上临界超过程的一类强极限的性质
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摘要
假设X={X_t, t≥0; P_μ}是局部紧可分距离空间E上的上临界超过程,Ф_0是X的均值半群的生成元的与第一特征值λ_0对应的正特征函数,则M_t:=e~(-λ_0t)<Ф_0, X_t>是非负鞅.令M_∞是M_t的极限,则M_∞是非退化的当且仅当L log L条件成立.当L log L条件不一定成立时,最近, Ren等(2017)证明了存在定义在[0,∞)上的非负函数γ_t及非退化随机变量W使得对任意E上非零Borel有限测度μ,有lim_(t→∞)γ_t
Suppose that X = {X_t, t≥0; P_μ} is a supercritical superprocess in a locally compact separable metric space E. Let Ф_0 be a positive eigenfunction corresponding to the first eigenvalue λ_0 of the generator of the mean semigroup of X. Then M_t:=e~(-λ_0t)<Ф_0, X_t> is a positive martingale. Let M_∞ be the limit of Mt. It is known that M_∞ is non-degenerate iff the L log L condition is satisfied. When the L log L condition may not be satisfied,Ren et al.(2017) recently proved that there exist a non-negative function γ_t on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E, lim_(t→∞) γ_t
Suppose that X = {X_t, t≥0; P_μ} is a supercritical superprocess in a locally compact separable metric space E. Let Ф_0 be a positive eigenfunction corresponding to the first eigenvalue λ_0 of the generator of the mean semigroup of X. Then M_t:=e~(-λ_0t)<Ф_0, X_t> is a positive martingale. Let M_∞ be the limit of Mt. It is known that M_∞ is non-degenerate iff the L log L condition is satisfied. When the L log L condition may not be satisfied,Ren et al.(2017) recently proved that there exist a non-negative function γ_t on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E, lim_(t→∞) γ_t
引文
1 Seneta E.On recent theorems concerning the supercritical Galton-Watson process.Ann Math Statist,1968,39:2098-2102
2 Heyde C C.Extension of a result of Seneta for the super-critical Galton-Watson process.Ann Math Statist,1970,41:739-742
3 Harris T E.The Theory of Branching Processes.Berlin:Springer,1963
4 Stigum B P.A theorem on the Galton-Watson process.Ann Math Statist,1966,37:695-698
5 Athreya K B.On the absolute continuity of the limit random variable in the supercritical Galton-Watson branching process.Proc Amer Math Soc,1971,30:563-565
6 Morters P,Ortgiese M.Small value probabilities via the branching tree heuristic.Bernoulli,2008,14:277-299
7 Biggins J D,Bingham N H.Large deviations in the supercritical branching process.Adv in Appl Probab,1993,25:757-772
8 Jones O D.Large deviations for supercritical multitype branching processes.J Appl Probab,2004,41:703-720
9 Hering H.The non-degenerate limit for supercritical branching diffusions.Duke Math J,1978,45:561-600
10 Ren Y-X,Song R,Zhang R.Supercritical superprocesses:Proper normalization and non-degenerate strong limit.Sci China Math,2019,in press
11 Li Z.Measure-Valued Branching Markov Processes.Heidelberg:Springer,2011
12 Dynkin E B.Superprocesses and partial differential equations.Ann Probab,1993,21:1185-1262
13 Ren Y-X,Song R,Zhang R.Central limit theorems for supercritical branching nonsymmetric Markov processes.Ann Probab,2017,45:564-623
14 Schaefer H H.Banach Lattices and Positive Operators.New York:Springer,1974
15 Ren Y-X,Song R,Zhang R.Limit theorems for some critical superprocesses.Illinois J Math,2015,59:235-276
16 Davies E B,Simon B.Ultracontractivity and the heat kernel for Schr¨odinger operators and Dirichlet Laplacians.JFunct Anal,1984,59:335-395
17 Liu R L,Ren Y-X,Song R.Llog L criterion for a class of superdiffusions.J Appl Probab,2009,46:479-496
18 Chen Z Q,Ren Y-X,Yang T.Skeleton decomposition and law of large numbers for supercritical superprocesses.Acta Appl Math,2019,159:225-285
19 Eckhoff M,Kyprianou A E,Winkel M.Spines,skeletons and the strong law of large numbers for superdiffusions.Ann Probab,2015,43:2594-2659
20 Kim P,Song R.Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains.Tohoku Math J(2),2008,60:527-547
21 Ren Y-X,Song R,Zhang R.Williams decomposition for superprocesses.Electron J Probab,2018,23:1-33
22 Seneta E.A Tauberian theorem of E.Landau and W.Feller.Ann Probab,1973,1:1057-1058
2 Heyde C C.Extension of a result of Seneta for the super-critical Galton-Watson process.Ann Math Statist,1970,41:739-742
3 Harris T E.The Theory of Branching Processes.Berlin:Springer,1963
4 Stigum B P.A theorem on the Galton-Watson process.Ann Math Statist,1966,37:695-698
5 Athreya K B.On the absolute continuity of the limit random variable in the supercritical Galton-Watson branching process.Proc Amer Math Soc,1971,30:563-565
6 Morters P,Ortgiese M.Small value probabilities via the branching tree heuristic.Bernoulli,2008,14:277-299
7 Biggins J D,Bingham N H.Large deviations in the supercritical branching process.Adv in Appl Probab,1993,25:757-772
8 Jones O D.Large deviations for supercritical multitype branching processes.J Appl Probab,2004,41:703-720
9 Hering H.The non-degenerate limit for supercritical branching diffusions.Duke Math J,1978,45:561-600
10 Ren Y-X,Song R,Zhang R.Supercritical superprocesses:Proper normalization and non-degenerate strong limit.Sci China Math,2019,in press
11 Li Z.Measure-Valued Branching Markov Processes.Heidelberg:Springer,2011
12 Dynkin E B.Superprocesses and partial differential equations.Ann Probab,1993,21:1185-1262
13 Ren Y-X,Song R,Zhang R.Central limit theorems for supercritical branching nonsymmetric Markov processes.Ann Probab,2017,45:564-623
14 Schaefer H H.Banach Lattices and Positive Operators.New York:Springer,1974
15 Ren Y-X,Song R,Zhang R.Limit theorems for some critical superprocesses.Illinois J Math,2015,59:235-276
16 Davies E B,Simon B.Ultracontractivity and the heat kernel for Schr¨odinger operators and Dirichlet Laplacians.JFunct Anal,1984,59:335-395
17 Liu R L,Ren Y-X,Song R.Llog L criterion for a class of superdiffusions.J Appl Probab,2009,46:479-496
18 Chen Z Q,Ren Y-X,Yang T.Skeleton decomposition and law of large numbers for supercritical superprocesses.Acta Appl Math,2019,159:225-285
19 Eckhoff M,Kyprianou A E,Winkel M.Spines,skeletons and the strong law of large numbers for superdiffusions.Ann Probab,2015,43:2594-2659
20 Kim P,Song R.Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains.Tohoku Math J(2),2008,60:527-547
21 Ren Y-X,Song R,Zhang R.Williams decomposition for superprocesses.Electron J Probab,2018,23:1-33
22 Seneta E.A Tauberian theorem of E.Landau and W.Feller.Ann Probab,1973,1:1057-1058