共查询到20条相似文献,搜索用时 10 毫秒
1.
Feng-Yu Wang 《Probability Theory and Related Fields》1997,108(1):87-101
Summary. This paper presents some explicit lower bound estimates of logarithmic Sobolev constant for diffusion processes on a compact
Riemannian manifold with negative Ricci curvature. Let Ric≧−K for some K>0 and d, D be respectively the dimension and the diameter of the manifold. If the boundary of the manifold is either empty or convex,
then the logarithmic Sobolev constant for Brownian motion is not less than
max
{(d d+2)
d
1 2(d+1)D
2
exp
[−1−(3d+2)D
2
K], (d−1 d+1)
d
K
exp
[−4D√d K]} .
Next, the gradient estimates of heat semigroups (including the Neumann heat semigroup and the Dirichlet one) are studied by
using coupling method together with a derivative formula modified from [11]. The resulting estimates recover or improve those
given in [7, 21] for harmonic functions.
Received: 19 September 1995 / In revised form 11 April 1996 相似文献
2.
A. F. Ramírez 《Probability Theory and Related Fields》1998,110(3):369-395
Summary. Let η be a diffusion process taking values on the infinite dimensional space T
Z
, where T is the circle, and with components satisfying the equations dη
i
=σ
i
(η) dW
i
+b
i
(η) dt for some coefficients σ
i
and b
i
, i∈Z. Suppose we have an initial distribution μ and a sequence of times t
n
→∞ such that lim
n
→∞μS
tn
=ν exists, where S
t
is the semi-group of the process. We prove that if σ
i
and b
i
are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf
i
,ησ
i
(η)>0, then ν is invariant.
Received: 12 September 1996 / In revised form: 10 November 1997 相似文献
3.
4.
Mountain pass type solutions for quasilinear elliptic equations 总被引:4,自引:0,他引:4
Ph. Clément M. García-Huidobro R. Manásevich K. Schmitt 《Calculus of Variations and Partial Differential Equations》2000,11(1):33-62
We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problem
where is a bounded domain in , , and the function is an increasing homeomorphism from onto . Under appropriate conditions on , , and the Orlicz-Sobolev conjugate of (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers),
we obtain the existence of nontrivial solutions which are of mountain pass type.
Received April 22, 1999 / Accepted June 11, 1999 / Published online April 6, 2000 相似文献
5.
The paper gives conditions under which the transition semigroup corresponding to a large class of semilinear equations on a Hilbert space transforms Borel functions onto Frechet differentia hies ones. 相似文献
6.
7.
We study solutions of first order partial differential relations Du∈K, where u:Ω⊂ℝ n →ℝ m is a Lipschitz map and K is a bounded set in m×n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of Du and second we replace Gromov’s P-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite. Received April 23, 1999 / final version received September 11, 1999 相似文献
8.
Luiz A.B. San Martin 《Mathematische Annalen》2001,321(3):587-600
Let be an affine symmetric space with a simple Lie group, an involutive automorphism of and an open subgroup of the -fixed point group . It is proved here that the existence of a proper semigroup with and implies that is of Hermitian type, as conjectured by Hilgert and Neeb [4]. When exists, it turns out that it leaves invariant an open -orbit in a minimal flag manifold of . A byproduct of our approach is an alternate proof of the maximality of the compression semigroup of an open orbit (see Hilgert
and Neeb [31]).
Received: 10 September 1999 / Published online: 23 July 2001 相似文献
9.
René L. Schilling 《Mathematische Annalen》1997,309(4):663-675
Under mild regularity assumptions on its domain the infinitesimal generator of a Feller process is known to be a pseudo-differential
operator. We give a simple condition on the symbol of the generator in order to characterize the smoothness of the sample
paths of real-valued Feller processes in terms of Besov spaces . Our result extends previous papers on the paths of Gaussian, symmetric -stable [6], [20], and Lévy processes [11].
Received: 31 May 1996 / Revised version: 10 December 1996 相似文献
10.
In this article we prove new results concerning the long-time behavior of random fields that are solutions in some sense
to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state
that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non-random
global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that
the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that
they set out to oscillate between them. In the first case we can also determine exactly the corresponding Lyapunov exponents.
In the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two
asymptotic states in a finite time with probability one. In both cases we also interpret our results in terms of stability
properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods
of the asymptotic states. Our methods of proof rest upon the use of a suitable regularization of the Brownian motion along
with a related Wong-Zaka? approximation procedure.
Received: 8 April 1997/Revised version: 30 January 1998 相似文献
11.
