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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, ∞).
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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.
We consider the parabolic Anderson problem ∂
t
u = κΔu + ξ(x)u on ℝ+×ℝ
d
with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases
of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order
term of the almost sure asymptotics of u(t, 0) as t→∞.
Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000 相似文献
3.
C. Boldrighini R. A. Minlos A. Pellegrinotti 《Probability Theory and Related Fields》1997,109(2):245-273
Summary We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ
t
(x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X
t
+1= y|X
t
= x,ξ
t
=η) =P
0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ
t
(x):(t,x) ∈ℤν+1} are i.i.d., that both P
0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P
0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X
t
, and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X
t
and a corresponding correction of order to the C.L.T.. Proofs are based on some new L
p
estimates for a class of functionals of the field.
Received: 4 January 1996/In revised form: 26 May 1997 相似文献
4.
Tomasz Komorowski 《Probability Theory and Related Fields》2001,121(4):525-550
We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂
t
u
ɛ
(t, x) = κΔ
x
(t, x) + 1/ɛV(t/ɛ2,xɛ) ·∇
x
u
ɛ
(t, x) with the initial condition u
ɛ(0,x) = u
0(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R
d
is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u
ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain
constant coefficient heat equation.
Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001 相似文献
5.
H. Brézis 《Israel Journal of Mathematics》1971,9(4):513-534
Let φ be a convex l.s.c. function fromH (Hilbert) into ] - ∞, ∞ ] andD(φ)={u ∈H; φ(u)<+∞}. It is proved that for everyu
0 ∈D(φ) the equation − (du/dt)(t ∈ ∂φ(u(t)),u(0)=u
0 has a solution satisfying ÷(du(t)/dt)÷ ≦(c
1/t)+c
2. The behavior ofu(t) in the neighborhood oft=0 andt=+∞ as well as the inhomogeneous equation (du(t)/dt)+∂φ(u(t)) ∈f(t) are then studied. Solutions of some nonlinear boundary value problems are given as applications.
相似文献
6.
We say that n independent trajectories ξ1(t),…,ξ
n
(t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ
i
(t
i
) and ξ
j
(t
j
) is at least ɛ, for some indices i, j and for all large enough t
1,…,t
n
, with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition
function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by
the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct
−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ(α) are asymptotically separated, where ξ(α) is the α-process generated by −(−Δ)α/2.
Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000
RID="*"
ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782
RID="**"
ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573 相似文献
7.
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
8.
Nils Svanstedt 《Applications of Mathematics》2008,53(2):143-155
Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic
behaviour of a sequence of realizations of the form ∂u
ɛ
ω
/ ∂t+1 / ɛ
3
C(T
3(x/ɛ
3)ω
3) · ∇u
ɛ
ω
− div(α(T
2(x/ɛ
2)ω
2, t) ∇u
ɛ
ω
) = f. It is shown, under certain structure assumptions on the random vector field C(ω
3) and the random map α(ω
1, ω
2, t), that the sequence {u
ɛ
ω
} of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f). 相似文献
9.
We study the existence and the properties of reduced measures for the parabolic equations ∂
t
u − Δu + g(u) = 0 in Ω × (0, ∞) subject to the conditions (P): u = 0 on ∂Ω × (0, ∞), u(x, 0) = μ and (P′): u = μ′ on ∂Ω × (0, ∞), u(x, 0) = 0, where μ and μ′ are positive Radon measures and g is a continuous nondecreasing function. 相似文献
10.
René L. Schilling 《Probability Theory and Related Fields》1998,112(4):565-611
Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C
c
∞(ℝ
n
)⊂D(A) and A|C
c
∞(ℝ
n
) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|∞≤c(1+|ξ|2) and |Imp(x,ξ)|≤c
0Rep(x,ξ). We show that the associated Feller process {X
t
}
t
≥0 on ℝ
n
is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour
of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β∞
x
:={λ>0:lim
|ξ|→∞
|
x
−
y
|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ∞
x
:={λ>0:liminf
|ξ|→∞
|
x
−
y
|≤2/|ξ|
|ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ
x
) that lim
t
→0
t
−1/λ
s
≤
t
|X
s
−x|=0 or ∞ according to λ>β∞
x
or λ<δ∞
x
. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].
Received: 21 July 1997 / Revised version: 26 January 1998 相似文献
11.
We prove that, starting at an initial metric
g(0)=e2u0(dx2+dy2)g(0)=e^{2u_{0}}(dx^{2}+dy^{2})
on ℝ2 with bounded scalar curvature and bounded u
0, the Ricci flow ∂
t
g(t)=−R
g(t)
g(t) converges to a flat metric on ℝ2. 相似文献
12.
