Abstract
Inhomogeneous Lévy processes in topological groups, defined by independent increments, were introduced in §1.4. More useful representation of these processes may be obtained on a Lie group G. The main purpose of this chapter is to present a martingale representation, which characterizes an inhomogeneous Lévy process in a Lie group by a triple (b, A, η) of a deterministic path b t in G, called a drift, a matrix function A(t) and a measure function η(t, ⋅), in close analogy with the representation of a homogeneous Lévy process.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, London (1986)
Feinsilver, P.: Processes with independent increments on a Lie group. Trans. Am. Math. Soc. 242, 73–121 (1978)
Heyer, H., Pap, G.: Martingale characterizations of increment processes in a local compact group. Infin. Dim. Anal. Quant. Probab. Relat. Top. 6, 563–595 (2003)
Itô, K.: Stochastic Processes. Lecture Notes Series No. 16, Aarhus University. Springer, Berlin (1969)
Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes, 2nd edn. Springer, Berlin (2003)
Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (2002)
Liao, M.: Inhomogeneous Lévy processes in Lie groups and homogeneous spaces. J. Theor. Probab. 27, 315–357 (2014)
Liao, M.: Fixed jumps of additive processes. Stat. Probab. Lett. 83, 820–823 (2013)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)
Stroock, D.W., Varadhan, S.R.S.: Limit theorems for random walks on Lie groups. Sankhy\(\bar {\mathrm{a}}\): Indian J. Stat. Ser. A 35, 277–294 (1973)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Liao, M. (2018). Inhomogeneous Lévy Processes in Lie Groups. In: Invariant Markov Processes Under Lie Group Actions. Springer, Cham. https://doi.org/10.1007/978-3-319-92324-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-92324-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92323-9
Online ISBN: 978-3-319-92324-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)