The multifractal spectrum of Brownian intersection local times

Let \ell be the projected intersection local time of two independent Brownian paths in R^d for d=2,3. We determine the lower tail of the random variable \ell(U), where U is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.Comment: Published at http://dx.doi.org/10.1214/009117905000000116 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Biased Random Walk on Spanning Trees of the Ladder Graph

We consider a specific random graph which serves as a disordered medium for a particle performing biased random walk. Take a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight $c$ for the (vertical) rungs. Now take a random walk on that spanning tree with a bias $\beta>1$ to the right. In contrast to other random graphs considered in the literature (random percolation clusters, Galton-Watson trees) this one allows for an explicit analysis based on a decomposition of the graph into independent pieces. We give an explicit formula for the speed of the biased random walk as a function of both the bias $\beta$ and the edge weight $c$. We conclude that the speed is a continuous, unimodal function of $\beta$ that is positive if and only if $\beta < \beta_c^{(1)}$ for an explicit critical value $\beta_c^{(1)}$ depending on $c$. In particular, the phase transition at $\beta_c^{(1)}$ is of second order. We show that another second order phase transition takes place at another critical value $\beta_c^{(2)}<\beta_c^{(1)}$ that is also explicitly known: For $\beta<\beta_c^{(2)}$ the times the walker spends in traps have second moments and (after subtracting the linear speed) the position fulfills a central limit theorem. We see that $\beta_c^{(2)}$ is smaller than the value of $\beta$ which achieves the maximal value of the speed. Finally, concerning linear response, we confirm the Einstein relation for the unbiased model ($\beta=1$) by proving a central limit theorem and computing the variance.Comment: 29 pages, 9 figure

Biodiversity of Catalytic Super-Brownian Motion

In this paper we investigate the structure of the equilibrium state of three-dimensional catalytic super-Brownian motion where the catalyst is itself a classical super-Brownian motion. We show that the reactant has an infinite local biodiversity or genetic abundance. This contrasts the finite local biodiversity of the equilibrium of classical super-Brownian motion. Another question we address is that of extinction of the reactant in finite time or in the long-time limit in dimensions d = 2,3. Here we assume that the catalyst starts in the Lebesgue measure and the reactant starts in a finite measure. We show that there is extinction in the long-time limit if d = 2 or 3. There is, however, no finite time extinction if d = 3 (for d = 2 this problem is left open). This complements a result of Dawson and Fleischmann (1997a) for d = 1 and again contrasts the behaviour of classical super-Brownian motion. As a key tool for both problems we show that in d = 3the reactant matter propagates everywhere in space immediately

Smooth density field of catalytic super-Brownian motion

Given an (ordinary) super-Brownian motion (SBM) ϱ on Rd of dimension d = 2, 3, we consider a (catalytic) SBM Xϱ on Rd with "local branching rates" ϱs(dx). We show that Xϱt is absolutely continuous with a density function ξϱt, say. Moreover, there exists a version of the map (t,z) ↦ ξϱt(z) which is C∞ and solves the heat equation off the catalyst ϱ, more precisely, off the (zero set of) closed support of the time-space measure ds ϱs(dx). Using self-similarity, we apply this result to answer the question of the long-term behavior of Xϱ in dimension d = 2 : If ϱ and Xϱ start with a Lebesgue measure, then XϱT converges (persistently) as T → ∞ towards a random multiple of Lebesgue measure