在Lie群作用下不变的Markov过程
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摘要
本文的目的是,依照经典的L′evy-Khinchin三参数表示的精神,论述在Lie群作用下不变的Markov过程的表示理论.对不变Markov过程,按其一般性,我们将在三个层面中进行讨论.首先是Lie群里平移不变的Markov过程,然后是一般流形中在可迁群作用下不变的Markov过程.这两类过程都称L′evy过程,具有三参数表示.第三类过程为在不可迁群作用下不变的Markov过程.在一定条件下,这类过程可分解为一横截群轨道的径过程和一沿着群轨道的角过程.后者为时间非齐次的不变Markov过程,具有依赖于时间的三参数表示.
We present a representation theory for invariant Markov processes under Lie group actions, in the spirit of the classic Levy-Khinchin representation. We will study invariant Markov processes at three diffierent levels of generality. First, we consider Markov processes in Lie groups that are invariant under translations,and then Markov processes in manifolds that are invariant under transitive group actions. These two types of processes are called L′evy processes, and they possess a triple representation. The third type of processes are Markov processes that are invariant under non-transitive group actions. Under certain conditions, such a process may be decomposed into a radial part, that is transversal to group orbits, and an angular part, that is along an orbit. The latter is a time inhomogeneous invariant Markov process, and may be represented by a time-dependent triple.
We present a representation theory for invariant Markov processes under Lie group actions, in the spirit of the classic Levy-Khinchin representation. We will study invariant Markov processes at three diffierent levels of generality. First, we consider Markov processes in Lie groups that are invariant under translations,and then Markov processes in manifolds that are invariant under transitive group actions. These two types of processes are called L′evy processes, and they possess a triple representation. The third type of processes are Markov processes that are invariant under non-transitive group actions. Under certain conditions, such a process may be decomposed into a radial part, that is transversal to group orbits, and an angular part, that is along an orbit. The latter is a time inhomogeneous invariant Markov process, and may be represented by a time-dependent triple.
引文
1王梓坤.马尔可夫过程和今日数学.长沙:湖南科学技术出版社,1999
2 Applebaum D.L′evy Processes and Stochastic Calculus,2nd ed.Cambridge:Cambridge University Press,2009
3 Bertoin J.L′evy Processes.Cambridge:Cambridge University Press,1996
4 Sato K.L′evy Processes and Infinitely Devisible Distributions(translated from 1990 Japanese original and revised by the author).Cambridge:Cambridge University Press,1999
5 Hunt G A.Semi-groups of measures on Lie groups.Trans Amer Math Soc,1956,81:264-293
6 Feinsilver P.Processes with independent increments on a Lie group.Trans Amer Math Soc,1978,242:73-121
7 Stroock D W,Varadhan S R S.Limit theorems for random walks on Lie groups.Sankhyˉa,1973,35:277-294
8 Heyer H,Pap G.Convolution hemigroups of bounded variation on a Lie projective group.J Lond Math Soc(2),1999,59:369-384
9 Liao M.A decomposition of Markov processes via group actions.J Theoret Probab,2009,22:164-185
10 Heyer H.Probability Measures on Locally Compact Groups.New York:Springer-Verlag,1977
11 Born E.An explicit L′evy-Hin?cin formula for convolution semigroups on locally compact groups.J Theoret Probab,1989,2:325-342
12 Dani S G,McCrudden M.Convolution roots and embeddings of probability measures on Lie groups.Adv Math,2007,209:198-211
13 Liao M.Convolution of probability measures on Lie groups and homogenous spaces.Potential Anal,2015,43:707-715
14 Diaconis P.Group Representations in Probability and Statistics.Lecture Notes-Monograph Series,vol.11.Group Representations in Probability and Statistics.Hayward:Institute of Mathematical Statistics,1988
15 Heyer H.Convolution semigroups of probability measures on Gelfand pairs.Expo Math,1983,1:3-45
16 Liao M.L′evy processes and Fourier analysison compact Lie groups.Ann Probab,2004,32:1553-1573
17 Liao M,Wang L.L′evy-Khinchin formula and existence of densities for convolution semigroups on symmetric spaces.Potential Anal,2007,27:133-150
18 Applebaum D.Probability on Compact Lie Groups.New York:Springer,2014
19 Liao M.L′evy Processes in Lie Groups.Cambridge:Cambridge University Press,2004
20 Liao M,Wang L.Limiting properties of L′evy processes in symmetric spaces of noncompact type.Stoch Dyn,2012,12:1250001
21 Kallenberg O.Foundations of Modern Probability,2nd ed.New York:Springer-Verlag,2002
