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We consider a measure-valued diffusion (i.e., a superprocess). It is determined by a couple \((L,\psi )\), where L is the infinitesimal generator of a strongly recurrent diffusion in \(\mathbb {R}^{d}\) and \(\psi \) is a branching mechanism assumed to be supercritical. Such processes are known, see for example, (Englander and Winter in Ann Inst Henri Poincare 42(2):171–185, 2006), to fulfill a law of large numbers for the spatial distribution of the mass. In this paper, we prove the corresponding central limit theorem. The limit and the CLT normalization fall into three qualitatively different classes arising from “competition” of the local growth induced by branching and global smoothing due to the strong recurrence of L. We also prove that the spatial fluctuations are asymptotically independent of the fluctuations of the total mass of the process.

Central limit theorems of the occupation time of a superprocess over a stochastic flow are proved. For the critical and higher dimensions d≥4, the limits are Gaussian variables. For d=3, the limit is conditional Gaussian. When the stochastic flow disappears, the results degenerate to those for the ordinary super-Brownian motion.

Consider the one-dimensional catalytic super-Brownian motion X (called the reactant) in the catalytic medium $$\varrho$$ which is an autonomous classical super-Brownian motion. We characterize $$(\varrho ,X)$$ both in terms of a martingale problem and (in dimension one) as solution of a certain stochastic partial differential equation. The focus of this paper is for dimension one the analysis of the longtime behavior via a mass-time-space rescaling. When scaling time by a factor of K, space is scaled by K η and mass by K −η. We show that for every parameter value η ≥ 0 the rescaled processes converge as K→ ∞ in path space. While the catalyst’s limiting process exhibits a phase transition at η = 1, the reactant’s limit is always the same degenerate process.

A superprocess with coalescing spatial motion is constructed in terms of one-dimensional excursions. Based on this construction, it is proved that the superprocess is purely atomic and arises as scaling limit of a special form of the superprocess with dependent spatial motion studied in Dawson {\it et al.} (2001) and Wang (1997, 1998).

Let η be a superprocess under the measure P. We show the existence of probability measures which are absolutely continuous with respect to P, and whose Randon–Nikodym derivatives are suitably normalized functions of the self intersection local time of η. These measures correspond to measure valued processes exhibiting a certain amount of (reinforcing) self interaction in terms of both the particle motion and the branching mechanism.

Existence and uniqueness of the solutions for some nonlinear evolution equations with measure-valued boundary conditions is established. This gives the existence of the collision local time and the collision measure for two independent (α1, d, β1) and (α2, d, β2) superprocesses without using any moment conditions on the mass processes. We obtain expressions for the Laplace transforms of the collision local time and the collision measure.

We consider systems of spatially distributed branching particles in Rd. The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on Rd of a superprocess in R+×Rd. The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.

A continuous super-Brownian motion $$X^Q $$ is constructed in which branching occurs only in the presence of catalysts which evolve themselves as a continuous super-Brownian motion $$Q$$ . More precisely, the collision local time $$L_{[W,Q]}$$ (in the sense of Barlow et al. (1)) of an underlying Brownian motion path W with the catalytic mass process $$Q$$ goerns the branching (in the sense of Dynkin's additive functional approach). In the one-dimensional case, a new type of limit behavior is encountered: The total mass process converges to a limit without loss of expectation mass (persistence) and with a nonzero limiting variance, whereas starting with a Lebesgue measure $$\ell$$ , stochastic convergence to $$\ell$$ occurs.

We start with a system of independent branching Brownian motions which, properly organized and normalized, generate a super Brownian motion in the high density limit. We introduce a weak interaction between the particles, that affects the diffusion but not the branching. The interaction is chosen in such a way that the infinite density limit is absolutely continuous with respect to the noninteracting system. We find that, despite the fact that the interaction mechanism never completely disappears, the limiting superprocesses areidentical. We study what actually happens to the interaction mechanism, or “ghost-process.”

We consider the branching treeT(n) of the first (n+1) generations of a critical branching process, conditioned on survival till time βn for some fixed β>0 or on extinction occurring at timekn withkn/n→β. We attach to each vertexv of this tree a random variableX(v) and define\(S(v) = \Sigma _{w \varepsilon \pi (0,v)} X(w)\), where π(0,v) is the unique path in the family tree from its root tov. FinallyMnis the maximal displacement of the branching random walkS(·), that isMn=max{S(v):v∈T(n)}. We show that if theX(v), v∈T(n), are i.i.d. with mean 0, then under some further moment conditionn−1/2Mn converges in distribution. In particular {n−1/2Mn}n⩾1 is a tight family. This is closely related to recent results about Aldous' continuum tree and Le Gall's Brownian snake.

Continuous super-Brownian motions in two and higher dimensions are known to have singular measure states. However, by weakening the branching mechanism in an irregular way they can be forced to have absolutely continuous states. The sufficient conditions we impose are identified in a couple of examples with irregularities in only one coordinate. This includes the case of branching restricted to some densely situated ensemble of hyperplanes.