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1.
Harris and Keane [Probab. Theory Related Fields 109 (1997) 27-37] studied absolute continuity/singularity of two probabilities on the coin-tossing space, one representing independent tosses of a fair coin, while in the other a biased coin is tossed at renewal times of an independent renewal process and a fair coin is tossed at all other times. We extend their results by allowing possibly different biases at the different renewal times. We also investigate the contiguity and asymptotic separation properties in this kind of set-up and obtain some sufficient conditions.Keywords:renewal process, absolute continuity, singularity, contiguity, asymptotic separation, martingale convergence theorem 相似文献
2.
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
3.
Thomas M. Liggett 《Probability Theory and Related Fields》1996,106(4):495-519
Summary. Branching random walks and contact processes on the homogeneous tree in which each site has d+1 neighbors have three possible types of behavior (for d≧ 2): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that
if there is local extinction, then the probability that an individual ever has a descendent at a site n units away from that individual’s location is at most d
− n/2
, while if there is global extinction, this probability is at most d
−n
. Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this
structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and
leave others open. In particular, we prove that for some values of the infection parameter λ, there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different
from those constructed by Durrett and Schinazi in a recent paper.
Received: 26 April 1996/In revised form: 20 June 1996 相似文献
4.
. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996),
we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p
2>p
1>p
c
, each infinite p
2-cluster contains an infinite p
1-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that
the probability θ
v
(p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p
c
. All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group.
Received: 22 December 1997 / Revised version: 1 July 1998 相似文献
5.
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
6.
Mario V. Wüthrich 《Probability Theory and Related Fields》1998,112(3):299-319
We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then the transverse fluctuation of the path is expected to be of order |y|ξ, whereas the distance fluctuation is of order |y|χ. Physics literature tells us that ξ and χ should satisfy a scaling identity 2ξ− 1 = χ. We give here rigorous results for
this conjecture.
Received: 31 December 1997 / Revised version: 14 April 1998 相似文献
7.
We prove the homogenization of convection-diffusion in a time-dependent, ergodic, incompressible random flow which has a
bounded stream matrix and a constant mean drift. We also prove two variational formulas for the effective diffusivity. As
a consequence, we obtain both upper and lower bounds on the effective diffusivity.
Received: 17 December 1996/Revised revision: 9 February 1998 相似文献
8.
Roberto H. Schonmann 《Probability Theory and Related Fields》1999,113(2):287-300
. A recent theorem by Häggström and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p 1<p 2≤1 and percolation occurs at level p 1, then every infinite cluster at level p 2 contains some infinite cluster at level p 1. Consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster, i.e., if at level p 1 there is a unique infinite cluster then the same holds at level p 2. These results are further generalized to graphs with a “uniform percolation” property. The threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls. 相似文献
9.
Tokuzo Shiga 《Probability Theory and Related Fields》1997,108(3):417-439
Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf;
d
in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate
the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0.
Received: 21 May 1996 / In revised form: 2 February 1997 相似文献
10.
Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of
their ergodic invariant measures. The question arises whether a.s.the process eventually stays close to one of these ergodic
states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that
there exists a convergence–determining class of distributions that is (strongly) preserved under the dynamics, we show that
the system is in fact recurrent in the above sense.
We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we
answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic
branching.
Received: 22 January 1999 / Revised version: 24 May 1999 相似文献
11.
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
12.
We consider symmetric simple exclusion processes with L=&ρmacr;N
d
particles in a periodic d-dimensional lattice of width N. We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far
away form equilibrium. It specifies in the scaling limit a density profile on the d-dimensional torus. We are interested in the large deviations of the empirical process, N
−
d
[∑
L
1δ
xi
(·)] as random variables taking values in the space of measures on D[0.1]. We prove a large deviation principle, with a rate function that is more or less universal, involving explicity besides
the initial profile, only such canonical objects as bulk and self diffusion coefficients.
Received: 7 September 1997 / Revised version: 15 May 1998 相似文献
13.
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
14.
15.
16.
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
in ?
N
and ?
N
×ℝ}+, with a∈L
∞, when 1<p≤p
0, where p
0 depends on N and the type of equation.
Received: 17 June 1999 相似文献
17.
We present an upper bound O(n
2
) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant,
and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion
process on a circle indeed mixes more rapidly than the corresponding symmetric process.
Received: 25 January 1999 / Revised version: 17 September 1999 / Published online: 14 June 2000 相似文献
18.
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
19.
Tobias Povel 《Probability Theory and Related Fields》1999,114(2):177-205
Consider Brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of ℝ
d
, d≥ 1, at the points of a Poisson cloud of constant intensity v <: 0. Assume that Brownian motion is absorbed instantaneously upon entering the obstacle set. In SZN-conf Sznitman has shown
that in d = 2, conditionally on the event that the process does not enter the obstacle set up to time t, the probability that Brownian motion remains within distance ∼t
1/4 from its starting point is going to 1 as t goes to infinity. We show that the same result holds true for d≥ 3, with t
1/4 replaced by t
1/(
d
+2). The proof is based on Sznitmans refined method of enlargement of obstacles [10] as well as on a quantitative isoperimetric
inequality due to Hall [4].
Received: 6 July 1998 相似文献
20.
Jean Bertoin 《Probability Theory and Related Fields》1999,114(1):97-121
We investigate the nature of the intersection of two independent regenerative sets. The approach combines Bochners subordination
and potential theory for a pair of Markov processes in duality.
Received: 21 November 1997 / Revised version: 31 August 1998 相似文献