This paper investigates the property of super-Brownian motion conditioned on non-extinction. The authors obtain a representation of Laplace functional for the weighted occupation time of this class of processes. By th...This paper investigates the property of super-Brownian motion conditioned on non-extinction. The authors obtain a representation of Laplace functional for the weighted occupation time of this class of processes. By this, they get a result about the distribution of the support of it.展开更多
We study in this paper the path properties of the Brownian motion and super-Brownian motion on the fractal structure-the Sierpinski gasket. At first some results about the limiting behaviour of its increments are obta...We study in this paper the path properties of the Brownian motion and super-Brownian motion on the fractal structure-the Sierpinski gasket. At first some results about the limiting behaviour of its increments are obtained and a kind of law of iterated logarithm is proved. Then A Lower bound of the spreading speed of its corresponding super-Brownian motion is obtained.展开更多
We prove a moderate deviation principle for a super-Brownian motion with immigration of all dimensions, and consequently fill the gap between the central limit theorem and large deviation principle.
SUPERPROCESS is one kind of measure-valued branching Markov processes. In the recentdecades many results of superprocesses on Euclidean spaces and on general abstract spaces havebeen obtained, but there is only little...SUPERPROCESS is one kind of measure-valued branching Markov processes. In the recentdecades many results of superprocesses on Euclidean spaces and on general abstract spaces havebeen obtained, but there is only little work on Riemannian manifold. In this note, wewill consider the super-Brownian motions on the hyperbolic space of dimension d and give anasymptotic result of this process (see Corollary 1 below). Iscoe investigated a similar prob-展开更多
In this paper we consider a super-Brownian motion X with branching mechanism k(x)za, where k(x) ＞ 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥‖x‖-1(0 ≤ l ＜∞) fo...In this paper we consider a super-Brownian motion X with branching mechanism k(x)za, where k(x) ＞ 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥‖x‖-1(0 ≤ l ＜∞) for sufficiently large x, then X has compact support property, and for dimension d = 1, if k(x) ≥ exp(-l‖x‖)(0 ≤ l ＜∞) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d ≥3: it is O(‖x‖-(a+1))in one dimension, O(‖x‖-2(log ‖x‖)-(a+1)) in two dimensions, and O(‖x‖2) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2△u =k(x)uα.展开更多
In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then stud...In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then study the large and moderate deviations associated with this central tendency.展开更多
We prove fluctuation limit theorems for the occupation times of super-Brownian motion with immigration. The weak convergence of the processes is established, which improves the results in references. The limiting proc...We prove fluctuation limit theorems for the occupation times of super-Brownian motion with immigration. The weak convergence of the processes is established, which improves the results in references. The limiting processes are Gaussian processes.展开更多
The Brownian motion and super-Brownian motion on the Sierpinski gasket are studied. Firstly it is proved that the local extinction property is possessed by the super-Brownian motion on this fractal structure. This fac...The Brownian motion and super-Brownian motion on the Sierpinski gasket are studied. Firstly it is proved that the local extinction property is possessed by the super-Brownian motion on this fractal structure. This fact is also true even in the presence of catalyst. Secondly it is proved that the paths of the Brownian motion on the Sierpinski gasket are dense in some sense.展开更多
Moderate deviations for the quenched mean of the super-Brownian motion with random immigration are proved for 3≤d≤6, which fills in the gap between central limit theorem(CLT)and large deviation principle(LDP).
This paper gives probabilistic expressions of theminimal and maximal positive solutions of the partial differential equation -1/2△v(x) + γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its c...This paper gives probabilistic expressions of theminimal and maximal positive solutions of the partial differential equation -1/2△v(x) + γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 ＜α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.展开更多
We establish the moderate deviation for the density process of the single point catalytic super-Brownian motion. The main tools are the abstract Gaertner-Ellis theorem, Dawson-Gaertner the- orem and the contraction pr...We establish the moderate deviation for the density process of the single point catalytic super-Brownian motion. The main tools are the abstract Gaertner-Ellis theorem, Dawson-Gaertner the- orem and the contraction principle. The rate function is expressed by the Fenchel-Legendre transform of log-exponential moment generation function.展开更多
We start from a super-Brownian motion with the branching mechanism presented by Dawson and Vinogradov. Its behaviour near extinction is studied in this paper, and the main result is that the diameter of the support te...We start from a super-Brownian motion with the branching mechanism presented by Dawson and Vinogradov. Its behaviour near extinction is studied in this paper, and the main result is that the diameter of the support tends to zero almost surely at the time of extinction.展开更多
The exit measures of super Brownian motions with branching mechanism ψ(z)=z α, 1<α≤2, from a bounded smooth domain D in R d+1 are known to be absolutely continuous with respect to the surface area on D if d&l...The exit measures of super Brownian motions with branching mechanism ψ(z)=z α, 1<α≤2, from a bounded smooth domain D in R d+1 are known to be absolutely continuous with respect to the surface area on D if d<2α-1, whereas in the case d>1+2α-1, they are singular. However, if the branching is restricted to a singular hyperplane, it is proved that they have absolutely continuous states for all d ≥1.展开更多
It is proved that the occupation time of the catalytic super-Brownian motion is absolutely continuous for d = 1, and the occupation density field is jointly continuous and jointly Holder continuous.
