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 ...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.展开更多
For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial...For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial quotients.In this paper,we establish the Hausdorff dimension of the exceptional set where the growth rate is a general function.展开更多
基金supported by Leshan Normal University Scientific Research Start-up Project for Introducing High-level Talents(Grand No.RC2024001).
文摘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.
基金Supported by Projects from Chongqing Municipal Science and Technology Commission(CSTB2022NSCQ-MSX0445)。
文摘For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial quotients.In this paper,we establish the Hausdorff dimension of the exceptional set where the growth rate is a general function.
