Introduction and main results 1.1 Aims of the paper Let a bunch of p independent Brownian motions W 1 ; : : : ; W p run in R d until their rst exit times T 1 ; : : : ; T p from a large ball, or, in the transient case, for innite time. By classical results of Dvoretzky, Erdos, Kakutani and Taylor the intersection of the paths of these motions, S = p \ i=1 x 2 R d : x = W i (t) for some t 2 [0; T i ) ; (1.1) contains points dierent from the starting point if and only if p < d=(d 2). By work of Geman, Horowitz and Rosen [GH84], in these cases the random set S of

Introduction and main results 1.1 Aims of the paper Intersections of Brownian motion or random walk paths have been studied for quite a long time in probability theory and statistical mechanics. One of the reasons for this interest is that the properties of the intersections are analogous to those of a number of more complicated models in equilibrium statistical physics. There is trivial behaviour in all dimensions exceeding a critical dimension, which in our case is d = 4, but below the critical dimension there are interesting critical exponents, which determine the universality class of the model and enter into most of its quantitative studies. Mostly nonrigorous techniques from mathematical physics, such as renormalisation group theory (see, e.g., [Ai85]) and conformal eld theory (see, e.g., [DK88]), have been applied to the model and, more recently, nding the intersection exponents of planar Brownian motion was one of the rst problems solved by the rigorous techniques based on th

Analogues of stepping-stone models are considered where the site-space is continuous, the migration process is a general Markov process, and the type{space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum-sites stepping-stone process has continuous sample paths. The case when the migration process is Brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show that a random compact metric space that is naturally associated to an infinite family of coalescing Brownian motions on the circle has Hausdorff and packing dimension both almost surely equal to 1/2 and, moreover, this space is capacity equivalent to the middle-1/2 Cantor set (and hence also to the Brownian zero set).

Consider a sequence of i.i.d. random variables, where each variable is refreshed (i.e., replaced by an independent variable with the same law) independently, according to a Poisson clock. At any fixed time t, the resulting sequence has the same law as at time 0, but there can be exceptional random times at which certain almost sure properties of the time 0 sequence are violated. We prove that there are no such exceptional times for the law of large numbers and the law of the iterated logarithm, so these laws are dynamically stable. However, there are times at which run lengths are exceptionally long, i.e., run tests are dynamically sensitive. We obtain a multifractal analysis of exceptional times for run lengths and for prediction. In particular, starting from an i.i.d. sequence of unbiased random bits, the random set of times t where ff log 2 (n) bits among the first n bits can be predicted from their predecessors, has Hausdorff dimension 1 \Gamma ff a.s. Finally, we consider simple random walk in the lattice Z d , and prove that transience is dynamically stable for d 5, and dynamically sensitive for d = 3; 4. Moreover, for d = 3; 4, the nonempty random set of exceptional times t where the walk is recurrent, has Hausdorff dimension (4 \Gamma d)=2 a.s. 1

this article. Define the function / as

In this paper we investigate fast particles in the range and support of super-Brownian motion in the historical setting. In this setting each particle of superBrownian motion alive at time t is represented by a path w : [0; t] ! R d and the state of historical super-Brownian motion is a measure on the set of paths. Typical particles have Brownian paths, however in the uncountable collection of particles in the range of a super-Brownian motion there are some which at exceptional times move faster than Brownian motion. We determine the maximal speed of all particles during a given time period E, which turns out to be a function of the packing dimension of E. A path w in the support of historical super-Brownian motion at time t is called a-fast if lim sup h#0 jw(t) \Gamma w(t \Gamma h)j= p h log(1=h) a. We calculate the Hausdorff dimension of the set of a-fast paths in the support and the range of historical super-Brownian motion. A valuable tool in the proofs is a uniform dimensio...

We determine the dimension spectrum of thick points of the state of a superBrownian motion in dimension d 3. Our method also yields improvements of a law of the iterated logarithm of Dawson and Perkins and a result of Barlow, Evans and Perkins about the most visited sites of super-Brownian motion. All these results involve a constant which can be characterized in terms of the upper tails of the associated equilibrium Palm distribution.

The aim of this paper is to give a survey of recent developments in the multifractal analysis of measures arising in the study of Brownian motion.

Dimensions are a tool to measure the size of mathematical objects...

Abstract not found