共查询到20条相似文献,搜索用时 31 毫秒
1.
For a bounded C
1,α domain in ℝ
d
we show that there exists a strong solution to the multidimensional Skorokhod equation and that weak uniqueness holds for
this equation. These results imply that pathwise uniqueness and strong uniqueness hold for the Skorokhod equation.
Received: 3 February 1999 / Revised version: 2 September 1999 /?Published online: 11 April 2000 相似文献
2.
Jean-François Delmas 《Probability Theory and Related Fields》1999,114(4):505-547
We consider a super-Brownian motion X. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting
behavior of the volume of the ɛ-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous
result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove
the support of X
t
is capacity-equivalent to [0, 1]2 in ℝd, d≥ 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1]4 in ℝd, d≥ 5.
Received: 7 April 1998 / Revised version: 2 October 1998 相似文献
3.
We present a new (1+ε)-spanner for sets of n points in ℝ
d
. Our spanner has size O(n/ε
d−1) and maximum degree O(log
d
n). The main advantage of our spanner is that it can be maintained efficiently as the points move: Assuming that the trajectories
of the points can be described by bounded-degree polynomials, the number of topological changes to the spanner is O(n
2/ε
d−1), and using a supporting data structure of size O(nlog
d
n), we can handle events in time O(log
d+1
n). Moreover, the spanner can be updated in time O(log n) if the flight plan of a point changes. This is the first kinetic spanner for points in ℝ
d
whose performance does not depend on the spread of the point set. 相似文献
4.
Let T:x↦2x (mod 1) be the doubling map of the circle ?=ℝ/ℤ. We construct a trigonometric polynomial f:?→ℝ with the following property: ∫f
dμ≥0 for every T-invariant probability measure μ, so that f is cohomologous to a non-negative Lipschitz function, yet f is not cohomologous to any non-negative C
1 function.
Oblatum 28-VI-2001 & 4-X-2001?Published online: 18 January 2002 相似文献
5.
We consider a stationary grain model Ξ in ℝ
d
with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula
for the(d− 1)-st quermass density.
Received: 22 November 1999 / Revised version: 2 November 2000 /?Published online: 14 June 2001 相似文献
6.
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
7.
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
8.
We prove a formula expressing the gradient of the phase function of a function f:ℝ
d
↦ℂ as a normalized first frequency moment of the Wigner distribution for fixed time. The formula holds when f is the Fourier transform of a distribution of compact support, or when f belongs to a Sobolev space H
d/2+1+ε
(ℝ
d
) where ε>0. The restriction of the Wigner distribution to fixed time is well defined provided a certain condition on its wave front
set is satisfied. Therefore we first need to study the wave front set of the Wigner distribution of a tempered distribution. 相似文献
9.
We obtain KSS, Strichartz and certain weighted Strichartz estimates for the wave equation on (ℝ
d
, g), d ≥ 3, when the metric g is non-trapping and approaches the Euclidean metric like 〈x〉−ρ
with ρ > 0. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial
data for ρ > 1 and d = 3. Also, we establish the Strauss conjecture when the metric is radial with ρ > 1 for d = 3. 相似文献
10.
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
11.
Jean-Stéphane Dhersin Jean-François Le Gall 《Probability Theory and Related Fields》1997,108(1):103-129
We prove a Wiener-type criterion for super-Brownian motion and the Brownian snake.If F is a Borel subset of ℝ
d
and x ∈ ℝ
d
, we provide a necessary and sufficientcondition for super-Brownian motion started at δ
x
to immediately hit the set F. Equivalently, this condition is necessary and sufficient for the hitting time of F by theBrownian snake with initial point x to be 0. A key ingredient of the proof isan estimate showing that the hitting probability of F is comparable, up to multiplicative constants,to the relevant capacity of F. This estimate, which is of independent interest, refines previous results due to Perkins and Dynkin. An important role is
played by additivefunctionals of the Brownian snake, which are investigated here via the potentialtheory of symmetric Markov
processes. As a direct application of our probabilisticresults, we obtain a necessary and sufficient condition for the existence
in a domain D of a positivesolution of the equation Δ; u = u
2
which explodes at a given point of ∂ D.
Received: 5 January 1996 / In revised form: 30 October 1996 相似文献
12.
Ante Mimica 《Potential Analysis》2010,32(3):275-303
In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ
d
. We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process
in ℝ
d
with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr
− d − α
, ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion
with subordinator with Laplace exponent ϕ(λ) = λ
α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2). 相似文献
13.
Let Ω be an open, simply connected, and bounded region in ℝ
d
, d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving the elliptic partial differential equation − Δu + γu = f over Ω with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable,
and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate
exponential convergence. 相似文献
14.
Summary. We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝ
d
. Compared to standard FPP on ℤ
d
, these models have some technical complications but also have the advantage of statistical isotropy. We prove two almost
sure results: a shape theorem (where isotropy implies an exact Euclidean ball for the asymptotic shape) and nonexistence of
certain doubly infinite geodesics (where isotropy yields a stronger result than in standard FPP).
Received: 21 May 1996 / In revised form: 19 November 1996 相似文献
15.
Summary. We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical
behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This
vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with
techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C
1,2([0,T]×ℝ
d
,(S)*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field,
i.e. ρ(t,x) is an L
2-random variable for all time and space parameters (t,x)∈[0,T]×ℝ
d
.
Received: 27 March 1995 / In revised form: 15 May 1997 相似文献
16.
N. A. Krasovskii A. M. Taras’ev 《Proceedings of the Steklov Institute of Mathematics》2010,269(1):174-185
We address the problem of optimal reconstruction of the values of a linear operator on ℝ
d
or ℤ
d
from approximate values of other operators. Each operator acts as the multiplication of the Fourier transform by a certain
function. As an application, we present explicit expressions for optimal methods of reconstructing the solution of the heat
equation (for continuous and difference models) at a given instant of time from inaccurate measurements of this solution at
other time instants. 相似文献
17.
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
18.
We investigate connections between radial Fourier multipliers on ℝ
d
and certain conical Fourier multipliers on ℝ
d+1. As an application we obtain a new weak type endpoint bound for the Bochner–Riesz multipliers associated with the light cone
in ℝ
d+1, where d≥4, and results on characterizations of L
p
→L
p,ν inequalities for convolutions with radial kernels. 相似文献
19.
We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the
form ∂
t
μ = L
*
μ for bounded Borel measures on ℝ
d
× (0, T), where L is a second order elliptic operator, for example, Lu = Dxu + ( b,?xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity
ò( ?tu + Lu )dm = 0 \int \left( {{\partial_t}u + Lu} \right)d\mu = 0 相似文献
20.
We prove that if u
1,u
2:(0,∞)×ℝ
d
→(0,∞) are sufficiently well-behaved solutions to certain heat inequalities on ℝ
d
then the function u:(0,∞)×ℝ
d
→(0,∞) given by
also satisfies a heat inequality of a similar type provided
. On iterating, this result leads to an analogous statement concerning n-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp n-fold Young convolution inequality and its reverse form.
Both authors were supported by EPSRC grant EP/E022340/1. 相似文献
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