This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm...This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.展开更多
Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian ...Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.展开更多
This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covarianc...This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.展开更多
In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t...In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.展开更多
In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its loca...In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.展开更多
Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP>...Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in ℤ<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n ∈ ℤ<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = Σ<SUB> m【j≤n </SUB>ζ<SUB> j </SUB>, σ<SUP>2</SUP>(|n−m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t 】 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 【 α 【 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.展开更多
基金Supported by NSFC(Grants Nos.11671115,11731012 and 11871425)NSF(Grant No.DMS-1855185)
文摘This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.
基金Supported by the National Natural Science Foundation of China （No.10471148）, Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei （No.2003A005）Acknowledgements. We wish to express our sincere thanks to Professor Xiao Yimin for suggesting the problem to me and for his subsequent encouragement and help.
文摘Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.
基金supported by an NSERC granta startup fund of University of Alberta
文摘This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
基金Supported by the National Natural Science Foundation of China(No.10571159)Specialized Research Fund for the Doctor Program of Higher Education(No.2002335090)
文摘In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.
基金supported by the National Natural Science Foundation of China (No. 10871177)the Ph. D.Programs Foundation of Ministry of Education of China (No. 20060335032)the Natural Science Foundation of Zhejiang Province of China (No. Y7080044)
文摘In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.
基金NSERC Canada grants of Miklos Csorgo and Barbara Szyszkowicz at Carleton University,Ottawa,and by KRF-2003-C00098NSERC Canada grants at Carleton University,Ottawa
文摘Let {ξ<SUB> j </SUB>; j ∈ ℤ<SUB>+</SUB><SUP> d </SUP>be a centered stationary Gaussian random field, where ℤ<SUB>+</SUB><SUP> d </SUP>is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ<SUP>d</SUP>, having nonnegative integer coordinates. For each j = (j <SUB>1 </SUB>, ..., jd) in ℤ<SUB>+</SUB><SUP> d </SUP>, we denote |j| = j <SUB>1 </SUB>... j <SUB>d </SUB>and for m, n ∈ ℤ<SUB>+</SUB><SUP> d </SUP>, define S(m, n] = Σ<SUB> m【j≤n </SUB>ζ<SUB> j </SUB>, σ<SUP>2</SUP>(|n−m|) = ES <SUP>2 </SUP>(m, n], S <SUB>n </SUB>= S(0, n] and S <SUB>0 </SUB>= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t 】 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 【 α 【 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.
