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1.
Summary. The super-Brownian motion X
ϱ in a super-Brownian medium ϱ constructed in [DF97a] is known to be persistent (no loss of expected mass in the longtime behaviour)
in dimensions one ([DF97a]) and three ([DF97b]). Here we fill the gap in showing that persistence holds also in the critical
dimension two. The key to this result is that in any dimension (d≤3), given the catalyst, the variance of the process is finite `uniformly in time'. This is in contrast to the `classical'
super-Brownian motion where this holds only in high dimensions (d≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction.
Received: 21 November 1996 / In revised form: 31 March 1997 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 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
4.
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
5.
Summary A new approach is provided to the super-Brownian motionX with a single point-catalyst
c
as branching rate. We start from a superprocessU with constant branching rate and spatial motion given by the 1/2-stable subordinator. We prove that the occupation density measure
c
ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatX
t is determined from
c
by an explicit representation formula. Heuristically, a mass
c
(ds) of particles leaves the catalyst at times and then evolves according to Itô's Brownian excursion measure. As a consequence of our representation formula, the density fieldx ofX satisfies the heat equation outside ofc, with a noisy boundary condition atc given by the singularly continuous random measure
c
. In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure
c
. 相似文献
6.
In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this
martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown
to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application
is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain.
The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general,
as the change of measure is given by a martingale which need not arise from a single harmonic function.
Received: 27 August 1998 / Revised version: 8 January 1999 相似文献
7.
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
8.
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
9.
Ito's rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with
small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices.
Received: 12 February 1997 / Revised version: 3 March 1999 相似文献
10.
L. Overbeck 《Probability Theory and Related Fields》1993,96(4):545-570
Summary We investigate classes of conditioned super-Brownian motions, namely H-transformsP
H
with non-negative finitely-based space-time harmonic functionsH(t, ). We prove thatH
H
is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functionsH(t, )=h(t, (1)) depending only on the total mass (1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, )=h(t, x) (dx) theH-transformP
H
as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion. 相似文献
11.
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
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 how to construct a canonical choice of stochastic area for paths of reversible Markov processes satisfying a weak
H?lder condition, and hence demonstrate that the sample paths of such processes are rough paths in the sense of Lyons. We
further prove that certain polygonal approximations to these paths and their areas converge in p-variation norm. As a corollary of this result and standard properties of rough paths, we are able to provide a significant
generalization of the classical result of Wong-Zakai on the approximation of solutions to stochastic differential equations.
Our results allow us to construct solutions to differential equations driven by reversible Markov processes of finite p-variation with p<4.
Received May 18, 2001 / final version received April 3, 2001?Published online April 8, 2002 相似文献
14.
This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate,
continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear
parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate.
Received: 12 May 1997 / Revised version: 10 January 1999 相似文献
15.
S. Fourati 《Probability Theory and Related Fields》1998,110(1):13-49
Summary. We prove a conjecture of J. Bertoin: a Lévy process has increase times if and only if the integral is finite, where G and H are the distribution functions of the minimum and the maximum of the Lévy process killed at an independent exponential time.
The “if” part of the statement had been obtained before by R. Doney. Our proof uses different techniques, from potential theory
and the general theory of processes, and is self-contained. Our results also show that if P(X
t
<0)≤1/2 for all t small enough, then the process does not have increase times.
Received: 4 May 1995/In revised form: 6 May 1997 相似文献
16.
Summary. Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian
paths in the hyperbolic plane ?
2
, and undergo binary fission(s) at rate λ > 0. It is shown that there is a phase transition in λ: For λ≦ 1/8 the number of particles in any compact region of ?
2
is eventually 0, w.p.1, but for λ > 1/8 the number of particles in any open set grows to ∞ w.p.1. In the subcritical case (λ≦ 1/8) the set Λ of all limit points in ∂?
2
(the boundary circle at ∞) of particle trails is a Cantor set, while in the supercritical case (λ > 1/8) the set Λ has full Lebesgue measure. For λ≦ 1/8 it is shown that w.p.1 the Hausdorff dimension of Λ is δ = (1−√1−8 λ)/2.
Received: 2 November 1995 / In revised form: 22 October 1996 相似文献
17.
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
18.
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
19.
John Hawkes 《Probability Theory and Related Fields》1998,112(1):1-11
Exact results are proved for the capacity of pullbacks of analytic sets by stable processes.
Received: 25 May 1988 / Revised version: 15 September 1997 相似文献
20.
Michel Talagrand 《Probability Theory and Related Fields》1998,110(2):109-176
Summary. The Sherrington–Kirkpatrick (SK) model for spin glasses is deceptively simple to state. Yet its rigorous study represents
a considerable challenge. We report here some modest progresses (obtained through elementary methods). Even in the supposedly
simple high temperature region, a number of basic questions remain unsolved.
Received: 7 December 1995 / In revised form: 6 March 1997 相似文献