22 Elworthy K D.Geometric aspects of diffusions on manifolds.In:Ecole d’Et′e de Probabilit′es de Saint-Flour XVII July1987.Lecture Notes in Mathematics,vol.1362.New York:Springer,1988,276-425
23 Ikeda N,Watanabe S.Stochastic Differential Equations and Diffusion Processes,2nd ed.Amsterdam:North-Holland Publishing;Tokyo:Kodansha,1989
24 Helgason S.Differential Geometry,Lie Groups,and Symmetric Spaces.Amsterdam:Academic Press,1978
25 Kobayashi S,Nomizu K.Foundations of Differential Geometry,Vols.I;II.New York:Interscience Publishers,1963;1969
26 Liao M.Inhomogeneous L′evy processes in Lie groups and homogeneous spaces.J Theoret Probab,2014,27:315-357
27 Helgason S.Groups and Geometric Analysis.Amsterdam:Academic Press,1984
28 tom Dieck T.Transformation Groups.Berlin:de Gruyter,1987
29 Ito K,Mckean H P Jr.Diffusion Processes and Their Sample Paths,Second Printing,Corrected.New York:SpringerVerlag,1974
30 Galmarino A R.Representation of an isotropic diffusion as a skew product.Z Wahrscheinlichkeit Verw Gebiete,1963,1:359-378
31 Evans S N,Hening A,Wayman E.When do skew-products exist?Electron J Probab,2015,20:1-14
2 Applebaum D.L′evy Processes and Stochastic Calculus,2nd ed.Cambridge:Cambridge University Press,2009
3 Bertoin J.L′evy Processes.Cambridge:Cambridge University Press,1996
4 Sato K.L′evy Processes and Infinitely Devisible Distributions(translated from 1990 Japanese original and revised by the author).Cambridge:Cambridge University Press,1999
5 Hunt G A.Semi-groups of measures on Lie groups.Trans Amer Math Soc,1956,81:264-293
6 Feinsilver P.Processes with independent increments on a Lie group.Trans Amer Math Soc,1978,242:73-121
7 Stroock D W,Varadhan S R S.Limit theorems for random walks on Lie groups.Sankhyˉa,1973,35:277-294
8 Heyer H,Pap G.Convolution hemigroups of bounded variation on a Lie projective group.J Lond Math Soc(2),1999,59:369-384
9 Liao M.A decomposition of Markov processes via group actions.J Theoret Probab,2009,22:164-185
10 Heyer H.Probability Measures on Locally Compact Groups.New York:Springer-Verlag,1977
11 Born E.An explicit L′evy-Hin?cin formula for convolution semigroups on locally compact groups.J Theoret Probab,1989,2:325-342
12 Dani S G,McCrudden M.Convolution roots and embeddings of probability measures on Lie groups.Adv Math,2007,209:198-211
13 Liao M.Convolution of probability measures on Lie groups and homogenous spaces.Potential Anal,2015,43:707-715
14 Diaconis P.Group Representations in Probability and Statistics.Lecture Notes-Monograph Series,vol.11.Group Representations in Probability and Statistics.Hayward:Institute of Mathematical Statistics,1988
15 Heyer H.Convolution semigroups of probability measures on Gelfand pairs.Expo Math,1983,1:3-45
16 Liao M.L′evy processes and Fourier analysison compact Lie groups.Ann Probab,2004,32:1553-1573
17 Liao M,Wang L.L′evy-Khinchin formula and existence of densities for convolution semigroups on symmetric spaces.Potential Anal,2007,27:133-150
18 Applebaum D.Probability on Compact Lie Groups.New York:Springer,2014
19 Liao M.L′evy Processes in Lie Groups.Cambridge:Cambridge University Press,2004
20 Liao M,Wang L.Limiting properties of L′evy processes in symmetric spaces of noncompact type.Stoch Dyn,2012,12:1250001
21 Kallenberg O.Foundations of Modern Probability,2nd ed.New York:Springer-Verlag,2002
22 Elworthy K D.Geometric aspects of diffusions on manifolds.In:Ecole d’Et′e de Probabilit′es de Saint-Flour XVII July1987.Lecture Notes in Mathematics,vol.1362.New York:Springer,1988,276-425
23 Ikeda N,Watanabe S.Stochastic Differential Equations and Diffusion Processes,2nd ed.Amsterdam:North-Holland Publishing;Tokyo:Kodansha,1989
24 Helgason S.Differential Geometry,Lie Groups,and Symmetric Spaces.Amsterdam:Academic Press,1978
25 Kobayashi S,Nomizu K.Foundations of Differential Geometry,Vols.I;II.New York:Interscience Publishers,1963;1969
26 Liao M.Inhomogeneous L′evy processes in Lie groups and homogeneous spaces.J Theoret Probab,2014,27:315-357
27 Helgason S.Groups and Geometric Analysis.Amsterdam:Academic Press,1984
28 tom Dieck T.Transformation Groups.Berlin:de Gruyter,1987
29 Ito K,Mckean H P Jr.Diffusion Processes and Their Sample Paths,Second Printing,Corrected.New York:SpringerVerlag,1974
30 Galmarino A R.Representation of an isotropic diffusion as a skew product.Z Wahrscheinlichkeit Verw Gebiete,1963,1:359-378
31 Evans S N,Hening A,Wayman E.When do skew-products exist?Electron J Probab,2015,20:1-14