Strong law of large numbers for supercritical superprocesses under second moment condition

Abstract

Consider a supercritical superprocess X = {X_t, t≥0} on a locally compact separable metric space (E,m). Suppose that the spatial motion of X is a Hunt process satisfying certain conditions and that the branching mechanism is of the form

\psi (x,\lambda )=-a(x)\lambda +b(x){\lambda }^{\mathrm{2}}+{\displaystyle {\int }_{(\mathrm{0},+\infty )}({\mathrm{e}}^{-\lambda y}-\mathrm{1}+\lambda y)n(x,\mathrm{d}y),}\hspace{1em}x\in E,\lambda >\mathrm{0},

where a\in {{B}}_b(E),b\in {{B}}_b^{+}(E), and n is a kernel from E to (0,+∞) satisfying . Put T_{t}f(x)={\mathbb{P}}_{{\delta }_x}\langle f,X_t\rangle. Suppose that the semigroup {T_t; t≥0}is compact. Let λ₀ be the eigenvalue of the (possibly non-symmetric) generator L of {T_t}that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let {\varphi }_{\mathrm{0}} and {\widehat{\varphi }}_{\mathrm{0}} be the eigenfunctions of L and \widehat{L}(the dual of L) associated with λ₀, respectively. Assume λ₀>0. Under some conditions on the spatial motion and the {\varphi }_{\mathrm{0}}-transform of the semigroup {T_t}, we prove that for a large class of suitable functions f,

\underset{t\to +\infty }{\lim }{\mathrm{e}}^{-{\lambda }_{\mathrm{0}}t}\langle f,X_t\rangle =W_{\infty }{\displaystyle {\int }_E{\widehat{\varphi }}_{\mathrm{0}}(y)f(y)m(\mathrm{d}y)},\hspace{1em}{\mathbb{P}}_{\mu }-\mathrm{a}.\mathrm{s}.,

for any finite initial measure μ on E with compact support, where W_∞ is the martingale limit defined by W_{\infty }:={\lim }_{t\to +\infty }{\mathrm{e}}^{-{\lambda }_{\mathrm{0}}t}\langle {\varphi }_{\mathrm{0}},X_t\rangle. Moreover, the exceptional set in the above limit does not depend on the initial measure μ and the function f.

Keywords

Superprocess / scaling limit theorem / Hunt process / spectral gap / h-transform / martingale measure

References

[1] Bass R F, Chen Z-Q. Brownian motion with singular drift. Ann Probab, 2013, 31: 791-817

[2] Chen Z-Q, Kim P, Song R. Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann Probab, 2012, 40: 2483-2538

[3] Chen Z-Q, Kim P, Song R. Dirichlet heat kernel estimates for rotationally symmetric Lévy processes. Proc Lond Math Soc, 2014, 109: 90-120

[4] Chen Z-Q, Kim P, Song R. Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbations. Trans Amer Math Soc, 2015, 367: 5237-5270

[5] Chen Z-Q, Kim P, Song R. Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components. J Reine Angew Math, Ahead of print,

[6] Chen Z-Q, Kumagai T. A priori Höolder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps. Rev Mat Iberoam, 2010, 26: 551-589

[7] Chen Z-Q, Ren Y-X, Wang H. An almost sure scaling limit theorem for Dawson-Watanabe superprocesses. J Funct Anal, 2008, 254: 1988-2019

[8] Chen Z-Q, Yang T. Dirichlet heat kernel estimates for fractional Laplacian under non-local perturbation. Preprint, 2015, arXiv: 1503.05302

[9] Davies E B, Simon B. Ultracontractivity and the kernel for Schrödinger operators and Dirichlet Laplacians. J Funct Anal, 1984, 59: 335-395

[10] Eckhoff M, Kyprianou A E, Winkel M. Spine, skeletons and the strong law of large numbers for superdiffusion. Ann Probab (To appear), arXiv: 1309.6196

[11] Engländer J. Law of large numbers for superdiffusions: The non-ergodic case. Ann Inst Henri Poincaré Probab Stat, 2009, 45: 1-6

[12] Engländer J, Harris S C, Kyprianou A E. Strong law of large numbers for branching diffusions. Ann Inst Henri Poincaré Probab Stat, 2010, 46(1): 279-298

[13] Engländer J, Turaev D. A scaling limit theorem for a class of superdiffusions. Ann Probab, 2002, 30: 683-722

[14] Engländer J, Winter A. Law of large numbers for a class of superdiffusions. Ann Inst Henri Poincaré Probab Stat, 2006, 42: 171-185

[15] Kim P, Song R. Two-sided estimates on the density of Brownian motion with singular drift. Illinois J Math, 2006, 50: 635-688

[16] Kim P, Song R. On dual processes of non-symmetric diffusions with measure-valued drifts. Stochastic Process Appl, 2008, 118: 790-817

[17] Kim P, Song R. Intrinsic ultracontractivity of non-symmetric diffusions with measurevalued drifts and potentials. Ann Probab, 2008, 36: 1904-1945

[18] Kim P, Song R. Stable process with singular drift. Stochastic Process Appl, 2014, 124: 2479-2516

[19] Kim P, Song R. Dirichlet heat kernel estimates for stable processes with singular drift in unbounded C^1,1 open sets. Potential Anal, 2014, 41: 555-581

[20] Kouritzin M A, Ren Y-X. A strong law of large numbers for super-stable Processes. Stochastic Process Appl, 2014, 121: 505-521

[21] Li Z. Measure-valued Branching Markov Processes. Heidelberg: Springer, 2011

[22] Liu R-L, Ren Y-X, Song R. Strong law of large numbers for a class of superdiffusions. Acta Appl Math, 2013, 123: 73-97

[23] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer, 1983

[24] Ren Y-X, Song R, Zhang R. Central limit theorems for supercritical superprocesses. Stochastic Process Appl, 2015, 125: 428-457

[25] Ren Y-X, Song R, Zhang R. Central limit theorems for supercritical branching nonsymmetric Markov processes. Ann Probab (to appear), arXiv: 1404.0116

[26] Ren Y-X, Song R, Zhang R. Functional central limit theorems for supercritical superprocesses. Preprint, 2014, arXiv: 1410.1598

[27] Ren Y-X, Song R, Zhang R. Limit theorems for some critical superprocesses. Preprint, 2014, arXiv: 1403.1342

[28] Sato K-I. Lévy Processes and Infinitely Divisible Distribution. Cambridge: Cambridge University Press, 1999

[29] Schaefer H H. Banach Lattices and Positive Operators. New York: Springer, 1974

[30] Stroock D W. Probability Theory. An Analytic View. 2nd ed. Cambridge: Cambridge University Press, 2011

[31] Wang L. An almost sure limit theorem for super-Brownian motion. J Theoret Probab, 2010, 23: 401-416

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