Search

We construct superprocesses with dependent spatial motion (SDSMs) in Euclidean spaces ℝd with d ≥ 1 and show that, even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on ℝd, their local times exist when d ≤ 3. A Tanaka formula of the local time is also derived. [ABSTRACT FROM AUTHOR]

We correct the statements of the non-critical convergence theorems in Gonzalez et al. (Probab Theory Rel Fields, 2022), principally correcting the recursive constants that appear in the limits. [ABSTRACT FROM AUTHOR]

We consider the Dawson–Watanabe superprocess obtained from a spatial motion with sub-Gaussian transition densities on a metric measure space with finite Hausdorff dimension, and examine the dimensions of the range and the set of times when the support intersects a given set, generalising results of Serlet and Tribe. As intermediate results, we prove existence of local times for the superprocess if the spectral dimension of the spatial motion satisfies d s < 4 , and prove that (2 − d s / 2) ∧ 1 is the critical Hölder-continuity exponent in the time variable. Furthermore, we prove a bound on moments of the integrated superprocess, and give uniform upper bounds on the mass the superprocess assigns to small balls, generalising a result of Perkins. [ABSTRACT FROM AUTHOR]

Suppose that X = (X t , t ≥ 0) is either a superprocess or a branching Markov process on a general space E, with non-local branching mechanism and probabilities P δ x , when issued from a unit mass at x ∈ E . For a general setting in which the first moment semigroup of X displays a Perron–Frobenius type behaviour, we show that, for k ≥ 2 and any positive bounded measurable function f on E, lim t → ∞ g k (t) E δ x [ ⟨ f , X t ⟩ k ] = C k (x , f) , where the constant C k (x , f) can be identified in terms of the principal right eigenfunction and left eigenmeasure and g k (t) is an appropriate deterministic normalisation, which can be identified explicitly as either polynomial in t or exponential in t, depending on whether X is a critical, supercritical or subcritical process. The method we employ is extremely robust and we are able to extract similarly precise results that additionally give us the moment growth with time of ∫ 0 t ⟨ f , X t ⟩ d s , for bounded measurable f on E. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our recent paper (Electron. J. Probab., 24(141), (2019)) and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes (Xt)t≥0 with branching mechanisms of infinite second moments. In the aforementioned paper, we proved stable central limit theorems for Xt(f) for some functions f of polynomial growth in three different regimes. However, we were not able to prove central limit theorems for Xt(f) for all functions f of polynomial growth. In this note, we show that the limiting stable random variables in the three different regimes are independent, and as a consequence, we get stable central limit theorems for Xt(f) for all functions f of polynomial growth. [ABSTRACT FROM AUTHOR]

Using a Tanaka representation of the local time for a class of superprocesses with dependent spatial motion, as well as sharp estimates from the theory of uniformly parabolic partial differential equations, the joint Hölder continuity in time and space of said local times is obtained in two and three dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

It is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP). In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever. Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open. [ABSTRACT FROM AUTHOR]

Suppose that X = {Xt, t ⩾ 0;ℙμ} is a supercritical superprocess in a locally compact separable metric space E. Let φ0 be a positive eigenfunction corresponding to the first eigenvalue λ0 of the generator of the mean semigroup of X. Then M t := e − λ 0 t 〈 ϕ 0 , X t 〉 is a positive martingale. Let M∞ be the limit of Mt. It is known (see Liu et al. (2009)) that M∞ is non-degenerate if and only if the L log L condition is satisfied. In this paper we are mainly interested in the case when the L log L condition is not satisfied. We prove that, under some conditions, there exist a positive function γt on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E, M t := e - λ 0 t 〈 φ 0 , X t 〉 We also give the almost sure limit of γt〈f, Xt〉 for a class of general test functions f. [ABSTRACT FROM AUTHOR]

The goal of this paper is twofold. First, we establish skeleton and spine decompositions for superprocesses whose underlying processes are general symmetric Hunt processes. Second, we use these decompositions to obtain weak and strong law of large numbers for supercritical superprocesses where the spatial motion is a symmetric Hunt process on a locally compact metric space E and the branching mechanism takes the form ψβ(x,λ)=−β(x)λ+α(x)λ2+∫(0,∞)(e−λy−1+λy)π(x,dy) with β∈Bb(E), α∈Bb+(E) and π being a kernel from E to (0,∞) satisfying supx∈E∫(0,∞)(y∧y2)π(x,dy)<∞. The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap. Our results cover many interesting examples of superprocesses, including super Ornstein-Uhlenbeck process and super stable-like process. The strong law of large numbers for supercritical superprocesses are obtained under the assumption that the strong law of large numbers for an associated supercritical branching Markov process holds along a discrete sequence of times, extending an earlier result of Eckhoff et al. (Ann. Probab. 43(5):2594-2659, 2015) for superdiffusions to a large class of superprocesses. The key for such a result is due to the skeleton decomposition of superprocess, which represents a superprocess as an immigration process along a supercritical branching Markov process. [ABSTRACT FROM AUTHOR]