You Peng CHEN Chun Hong XIE 《数学学报(英文版)》2006,22(5):1297-1304
This paper deals with the strongly coupled parabolic system ut = v^m△u, vt = u^n△v, (x, t) ∈Ω × (0,T) subject to nonlinear boundary conditions 偏du/偏dη = u^αv^p, 偏du/偏dη= u^qv^β, (x, t) ∈ 偏dΩ × (0, T), where Ω 包含 RN is a bounded domain, m, n are positive constants and α,β, p, q are nonnegative constants. Global existence and nonexistence of the positive solution of the above problem are studied and a new criterion is established. It is proved that the positive solution of the above problem exists globally if and only if α 〈 1,β 〈 1 and (m +p)(n + q) ≤ (1 - α)(1 -β). 相似文献
13.
Joo-Paulo Dias Mrio Figueira Luis Sanchez 《Mathematical Methods in the Applied Sciences》1998,21(12):1107-1113
In this paper we consider the Cauchy problem for the equation ∂u/∂t + u ∂u/∂x + u/x = 0 for x > 0, t ⩾ 0, with u(x, 0) = u0−(x) for x < x0, u(x, 0) = u0+(x) for x > x0, u0−(x0) > u0+(x0). Following the ideas of Majda, 1984 and Lax, 1973, we construct, for smooth u0− and u0+, a global shock front weak solution u(x, t) = u−(x, t) for x < ϕ(t), u(x, t) = u+(x, t) for x > ϕ(t), where u− and u+ are the strong solutions corresponding (respectively) to u0− and u0+ and the curve t → ϕ(t) is defined by dϕ/dt (t) = 1/2[u−(ϕ(t), t) + u+(ϕ(t), t)], t ⩾ 0 and ϕ(0) = x0. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd. 相似文献
14.
Maria E. Schonbek 《Mathematische Annalen》2006,336(3):505-538
This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u
t
−Δu+b(x)·∇(u|u|
q
−1)=f(x, t) in ℝ
n
×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u
0 is supposed to be in an appropriate L
p
space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant 相似文献
15.
Jean-René Licois 《Journal d'Analyse Mathématique》1995,66(1):1-36
LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar
P-B. We prove that any non-negative solutionu ofu
tt+Δgu=f(u) onM x ℝ+ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ
g
Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C
2(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that
for every λ the map (u(0,·),u
t(0,·))→(u(t,·), u
t(t,·)) defines a dynamical system onW
1/2(M)⊂C(M)×L
2(M) which possesses a compact maximal attractor.
相似文献
16.
The main focus in this paper is on homogenization of the parabolic problem ∂
t
uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ
r
)∇u
ɛ
) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made. 相似文献
17.
Carl Mueller 《Probability Theory and Related Fields》1998,110(1):51-68
Summary. Let ? be the circle [0,J] with the ends identified. We prove long-time existence for the following equation.
Here, =(t,x) is 2-parameter white noise, and we assume that u
0(x) is a continuous function on ?. We show that if g(u) grows no faster than C
0(1+|u|)γ for some γ<3/2, C
0>0, then this equation has a unique solution u(t,x) valid for all times t>0.
Received: 27 November 1996 / In revised form: 28 July 1997 相似文献
18.
Pierre Collet Servet Martínez Jaime San Martín 《Probability Theory and Related Fields》2000,116(3):303-316
We study the asymptotic behaviour of the transition density of a Brownian motion in ?, killed at ∂?, where ?
c
is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density p
?
t
(x, y) behaves, for large t, as
u(x)u(y)(t(log t)2)−1 for x, y∈?, where u is the unique positive harmonic function vanishing on (∂?)
r
, such that u(x) ∼ log ∣x∣.
Received: 29 January 1999 / Revised version: 11 May 1999 相似文献
19.
Let {ξ(t), t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E
ξ(t) ≡ 0, D
ξ(t) ≡ 1, and continuous trajectories defined on the m-dimensional interval
T ì \mathbbRm T \subset {\mathbb{R}^m} . The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{−v(t) < ξ(t) < u(t), t ∈ T}, when, for all t ∈ T, u(t), v(t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions
of empirical processes and fields, Sov. Math., Dokl., 45(1):226–228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e−Q
+ Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R(t) = cov(ξ′(t), ξ′(t)). 相似文献
20.
A. I. Kozhanov 《Mathematical Notes》1999,65(1):59-63
A comparison principle for solutions of the first initial boundary value problem for the generalized Boussinesque equation
with a nonlinear sourceu
t-Δψ(u)-Δu
t+q(u)=0 is established. By using this comparison principle, we prove new existence and nonexistence theorems for solutions of the
first initial boundary value problem in the case of power-law functions ψ (ξ) andq (ξ).
Translated fromMathematicheskie Zametki, Vol. 65, No. 1, pp. 70–75, January, 1999. 相似文献