Let X^H = {X^H（8）,8∈ R^N1} and XK = {X^K（t）,t ∈R^2} be two independent anisotropic Gaussian random fields with values in R^d with indices H = （H1,... ,HN1） ∈ （0, 1）^N1, K = （K1,..., KN2）∈ （0, 1）^N2, r...Let X^H = {X^H（8）,8∈ R^N1} and XK = {X^K（t）,t ∈R^2} be two independent anisotropic Gaussian random fields with values in R^d with indices H = （H1,... ,HN1） ∈ （0, 1）^N1, K = （K1,..., KN2）∈ （0, 1）^N2, respectively. Existence of intersections of the sample paths of XH and XK is studied. More generally, let E1 R^N1, E2 R^N2 and F R^d be Borel sets. A necessary condition and a sufficient condition for P{（X^H（E1） ∩ X^K（E2）） ∩ F ≠ Ф} 〉 0 in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 x E2 x F in the metric space （R^N1＋N2＋d, ρ） are proved, whereρ is a metric defined in terms of H and K. These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets.展开更多
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
基金supported by Zhejiang Provincial Natural Science Foundation of China(Grant No. Y6100663)National Science Foundation of US (Grant No. DMS-1006903)
文摘Let X^H = {X^H（8）,8∈ R^N1} and XK = {X^K（t）,t ∈R^2} be two independent anisotropic Gaussian random fields with values in R^d with indices H = （H1,... ,HN1） ∈ （0, 1）^N1, K = （K1,..., KN2）∈ （0, 1）^N2, respectively. Existence of intersections of the sample paths of XH and XK is studied. More generally, let E1 R^N1, E2 R^N2 and F R^d be Borel sets. A necessary condition and a sufficient condition for P{（X^H（E1） ∩ X^K（E2）） ∩ F ≠ Ф} 〉 0 in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 x E2 x F in the metric space （R^N1＋N2＋d, ρ） are proved, whereρ is a metric defined in terms of H and K. These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets.
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-
