A super-Brownian motion with a locally infinite catalytic mass

Summary. A super-Brownian motion in with “hyperbolic” branching rate , is constructed, which symbolically could be described by the formal stochastic equation (with a space-time white noise ). Starting at this superprocess will never hit the catalytic center: There is an increasing sequence of Brownian stopping times strictly smaller than the hitting time of such that with probability one Dynkin's stopped measures vanish except for finitely many Received: 27 November 1995 / In revised form: 24 July 1996     

Persistence of a two-dimensional super-Brownian motion in a catalytic medium

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     

Rescaled contact processes converge to super-Brownian motion in two or more dimensions

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     

Wiener’s test for super-Brownian motion and the Brownian snake

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     

Conditioned super-Brownian motion

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.     

An extension of Hawkes theorem on the Hausdorff dimension of a Galton–Watson tree

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 Ω_μ 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 (Ω_μ) in the metric d is

Hyperbolic branching Brownian motion

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     

Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion

Any solution of the functional equation

Pathwise construction of additive H-transforms of super-Brownian motion

Summary The pathwise construction of additiveH-transforms of the super-Brownian motion is carried out as a modification of Le Gall's construction of superprocesses. It provides then the explicit conditioning of the super-Brownian motion on its exit behaviour at its Martin boundary, which yields an additiveH-transforms of the super-Brownian motion. The condition turns out to be that the space-time point of death of the super-Brownian motion converges in the Martin topology of the Brownian motion.Supported by an EC-Individual-Fellowship under Contract No. ERBCHBICT930682 and the SFB 256 of the University of Bonn, Germany     

Brownian motion in cones

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     

On the conditioned exit measures of super Brownian motion

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     

Cone paths for the planar Brownian snake

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     

A new approach to the single point catalytic super-Brownian motion

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 .     

Limiting angle of Brownian motion on certain manifolds

Summary. Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m ≧ 3. If, outside a fixed compact set, the sectional curvatures are bounded above by a negative constant multiple of the inverse of the square of the geodesic distance from a fixed point and below by another negative constant multiple of the square of the geodesic distance, then the angular part of Brownian motion on M tends to a limit as time tends to infinity, and the closure of the support of the distribution of this limit is the entire S ^m−1 . This improves a result of Hsu and March. Received: 7 December 1994/In revised form: 2 September 1995     

Relative entropy and mixing properties of infinite dimensional diffusions

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 ,η}σ _i (η)>0, then ν is invariant. Received: 12 September 1996 / In revised form: 10 November 1997     

Increment sizes of the principal value of Brownian local time

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     

Some dimension results for super-Brownian motion

Summary The Dawson-Watanabe super-Brownian motion has been intensively studied in the last few years. In particular, there has been much work concerning the Hausdorff dimension of certain remarkable sets related to super-Brownian motion. We contribute to this study in the following way. Let (Y _t)_t0 be a super-Brownian motion on ^d (d2) andH be a Borel subset of ^d . We determine the Hausdorff Dimension of {t0; SuppY _tHØ}, improving and generalizing a result of Krone. We also obtain a new proof of a result of Tribe which gives, whend4, the Hausdorff dimension of SuppY _t as a function of the dimension ofB.     

Symmetries in the stochastic calculus of variations

Summary. Given a stochastic action integral we define a notion of invariance of this action under a family of one parameter space-time transformations and a notion of prolonged transformations which extend the existing analogs in classical calculus of variations. We prove that a family of prolonged transformations leaves the action integral invariant if and only if it leaves invariant the heat equation associated to it as well as the structure of the extremals. We then prove a stochastic version of Noether theorem: to each family of transformations leaving the action invariant (or symmetries) we can associate a function which gives a martingale when taken along a process minimizing the action under endpoint constraints. Received: 29 June 1996 / In revised form: 19 July 1996     

Tangent measure distributions of fractal measures

Tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in Euclidean spaces. The idea is, loosely speaking, to attach to every point of the set a family of random measures, called the -dimensional tangent measure distributions at the point, which describe asymptotically the -dimensional scenery seen by an observer zooming down towards this point. This tool has been used by Bandt [BA] and Graf [G] to study the regularity of the local geometry of self similar sets, but in this paper we show that its scope goes much beyond this situation and, in fact, it may be used to describe a strong regularity property possessed by every measure: We show that, for every measure on a Euclidean space and any dimension , at -almost every point, all -dimensional tangent measure distributions are Palm measures. This means that the local geometry of every dimension of general measures can be described – like the local geometry of self similar sets – by means of a family of statistically self similar random measures. We believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities. Received: 27 November 1996 / Revised version: 27 February 1998