The catalytic super_Brownian motion has been considered. If both the catalytic medium process Q and CSBM started with Lebesgue measure λ, the central limit theorem for occupation time of CSBM has been obtained in dim...The catalytic super_Brownian motion has been considered. If both the catalytic medium process Q and CSBM started with Lebesgue measure λ, the central limit theorem for occupation time of CSBM has been obtained in dimension 3 for P-λ_a.s.Q.展开更多
文摘This paper investigates the property of super-Brownian motion conditioned on non-extinction. The authors obtain a representation of Laplace functional for the weighted occupation time of this class of processes. By this, they get a result about the distribution of the support of it.
文摘We study in this paper the path properties of the Brownian motion and super-Brownian motion on the fractal structure-the Sierpinski gasket. At first some results about the limiting behaviour of its increments are obtained and a kind of law of iterated logarithm is proved. Then A Lower bound of the spreading speed of its corresponding super-Brownian motion is obtained.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.10071008 and 10121101).
文摘We prove a moderate deviation principle for a super-Brownian motion with immigration of all dimensions, and consequently fill the gap between the central limit theorem and large deviation principle.
文摘SUPERPROCESS is one kind of measure-valued branching Markov processes. In the recentdecades many results of superprocesses on Euclidean spaces and on general abstract spaces havebeen obtained, but there is only little work on Riemannian manifold. In this note, wewill consider the super-Brownian motions on the hyperbolic space of dimension d and give anasymptotic result of this process (see Corollary 1 below). Iscoe investigated a similar prob-
基金This work was supported by the N ational Natural Science Foundation of China(Grant Nos.10001020 and 10131040).
文摘In this paper we consider a super-Brownian motion X with branching mechanism k(x)za, where k(x) ＞ 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥‖x‖-1(0 ≤ l ＜∞) for sufficiently large x, then X has compact support property, and for dimension d = 1, if k(x) ≥ exp(-l‖x‖)(0 ≤ l ＜∞) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d ≥3: it is O(‖x‖-(a+1))in one dimension, O(‖x‖-2(log ‖x‖)-(a+1)) in two dimensions, and O(‖x‖2) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2△u =k(x)uα.
基金the National Natural Science Foundation of China (Grant No.10121101)
文摘In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then study the large and moderate deviations associated with this central tendency.
基金supported by National Natural Science Foundation of China (Grant No.10721091)
文摘We prove fluctuation limit theorems for the occupation times of super-Brownian motion with immigration. The weak convergence of the processes is established, which improves the results in references. The limiting processes are Gaussian processes.
文摘The Brownian motion and super-Brownian motion on the Sierpinski gasket are studied. Firstly it is proved that the local extinction property is possessed by the super-Brownian motion on this fractal structure. This fact is also true even in the presence of catalyst. Secondly it is proved that the paths of the Brownian motion on the Sierpinski gasket are dense in some sense.
基金the Program for New Century Excellent Talents in University (Grant No. 05-0143)the National Natural Science Foundation of China (Grant No. 10721091)
文摘Moderate deviations for the quenched mean of the super-Brownian motion with random immigration are proved for 3≤d≤6, which fills in the gap between central limit theorem(CLT)and large deviation principle(LDP).
基金supported by the National Natural Science Foundation of China(Grant No.10471003)Foundation for Authors Awarded Excellent Ph.D.Dissertation.
文摘This paper gives probabilistic expressions of theminimal and maximal positive solutions of the partial differential equation -1/2△v(x) + γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 ＜α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.
基金Supported by National Natural Science Foundation of China (Grant No. 11071021)
文摘We establish the moderate deviation for the density process of the single point catalytic super-Brownian motion. The main tools are the abstract Gaertner-Ellis theorem, Dawson-Gaertner the- orem and the contraction principle. The rate function is expressed by the Fenchel-Legendre transform of log-exponential moment generation function.
基金Research supported by Tianyuan FoundationPostdoctoral Foundation
文摘We start from a super-Brownian motion with the branching mechanism presented by Dawson and Vinogradov. Its behaviour near extinction is studied in this paper, and the main result is that the diameter of the support tends to zero almost surely at the time of extinction.
文摘The exit measures of super Brownian motions with branching mechanism ψ(z)=z α, 1<α≤2, from a bounded smooth domain D in R d+1 are known to be absolutely continuous with respect to the surface area on D if d<2α-1, whereas in the case d>1+2α-1, they are singular. However, if the branching is restricted to a singular hyperplane, it is proved that they have absolutely continuous states for all d ≥1.
文摘It is proved that the occupation time of the catalytic super-Brownian motion is absolutely continuous for d = 1, and the occupation density field is jointly continuous and jointly Holder continuous.
文摘The catalytic super_Brownian motion has been considered. If both the catalytic medium process Q and CSBM started with Lebesgue measure λ, the central limit theorem for occupation time of CSBM has been obtained in dimension 3 for P-λ_a.s.Q.
