We consider the one-dimensional KPP-equation driven by space-time white noise and extend the construction of travelling wave solutions arising from initial data f 0 (x) = 1 ∧ (−x ∨ 0) from [17] to non-negative continuous functions with compact support. As an application the existence of travelling wave solutions is used to prove that the support of any solution is recurrent. As a by-product, several upper measures are introduced that allow for a stochastic domination of any solution to the SPDE at a fixed point in time.

This paper deals with the large deviations behavior of a stochastic process called thinned Lévy process. This process appeared recently as a stochastic-process limit in the context of critical inhomogeneous random graphs [3]. The process has a strong negative drift, while we are interested in the rare event of the process being positive at large times. To characterize this rare event, we identify a tilted measure. This presents some challenges inherent to the power-law nature of the thinned Lévy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned Lévy process this involves a Riemann sum that is hard to control. We choose to approximate the Riemann sum by its limiting integral, derive the first-order correction term, and prove that the tilt that follows from the corresponding approximate variational problem is sufficient to establish the large deviations results.

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299–2361, 2012 ). It was proved that when the degrees obey a power law with exponent , the sequence of clusters ordered in decreasing size and multiplied through by converges as to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u , as a function of u . This extends a related result of Pittel (J Combin Theory Ser B 82(2):237–269, 2001 ) for the Erdős–Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

We give criteria on the existence of a so-called mark function in the context of marked metric measure spaces (mmm-spaces). If an mmm-space admits a mark function, we call it functionally-marked metric measure space (fmm-space). This is not a closed property in the usual marked Gromov-weak topology, and thus we put particular emphasis on the question under which conditions it carries over to a limit. We obtain criteria for deterministic mmm-spaces as well as random mmm-spaces and mmm-space-valued processes. As an example, our criteria are applied to prove that the tree-valued Fleming-Viot dynamics with mutation and selection from [Depperschmidt, Greven, Pfaffelhuber, Ann. Appl. Probab. '12] admits a mark function at all times, almost surely. Thereby, we fill a gap in a former proof of this fact, which used a wrong criterion. Furthermore, the subspace of fmm-spaces, which is dense and not closed, is investigated in detail. We show that there exists a metric that induces the marked Gromov-weak topology on this subspace and is complete. Therefore, the space of fmm-spaces is a Polish space. We also construct a decomposition into closed sets which are related to the case of uniformly equicontinuous mark functions.Comment: 22 pages. Journal version, only minor change

E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractWe consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth-and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [9], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present article is to verify a compact containment condition in the more general setup of Polish trait-spaces and general mutation kernels that allow for a dependence on the parent's trait. As a by-product, a result on the paths of individuals is obtained. An application to evolving genealogies on marked metric measure spaces is mentioned where genealogical distance, counted in terms of the number of births without mutation, can be regarded as a trait. Because of the use of exponential times in the modeling of birth-and deathevents the analysis of the modulus of continuity of the trait-history of a particle plays a major role in obtaining appropriate bounds.