Hausdorff Dimension of Range and Graph for General Markov Processes  

Markov过程象集和图集的Hausdorff维数

作　　者：CHEN Zhi-He 陈芷禾(乐山师范学院数理学院,乐山614000)

机构地区：[1]School of Mathematics and Physics,Leshan Normal University,Leshan,614000,China

出　　处：《应用概率统计》2024年第6期942-956,共15页Chinese Journal of Applied Probability and Statistics

基　　金：supported by Leshan Normal University Scientific Research Start-up Project for Introducing High-level Talents(Grand No.RC2024001).

摘　　要：We establish the Hausdorff dimension of the graph of general Markov processes on Rd based on some probability estimates of the processes staying or leaving small balls in small time.In particular,our results indicate that,for symmetric diffusion processes(withα=2)or symmetricα-stable-like processes(withα∈(0,2))on Rd,it holds almost surely that dimH GrX([0,1])=1{α<1}+(2−1/α)1{α≥1,d=1}+(d∧α)1{α≥1,d≥2}.We also systematically prove the corresponding results about the Hausdorff dimension of the range of the processes.在假定小时间内过程停留或离开小球概率估计的条件下,我们建立R^(d)上一般Markov过程图集的Hausdorff维数.特别地,我们的结果表明在Rd上对称扩散过程(α=2)或对称α-stable型过程(α∈(0,2)),几乎处处有dimH GrX([0,1])=1{α<1}+(2−1/α)1{α≥1,d=1}+(d∧α)1{α≥1,d≥2}.同时我们也系统地证明了Markov过程象集Hausdorff维数的相关结果.

关 键 词：Markov process Hausdorff dimension RANGE GRAPH 

分 类 号：O211.6[理学—概率论与数理统计]