Xi-Nan Ma 《Mathematische Zeitschrift》2002,240(1):1-11
We study solutions of the nonlinear elliptic equation on a bounded domain in . It is shown that the set of points where the graph of the solution has negative Gauss curvature always extends to the boundary, unless it is empty.
The meethod uses an elliptic equation satisfied by an auxiliary function given by the product of the Hessian determinant and
a suitable power of the solutions. As a consequence of the result, we give a new proof for power concavity of solutions to
certain semilinear boundary value problems in convex domains.
Received: 12 January 2000; in final form: 15 March 2001 / Published online: 4 April 2002 相似文献
12.
For a natural number k, define an oriented site percolation on ℤ2 as follows. Let x
i
, y
j
be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ2
closed if x
i
= y
j
, and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open.
I.e., there is an infinite path P = (i
0, j
0)(i
1, j
1) · · · such that 0 = i
0≤i
1≤· · ·, 0 = j
0≤j
1≤· · ·, and each site (i
n
, j
n
) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive.
Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x
i
) and (y
j
). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods.
Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000 相似文献
13.
14.
John Urbas 《Mathematische Zeitschrift》2001,236(3):625-641
We derive a monotonicity formula for smooth solutions u of degenerate two dimensional Monge-Ampère equations, and use this to obtain a local H?lder gradient estimate, depending
on for some .
Received August 9, 1999; in final form December 8, 1999/ Published online December 8, 2000 相似文献
15.
Shinji Adachi Kazunaga Tanaka 《Calculus of Variations and Partial Differential Equations》2000,11(1):63-95
We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000 相似文献
16.
Erwan Saint Loubert Bié 《Probability Theory and Related Fields》1998,111(2):287-321
Résumé . Nous étudions une équation aux dérivées partielles stochastique (EDPS), de type parabolique, posée sur ℝ
d
, d entier, et conduite par un bruit poissonnien, compensé ou non. La première partie de ce travail montre l'existence et l'unicité
d'une solution progressivement mesurable. Les techniques employées sont proches de celles utilisées pour résoudre les équations
analogues conduites par un bruit blanc. La seconde partie donne des conditions, portant sur l'intensité du bruit poissonnien,
et permettant d'assurer certaines régularités, en espace ou bien en temps, pour le processus solution.
Summary. We study a Stochastic Partial Differential Equation, of parabolic type, set on ℝ d , with d∈ℕ. This equation is driven by a Poisson random measure, either compensated or not. The first part of this work shows existence and uniqueness of a progressively measurable solution. The technics involved are close to those used to deal with analogous equations driven by a Gaussian noise. The second part gives some criterions on the intensity of the Poisson random measure, in order to ensure some smoothness, either in space or in time, for the solution of this equation.
Received: 7 April 1997/In revised form: 20 January 1998 相似文献
17.
David Steinsaltz 《Probability Theory and Related Fields》1997,107(1):99-121
Summary. A self-modifying random walk on is derived from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed. This
process converges almost surely to a random variable which is totally singular with respect to Lebesgue measure, and which
is supported on a subset of having Hausdorff dimension less than , which we calculate by a theorem of Billingsley. By generating function techniques we then calculate the exponential rate
of convergence of the process to its limit point, which may be taken as a bound for the convergence of the measure in the
Wasserstein metric. We describe how the process may viewed as a random walk on the space of monotone piecewise linear functions,
where moves are taken by successive compositions with a randomly chosen such function.
Received: 20 November 1995 / In revised form: 14 May 1996 相似文献
18.
J. Chabrowski M. Willem 《Calculus of Variations and Partial Differential Equations》2002,15(4):421-431
We investigate the effect of the coefficient of the critical nonlinearity for the Neumann problem on the existence of least
energy solutions. As a by-product we establish a Sobolev inequality with interior norm.
Received: 26 April 2000 / Accepted: 25 February 2001 / Published online: 5 September 2002 相似文献
19.
Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability
one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundaryΩ of the tree. The random subset Λ of the boundary consisting of all ends of the tree in which the population survives, called
the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric
boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure μ supported by Ω such that the Hausdorff
dimension of Λ∩Ωμ, where Ωμ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ωμ at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of Λ and Λ∩Ωμ, and it is shown that the log Hausdorff dimension of Λ has critical exponent 1/2 at the phase separation point.
Received: 30 June 1998 / Revised version: 10 March 1999 相似文献
20.
Olivier Raimond 《Probability Theory and Related Fields》1997,107(2):177-196
Summary. In this paper we study a self-attracting diffusion in the case of a constant self-attraction and for dimension larger than
two. We prove that this process converges almost surely.
Received: 27 March 1995 / In revised form: 22 May 1996 相似文献