Stochastic solutions provide new rigorous results for nonlinear PDE’s and, through its local nongrid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions: McKean’s and superprocesses. In favour of superprocesses is the fact that they handle arbitrary boundary conditions. However, when restricted to measures, superprocesses can only be used to generate solutions for a limited class of nonlinear PDE’s. A new class of superprocesses, namely superprocesses on ultradistributions, is proposed to extend the stochastic solution approach to a wider class of PDE’s. [ABSTRACT FROM AUTHOR]

Abstract: We perform various changes of measure in the lookdown particle system of Donnelly and Kurtz. The first example is a product type -transform related to conditioning a Generalized Fleming–Viot process without mutation on coexistence of some genetic types in remote time. We give a pathwise construction of this -transform by just “forgetting” some reproduction events in the lookdown particle system. We also provide an intertwining relationship for the Wright–Fisher diffusion and explicit the associated pathwise decomposition. The second example, called the linear or additive -transform, concerns a wider class of measure-valued processes with spatial motion. Applications include a simple description of the additive -transform of the Generalized Fleming–Viot process and an immortal particle representation for the additive -transform of the Dawson–Watanabe process. [Copyright &y& Elsevier]

We offer a probabilistic treatment of die classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the reasoning in FCyprianou (Ann. Inst. Henri Poincare Prohab. Stat. 40 (2004) 53-72) for branching Brownian motion, die current paper offers a number of new insights. Our analysis incorporates the roleof Seneta-Heydenorming which, in the current setting, draws on classical work of Grey (J.AppL Prabab. II (1974)669-677). We give 3.path-wise explanation of Evans' immortal particle picture (the spine decomposition) which uses the Dynkin-Kuznetsov N-measure as a key ingredient. Moreover, in the spirit of Neveu's stopping lines we make repeated use of Dynkin's exit measures. Additional complications arise from the general nature of the branching mechanism. As a consequence of the analysis we also offer an exact X(log X)2 moment dichotomy for the almost sure convergence of the so-called derivative martingale at its critical parameter to a non-trivial limit. This differs to the case of branching Brownian motion {Aim. Inst. Henri Poincare Prabab, Stat. 40 (2004) 53-72), and branching random walk {Adv. in Appl. Prabab. 36 (2004) 544-581), where a moment 'gap' appears in the necessary and sufficient conditions. Our probabilistic treatment allows us to replicate known existence, uniqueness and asymptotic results for the travelling wave equation, which is related to a super-Brownian motion. [ABSTRACT FROM AUTHOR]

Let ( X) be a super-Brownian motion in a bounded domain D in ℝ. The random measure Y(·) = ∫ X(·) dt is called the total weighted occupation time of ( X). We consider the regularity properties for the densities of a class of Y. When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y( dx)-measure. [ABSTRACT FROM AUTHOR]

Abstract: We develop an idea of Evans and O’Connell (1994) , Engländer and Pinsky (1999) and Duquesne and Winkel (2007) by giving a pathwise construction of the so-called ‘backbone’ decomposition for supercritical superprocesses. Our results also complement a related result for critical -superprocesses given in Etheridge and Williams (2003) . Our approach relies heavily on the use of Dynkin–Kuznetsov -measures. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

The stochastic construction of solutions for nonlinear partial differential equations is discussed. The technique provides new exact solutions and efficient numerical codes for localized solutions and parallel computing. A brief review of known solutions and some new results for the Poisson-Vlasov and Euler equations are included. [ABSTRACT FROM AUTHOR]

Abstract: The martingale problem for superprocesses with parameters is studied where may not be absolutely continuous with respect to the Lebesgue measure. This requires a generalization of the concept of martingale problem: we show that for any process which partially solves the martingale problem, an extended form of the liftings defined in [E.B. Dynkin, S.E. Kuznetsov, A.V. Skorohod, Branching measure-valued processes, Probab. Theory Related Fields 99 (1995) 55–96] exists; these liftings are part of the statement of the full martingale problem, which is hence not defined for processes who fail to solve the partial martingale problem. The existence of a solution to the martingale problem follows essentially from Itô’s formula. The proof of uniqueness requires that we find a sequence of -superprocesses “approximating” the -superprocess, where has the form . Using an argument in [N. El Karoui, S. Roelly-Coppoletta, Propriété de martingales, explosion et représentation de Lévy–Khintchine d’une classe de processus de branchement à valeurs mesures, Stochastic Process. Appl. 38 (1991) 239–266], applied to the -superprocesses, we prove, passing to the limit, that the full martingale problem has a unique solution. This result is applied to construct superprocesses with interactions via a Dawson–Girsanov transformation. [Copyright &y& Elsevier]

We extend the class of (ξ, ψ, K)-superprocesses known so far by applying a simple transformation induced by a “weight function” for the one-particle motion. These transformed superprocesses may exist under weak conditions on the branching parameters, and their state space automatically extends to a certain space of possibly infinite Radon measures. It turns out that a number of superprocesses which were so far not included in the general theory fall into this class. For instance, the hyperbolic branching catalyst of Fleischmann and Mueller(12) is included and we are able to extend it to the case of β-branching. In the second part of this paper, we discuss regularity properties of our processes. Under the assumption that the one-particle motion is a Hunt process, we show that our superprocesses possess right versions having càdlàg paths with respect to a natural topology on the state space. The proof uses an approximation with branching particle systems on Skorohod space. [ABSTRACT FROM AUTHOR]