We investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér’s condition. We prove moderate deviation principles in dimensions d ≥ 2, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case d ≥ 4 we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In d ≥ 3, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in d = 2 we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

It has been well known for a long time that the measure states of the process in the title are absolutely continuous at any fixed time provided that the dimension of space is small enough. However, besides the very special case of one-dimensional continuous super-Brownian motion, properties of the related density functions were not well understood. Only in 2003, Mytnik and Perkins 21 revealed that in the Brownian motion case and if the branching is discontinuous, there is a dichotomy for the densities: Either there are continuous versions of them or they are locally unbounded. We recently showed that the same type of fixed time dichotomy holds also in the case of discontinuous motion. Moreover, the continuous versions are locally Hölder continuous, and we determined the optimal index for them. Finally, we determine the optimal index of Hölder continuity at given space points which is strictly larger than the optimal index of local Hölder continuity.

A Hölder regularity index at given points for density states of ( α ,1, β )-superprocesses with α >1+ β is determined. It is shown that this index is strictly greater than the optimal index of local Hölder continuity for those density states.

For 0 < alpha <= 2, a super-alpha-stable motion X in R(d) with branching of index 1 + beta is an element of (1,2) is considered. Fix arbitrary t > 0. If d < alpha/beta, a dichotomy for the density function of the measure X(t) holds: the density function is locally Holder continuous if d = 1 and alpha > 1 + beta but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Holder index.

A H"older regularity index at given points for density states of (alpha,1,beta)-superprocesses with alpha>1+beta is determined. It is shown that this index is strictly greater than the optimal index of local H"older continuity for those density states.

Par un changement d'echelle bien connu, on obtient que les processus de Galton-Watson supercritiques sur Z convergent vers une variable aleatoire non-degeneree W. Nous considerons les estimees asymptotiques a gauche (pres de l'origine) de la distribution. Dans le cas Bottcher (quand il y a au moins deux progenitures en chaque point), nous obtenons l'asymptotique exacte presentant un comportement oscillatoire (Theoreme 1). Sous une autre hypothese raisonnable, les oscillations s'annulent (Corollaire 2). Pour le cas Bottcher, nous presentons un resultat sur la probabilite des grandes deviations, ameliore en exprimant l'asymptotique exacte sous un scaling logarithmique (Theoreme 7). En imposant d'autres conditions, nous obtenons des asymptotiques plus raffinees (Theoreme 8), c'est-a-dire sans log-scaling.

For 0 < α ≤ 2, a super-α-stable motion X in R with branching of index 1 + β ∈ (1, 2) is considered. If d < α/β, a dichotomy for the density of states Xt at fixed times t > 0 holds: the density function is locally Hölder continuous if d = 1 and α > 1 + β, but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal Hölder index.

We investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramer's condition. We prove moderate deviation principles in dimensions two and larger, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case of dimension four and larger we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In dimension three and larger, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in dimension two we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

Consider a critical Galton–Watson process $Z=\{Z_n: n=0,1,\ldots\}$ of index $1+\alpha$, $\alpha\in(0,1]$. Let $S_k(j)$ denote the sum of the $Z_{n}$ with n in the window $[k,\ldots,k+j)$ and let $M_{m}(j)$ be the maximum of the $S_{k}(j)$ with k moving in $[0,m-j]$. We describe the asymptotic behavior of the expectation ${\bf E} M_m(j)$ if the window width $j=j_{m}$ is such that $j/m\to\eta\in$ $[0,1]$ as $m\uparrow\infty$. This will be achieved via establishing the asymptotic behavior of the tail of the distribution of the random variable $M_{\infty}(j)$.

In a recent work, Fleischmann and Mueller (2004) showed the existence of a super-Brownian motion in R^d, d=2,3, with extra birth at the origin. Their construction made use of an analytical approach based on the fundamental solution of the heat equation with a one point potential worked out by Albeverio et al. (1995). The present note addresses two properties of this measure-valued process in the three-dimensional case, namely the scaling of the process and the large scale behavior of its mean.

There is a well-known sequence of constants c_n describing the growth of supercritical Galton-Watson processes Z_n. With 'lower deviation probabilities' we refer to P(Z_n=k_n) with k_n=o(c_n) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Z_{n+1}/Z_n. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramer method for proving large deviations of sums of independent variables to our needs.

In this paper, we study the large deviation behavior of sums S_Z_n of i.i.d. random variables X i , where Z n is the n th generation of a supercritical Galton–Watson process. We assume the finiteness of the moments EX_1^2 and EZ 1 logZ 1 . The underlying interplay of large deviation probabilities of partial sums of the X i and of lower deviation probabilities of Z is clarified. Here, we heavily use lower deviation probability results on Z we recently published in [7].

We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a 'Gaussian' situation to stable fluctuations of index 1+gamma, where gamma is an index associated to the medium.

In this paper we study the large deviation behavior of sums of i.i.d. random variables X_i defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX_1^2 and EZ_1log Z_1. The underlying interplay of the partial sums of the X_i and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on Z we recently published in [FW06].

Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the unique invariant measure of the diffusion process, as a function of the attraction point. Such mappings arise in the analysis of infinite systems of diffusions indexed by the hierarchical group, with a linear attractive interaction between the components. In this context, the mappings are called renormalization transformations. We consider such maps for catalytic Wright-Fisher diffusions. These are diffusions on the unit square where the first component (the catalyst) performs an autonomous Wright-Fisher diffusion, while the second component (the reactant) performs a Wright-Fisher diffusion with a rate depending on the first component through a catalyzing function. We determine the limit of rescaled iterates of renormalization transformations acting on the diffusion matrices of such catalytic Wright-Fisher diffusions.

Recently a spatial version of Neveu’s (1992) continuous-state branching process was constructed by Fleischmann and Sturm (2004). This superprocess with infinite mean branching behaves quite differently from usual supercritical spatial branching processes. In fact, at macroscopic scales, the mass renormalized to a (random) probability measure is concentrated in a single space point which randomly fluctuates according to the underlying symmetric stable motion process.