Spine decomposition and L log L criterion for superprocesses with non-local branching mechanisms

In this paper, we provide a pathwise spine decomposition for superprocesses with both local and non-local branching mechanisms under a martingale change of measure. This result complements the related results obtained in Evans (1993), Kyprianou et al. (2012) and Liu, Ren and Song (2009) for superprocesses with purely local branching mechanisms and in Chen, Ren and Song (2016) and Kyprianou and Palau (2016) for multitype superprocesses. As an application of this decomposition, we obtain necessary/sufficient conditions for the limit of the fundamental martingale to be non-degenerate. In particular, we obtain extinction properties of superprocesses with non-local branching mechanisms as well as a Kesten-Stigum $L\log L$ theorem for the fundamental martingale.

[1]  R. Song,et al.  Weak extinction versus global exponential growth of total mass for superdiffusions , 2013, 1301.6842.

[2]  Zhan Shi,et al.  Weak convergence for the minimal position in a branching random walk: A simple proof , 2010, Period. Math. Hung..

[3]  Zhen-Qing Chen,et al.  Skeleton Decomposition and Law of Large Numbers for Supercritical Superprocesses , 2017, Acta Applicandae Mathematicae.

[4]  Zhi-Ming Ma,et al.  Perturbation of Dirichlet forms—Lower semiboundedness, closability, and form cores , 1991 .

[5]  Zhan Shi,et al.  The Seneta–Heyde scaling for the branching random walk , 2011, 1102.0217.

[6]  J. D. Biggins,et al.  Measure change in multitype branching , 2004, Advances in Applied Probability.

[7]  Russell Lyons,et al.  A Conceptual Proof of the Kesten-Stigum Theorem for Multi-Type Branching Processes , 1997 .

[8]  Extinction properties of multi-type continuous-state branching processes , 2016, Stochastic Processes and their Applications.

[9]  Joanna Mitro.  General theory of markov processes , 1991 .

[10]  Zhen-Qing Chen,et al.  Two-sided heat kernel estimates for censored stable-like processes , 2010 .

[11]  Ting Yang,et al.  Law of large numbers for supercritical superprocesses with non-local branching , 2018, Stochastic Processes and their Applications.

[12]  Renming Song,et al.  Heat kernel estimates for the Dirichlet fractional Laplacian , 2009 .

[13]  Zhen-Qing Chen.  Gaugeability and conditional gaugeability , 2002 .

[14]  Russell Lyons,et al.  A Simple Path to Biggins’ Martingale Convergence for Branching Random Walk , 1998, math/9803100.

[15]  H. Hering.  Uniform primitivity of semigroups generated by perturbed elliptic differential operators , 1978, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  L log L criterion for a class of multitype superdiffusions with non-local branching mechanisms , 2017, Science China Mathematics.

[17]  Laurence Marsalle,et al.  Limit theorems for Markov processes indexed by continuous time Galton--Watson trees , 2009, 0911.1973.

[18]  R. Song,et al.  INTRINSIC ULTRACONTRACTIVITY OF NON-SYMMETRIC DIFFUSION SEMIGROUPS IN BOUNDED DOMAINS , 2008 .

[19]  A. E. Kyprianou,et al.  Supercritical super-Brownian motion with a general branching mechanism and travelling waves , 2010, 1005.3659.

[20]  H. Kesten,et al.  A Limit Theorem for Multidimensional Galton-Watson Processes , 1966 .

[21]  R. Pinsky.  ON THE CONSTRUCTION AND SUPPORT PROPERTIES OF MEASURE-VALUED DIFFUSIONS ON D ⊆ R d WITH SPATIALLY DEPENDENT BRANCHING , 2001 .

[22]  M. Fukushima,et al.  Dirichlet forms and symmetric Markov processes , 1994 .

[23]  S. Evans.  Two representations of a conditioned superprocess , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[24]  Andreas E. Kyprianou,et al.  Local extinction versus local exponential growth for spatial branching processes , 2004 .

[25]  Zenghu Li,et al.  Measure-Valued Branching Markov Processes , 2010, Probability Theory and Stochastic Modelling.

[26]  Matthew I. Roberts,et al.  Measure changes with extinction , 2008, 0811.1696.

[27]  Shinzo Watanabe,et al.  A Construction of Markov Processes by Piecing Out , 1966 .

[28]  S. Evans,et al.  Weighted Occupation Time for Branching Particle Systems and a Representation for the Supercritical Superprocess , 1994, Canadian Mathematical Bulletin.

[29]  Rick Durrett.  Probability , 2019 .

[30]  J. Zabczyk,et al.  Ergodicity for Infinite Dimensional Systems: Invariant measures for stochastic evolution equations , 1996 .

[31]  M. Takeda,et al.  Absolute continuity of symmetric Markov processes , 2004, math/0410108.

[32]  H. H. Schaefer.  Banach Lattices and Positive Operators , 1975 .

[33]  L. Rogers.  GENERAL THEORY OF MARKOV PROCESSES , 1989 .

[34]  Russell Lyons,et al.  Conceptual proofs of L log L criteria for mean behavior of branching processes , 1995 .

[35]  R. Song,et al.  L Log L Criterion for a Class of Superdiffusions , 2009, Journal of Applied Probability.

[36]  R. Pinsky,et al.  On the Construction and Support Properties of Measure-Valued Diffusions on $D \subseteq \mathbb{R}^d$ with Spatially Dependent Branching , 1999 .

[37]  Asymptotic behavior of critical irreducible multi-type continuous state and continuous time branching processes with immigration , 2014, 1404.2242.

[38]  Zhen-Qing Chen,et al.  Heat kernel estimates for stable-like processes on d-sets , 2003 .

[39]  K. Kaleta,et al.  Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of L\'{e}vy processes , 2012, 1209.4220.

[40]  Zhen-Qing Chen,et al.  Conditional gauge theorem for non-local Feynman-Kac transforms , 2003 .

[41]  R. Durrett.  Probability: Theory and Examples , 1993 .

[42]  Super-Brownian Motion: Lp-Convergence of Martingales Through the Pathwise Spine Decomposition , 2011, 1106.2678.

[43]  Zhen-Qing Chen,et al.  Law of Large Numbers for Branching Symmetric Hunt Processes with Measure-Valued Branching Rates , 2015, 1503.00106.

[44]  R. Hardy,et al.  A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales , 2009 .

[45]  Nonlocal Branching Superprocesses and Some Related Models , 2002, math/0606616.

[46]  Zhen-Qing Chen,et al.  Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation , 2011, 1112.3401.

[47]  福島 正俊,et al.  Symmetric Markov Processes, Time Change, and Boundary Theory , 2011 .

[48]  Yueyun Hu,et al.  Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees , 2007, math/0702799.

[49]  P. Stollmann,et al.  Perturbation of Dirichlet forms by measures , 1996 .

[50]  Zhen-Qing Chen,et al.  Drift transforms and Green function estimates for discontinuous processes , 2003 .