research partially supported by NSF grant no. : 0204999 相似文献
13.
Basudeb Dhara 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):401-410
Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x
1, ..., x
n
) a nonzero multilinear polynomial over C. Suppose that x
s
d(x)x
t
∈ Z(R) for all x ∈ {d(f(x
1, ..., x
n
))|x
1, ..., x
n
∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ
C = eRC for some idempotent e in the socle of RC and f(x
1, ..., x
n
)
N
is central-valued in eRCe, where N = s + t + 1.
相似文献
14.
We consider the singular Cauchy problem
, where x: (0, τ) → ℝ, g: (0, τ) → (0, + ∞), h: (0, τ) → (0, + ∞), g(t) ≤ t, and h(t) ≤ t, t ∈ (0, τ), for linear, perturbed linear, and nonlinear equations. In each case, we prove that there exists a nonempty set
of continuously differentiable solutions x: (0, ρ] → (ρ is sufficiently small) with required asymptotic properties.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1344–1358, October, 2005. 相似文献
15.
A. Yu. Pilipenko 《Ukrainian Mathematical Journal》2005,57(8):1262-1274
We consider the properties of a random set ϕ
t
(ℝ
+
d
), where ϕ
t
(x) is a solution of a stochastic differential equation in ℝ
+
d
with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ
t
(ℝ
+
d
) and prove that the Hausdorff dimension of the boundary ∂ϕ
t
(ℝ
+
d
) does not exceed d − 1.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005. 相似文献
16.
András Máthé 《Israel Journal of Mathematics》2008,164(1):285-302
We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H
s
) and (ℝ, B, H
t
) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H
d
is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss.
To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d.
We also prove this statement in a more general form: If A ⊂ ℝn is Borel and ƒ: A → ℝm is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A).
Partially supported by the Hungarian Scientific Research Fund grant no. T 49786. 相似文献
17.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
18.
Let Ω be a Greenian domain in ℝ d , d≥2, or—more generally—let Ω be a connected -Brelot space satisfying axiom D, and let u be a numerical function on Ω, , which is locally bounded from below. A short proof yields the following result: The function u is the infimum of its superharmonic majorants if and only if each set {x:u(x)>t}, t∈ℝ, differs from an analytic set only by a polar set and , whenever V is a relatively compact open set in Ω and x∈V. 相似文献
19.
This paper is concerned with BV periodic solutions for multivalued perturbations of an evolution equation governed by the sweeping process (or Moreau's process). The perturbed equation has the form –DuN
C
(t)(u(t))+F(t,u(t)), whereC is a closed convex valued continuousT-periodic multifunction from [0,T] to
d
,N
C
(t)(u(t)) is the normal cone ofC(t) atu(t),F: [0,T]×
d
d
is a compact convex valued multifunction and Du is the differential measure of the periodic BV solutionu. Several existence results for this differential inclusion are stated under various assumptions on the perturbationF. 相似文献
20.
Nadya Gurevich 《Israel Journal of Mathematics》2001,121(1):125-141
A theorem of Bourgain states that the harmonic measure for a domain in ℝ
d
is supported on a set of Hausdorff dimension strictly less thand [2]. We apply Bourgain’s method to the discrete case, i.e., to the distribution of the first entrance point of a random walk
into a subset of ℤ
d
,d≥2. By refining the argument, we prove that for allβ>0 there existsρ(d,β)<d andN(d,β), such that for anyn>N(d,β), anyx ∈ ℤ
d
, and anyA ⊂ {1,…,n}
d
•{y∈ℤ whereν
A,x
(y) denotes the probability thaty is the first entrance point of the simple random walk starting atx intoA. Furthermore,ρ must converge tod asβ → ∞.
Supported by Swiss NF grant 20-55648.98. 相似文献