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1.
Any solution of the functional equation
where B is a Brownian motion, behaves like a reflected Brownian motion, except when it attains a new maximum: we call it an α-perturbed
reflected Brownian motion. Similarly any solution of
behaves like a Brownian motion except when it attains a new maximum or minimum: we call it an α,β-doubly perturbed Brownian
motion. We complete some recent investigations by showing that for all permissible values of the parameters α, α and β respectively,
these equations have pathwise unique solutions, and these are adapted to the filtration of B.
Received: 7 November 1997 / Revised version: 13 July 1998 相似文献
2.
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
3.
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
4.
Summary. We study `perturbed Brownian motions', that can be, loosely speaking, described as follows: they behave exactly as linear
Brownian motion except when they hit their past maximum or/and maximum where they get an extra `push'. We define with no restrictions
on the perturbation parameters a process which has this property and show that its law is unique within a certain `natural
class' of processes. In the case where both perturbations (at the maximum and at the minimum) are self-repelling, we show
that in fact, more is true: Such a process can almost surely be constructed from Brownian paths by a one-to-one measurable
transformation. This generalizes some results of Carmona-Petit-Yor and Davis. We also derive some fine properties of perturbed
Brownian motions (Hausdorff dimension of points of monotonicity for example).
Received: 17 May 1996 / In revised form: 21 January 1997 相似文献
5.
Summary. We study the asymptotic behavior of Brownian motion and its conditioned process in cones using an infinite series representation
of its transition density. A concise probabilistic interpretation of this series in terms of the skew product decomposition
of Brownian motion is derived and used to show properties of the transition density.
Received: 2 April 1996 / In revised form: 21 December 1996 相似文献
6.
Hirofumi Osada 《Probability Theory and Related Fields》1998,112(1):53-90
We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of
the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of
a tagged particle. We will do this for all activities, z>0, of Gibbs measures; in particular, for large z– the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian
balls.
Received: 22 September 1997 / Revised version: 15 January 1998 相似文献
7.
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
8.
John Verzani 《Probability Theory and Related Fields》1997,107(4):517-526
Summary. For the Brownian path-valued process of Le Gall (or Brownian snake) in , the times at which the process is a cone path are considered as a function of the size of the cone and the terminal position
of the path. The results show that the paths for the path-valued process have local properties unlike those of a standard
Brownian motion.
Received: 29 January 1996 / In revised form: 21 June 1996 相似文献
9.
Endre Csáki Miklós Csörgő Antónia Földes Zhan Shi 《Probability Theory and Related Fields》2000,117(4):515-531
Let W be a standard Brownian motion, and define Y(t)= ∫0
t
ds/W(s) as Cauchy's principal value related to local time. We determine: (a) the modulus of continuity of Y in the sense of P. Lévy; (b) the large increments of Y.
Received: 1 April 1999 / Revised version: 27 September 1999 / Published online: 14 June 2000 相似文献
10.
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
11.
The number of infinite clusters in dynamical percolation 总被引:2,自引:2,他引:0
Summary. Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure
as its stationary distribution. In previous work with O. H?ggstr?m, we found conditions for existence of infinite clusters
at exceptional times. Here we show that for ℤ
d
, with p>p
c
, a.s. simultaneously for all times there is a unique infinite cluster, and the density of this cluster is θ(p). For dynamical percolation on a general tree Γ, we show that for p>p
c
, a.s. there are infinitely many infinite clusters at all times. At the critical value p=p
c
, the number of infinite clusters may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we
find the Hausdorff dimension of the set T
k
of times where the number of infinite clusters is k, and obtain sharp capacity criteria for a given time set to intersect T
k
. The proof of this capacity criterion is based on a new kernel truncation technique.
Received: 5 May 1997 / In revised form: 24 November 1997 相似文献
12.
Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2
−n
where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of
types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure
μ on Ω, define Ωμ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ−1 (Ωμ) in the metric d is
almost surely on the event of nonextinction, where h(μ) is the entropy of the measure μ and q(i, j) is the mean number of type-j offspring of a type-i individual. This extends a theorem of HAWKES [5], which shows that the Hausdorff dimension of the entire boundary at infinity is log2 α, where α is the Malthusian parameter.
Received: 30 June 1998 / Revised: 4 February 1999 相似文献
13.
We show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swindle (1989) by replacing their order of magnitude estimates of how close the critical value is to 1 with sharp asymptotics. Received: 2 February 1998 / Revised version: 28 August 1998 相似文献
14.
15.
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
16.
17.
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
18.
Burgess Davis 《Probability Theory and Related Fields》1999,113(4):501-518
Let b
t
be Brownian motion. We show there is a unique adapted process x
t
which satisfies dx
t
= db
t
except when x
t
is at a maximum or a minimum, when it receives a push, the magnitudes and directions of the pushes being the parameters of
the process. For some ranges of the parameters this is already known. We show that if a random walk close to b
t
is perturbed properly, its paths are close to those of x
t
.
Received: 15 October 1997 / Revised version: 18 May 1998 相似文献
19.
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
20.
Hideki Tanemura 《Probability Theory and Related Fields》1997,109(2):275-299
Summary. Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ
d
are studied. Introducing a linear operator L
0 defined on a space of smooth local functions, we show the uniqueness of Dirichlet forms associated with self adjoint Markovian
extensions of L
0. We also discuss the ergodicity of the reversible process associated with the Dirichlet form.
Received: 18 July 1996/In revised form: 13 February 1997 相似文献