Dawson Watanabe Superprocesses Research Articles

Measure-valued affine and polynomial diffusions

We introduce a class of measure-valued processes, which – in analogy to their finite dimensional counterparts – will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e. a representation of the conditional marginal moments via a system of finite dimensional linear PDEs. Furthermore, we characterize the corresponding infinitesimal generators obtaining a representation analogous to polynomial diffusions on R+m, in cases where their domain is large enough. In general the infinite dimensional setting allows for richer specifications strictly beyond this representation. As a special case, we recover measure-valued affine diffusions, sometimes also called Dawson–Watanabe superprocesses. From a mathematical finance point of view, the polynomial framework is especially attractive since it allows to transfer many famous finite dimensional models and their tractability properties to an infinite dimensional measure-valued setting.

A control problem with fuel constraint and Dawson–Watanabe superprocesses

We solve a class of control problems with fuel constraint by means of the log-Laplace transforms of $J$-functionals of Dawson–Watanabe superprocesses. This solution is related to the superprocess solution of quasilinear parabolic PDEs with singular terminal condition. For the probabilistic verification proof, we develop sharp bounds on the blow-up behavior of log-Laplace functionals of $J$-functionals, which might be of independent interest.

Lebesgue approximation of [formula omitted]-superprocesses

Let ξ=(ξt) be a locally finite (2,β)-superprocess in Rd with β<1 and d>2/β. Then for any fixed t>0, the random measure ξt can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the ε-neighborhoods of suppξt. This extends the Lebesgue approximation of Dawson–Watanabe superprocesses. Our proof is based on a truncation of (α,β)-superprocesses and uses bounds and asymptotics of hitting probabilities.

Local conditioning in Dawson–Watanabe superprocesses

Consider a locally finite Dawson–Watanabe superprocess $\xi=(\xi_{t})$ in $\mathsf{R}^{d}$ with $d\geq2$. Our main results include some recursive formulas for the moment measures of $\xi$, with connections to the uniform Brownian tree, a Brownian snake representation of Palm measures, continuity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of $\xi_{t}$ by a stationary cluster $\tilde{\eta}$ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of $\xi_{t}$ for a fixed $t>0$, given that $\xi_{t}$ charges the $\varepsilon$-neighborhoods of some points $x_{1},\ldots,x_{n}\in\mathsf{R}^{d}$. In the limit as $\varepsilon\to0$, the restrictions to those sets are conditionally independent and given by the pseudo-random measures $\tilde{\xi}$ or $\tilde{\eta}$, whereas the contribution to the exterior is given by the Palm distribution of $\xi_{t}$ at $x_{1},\ldots,x_{n}$. Our proofs are based on the Cox cluster representations of the historical process and involve some delicate estimates of moment densities.

FLEMING–VIOT PROCESSES IN AN ENVIRONMENT

We consider a new type of lookdown processes where spatial motion of each individual is influenced by an individual noise and a common noise, which could be regarded as an environment. Then a class of probability measure-valued processes on real line ℝ is constructed. The sample path properties are investigated: the values of this new type process are either purely atomic measures or absolutely continuous measures according to the existence of individual noise. When the process is absolutely continuous with respect to Lebesgue measure, we derive a new stochastic partial differential equation for the density process. At last we show that such processes also arise from normalizing a class of measure-valued branching diffusions in a Brownian medium as the classical result that Dawson–Watanabe superprocesses, conditioned to have total mass one, are Fleming–Viot superprocesses.

Distribution and propagation properties of superprocesses with general branching mechanisms

A theorem of Evans and Perkins (1991) on the absolute con- tinuity of distributions of Dawson-Watanabe superprocesses is extended to general branching mechanisms. This result is then used to establish the prop- agation properties of the superprocesses following Evans and Perkins (1991) and Perkins (1990).

Some local approximations of Dawson–Watanabe superprocesses

Let ξ be a Dawson–Watanabe superprocess in ℝd such that ξt is a.s. locally finite for every t≥0. Then for d≥2 and fixed t>0, the singular random measure ξt can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the ɛ-neighborhoods of supp ξt. When d≥3, the local distributions of ξt near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure ξ. By contrast, the corresponding distributions for d=2 are locally invariant. Further results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities. Our main proofs are based on a detailed analysis of the historical structure of ξ.

An almost sure scaling limit theorem for Dawson–Watanabe superprocesses

We establish a scaling limit theorem for a large class of Dawson–Watanabe superprocesses whose underlying spatial motions are symmetric Hunt processes, where the convergence is in the sense of convergence in probability. When the underling process is a symmetric diffusion with C b 1 -coefficients or a symmetric Lévy process on R d whose Lévy exponent Ψ ( η ) is bounded from below by c | η | α for some c > 0 and α ∈ ( 0 , 2 ) when | η | is large, a stronger almost sure limit theorem is established for the superprocess. Our approach uses the principal eigenvalue and the ground state for some associated Schrödinger operator. The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap.

Some Problems of Local Hitting, Scaling, and Conditioning

The aim of this paper is to convey the basic ideas regarding some local hitting and conditioning properties of random measures. Some very general results are obtainable in the special case of simple point processes. Though much of this theory has no counterpart for general random measures, similar results do exist in two cases of special interest—for local time random measures and for Dawson–Watanabe superprocesses. This is an informal account of some of the basic hitting and conditioning properties common to all three cases. Precise statements and proofs will be provided elsewhere.

A LIMIT THEOREM FOR INTERACTING MEASURE-VALUED BRANCHING PROCESSES

It is well known that Fleming-Viot superprocesses can be obtained from the Dawson-Watanabe superprocesses by conditioning the latter to have constant total mass. The same question is investigated for measure-valued branching processes with interacting intensity independent of the geographical position. It is showed that a sequence of conditioned probability laws of this kind of interacting measure-valued branching processes also approximates to the probability law of Fleming-Viot superprocesses.

Immigration structures associated with Dawson-Watanabe superprocesses

The immigration structure associated with a measure-valued branching process may be described by a skew convolution semigroup. For the special type of measure-valued branching process, the Dawson-Watanabe superprocess, we show that a skew convolution semigroup corresponds uniquely to an infinitely divisible probability measure on the space of entrance laws for the underlying process. An immigration process associated with a Borel right superprocess does not always have a right continuous realization, but it can always be obtained by transformation from a Borel right one in an enlarged state space.

A probabilistic approach to blow-up of a semilinear heat equation

We introduce a probabilistic approach to the study of blow-up of positive solutions to a class of semilinear heat equations. This then gives a representation of the coefficients in the power series expansion of the solutions. In a special case, this approach leads to a path-valued Markov process which can also be understood via the theory of Dawson-Watanabe superprocesses. We demonstrate the utility of the approach by proving a result on ‘complete blow-up’ of solutions.