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Random transformations and invariance of semimartingales on Lie groups. (English) Zbl 1470.60145

Summary: Invariance properties of semimartingales on Lie groups under a family of random transformations are defined and investigated, generalizing the random rotations of the Brownian motion. A necessary and sufficient explicit condition characterizing semimartingales with this kind of invariance is given in terms of their stochastic characteristics. Non-trivial examples of symmetric semimartingales are provided and applications of this concept to stochastic analysis are discussed.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
22E30 Analysis on real and complex Lie groups
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[1] S. Albeverio, A. B. Cruzeiro and D. Holm, Stochastic Geometric Mechanics, Springer Proc. Math. Stat. 202, Springer, Cham, 2017. · Zbl 1386.70002
[2] S. Albeverio, F. C. De Vecchi, P. Morando and S. Ugolini, Weak symmetries of stochastic differential equations driven by semimartingales with jumps, Electron. J. Probab. 25 (2020), Paper No. 44. · Zbl 1465.60046
[3] S. Albeverio and S.-M. Fei, Symmetry, integrable chain models and stochastic processes, Rev. Math. Phys. 10 (1998), no. 6, 723-750. · Zbl 0929.60089
[4] S. Albeverio and M. Gordina, Lévy processes and their subordination in matrix Lie groups, Bull. Sci. Math. 131 (2007), no. 8, 738-760. · Zbl 1140.60006
[5] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge Stud. Adv. Math. 116, Cambridge University, Cambridge, 2009. · Zbl 1200.60001
[6] D. Applebaum and S. Cohen, Stochastic parallel transport along Lévy flows of diffeomorphisms, J. Math. Anal. Appl. 207 (1997), no. 2, 496-505. · Zbl 0881.58074
[7] D. Applebaum and A. Dooley, A generalised Gangolli-Lévy-Khintchine formula for infinitely divisible measures and Lévy processes on semi-simple Lie groups and symmetric spaces, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 2, 599-619. · Zbl 1353.60007
[8] D. Applebaum and A. Estrade, Isotropic Lévy processes on Riemannian manifolds, Ann. Probab. 28 (2000), no. 1, 166-184. · Zbl 1044.60035
[9] D. Applebaum and F. Tang, Stochastic flows of diffeomorphisms on manifolds driven by infinite-dimensional semimartingales with jumps, Stochastic Process. Appl. 92 (2001), no. 2, 219-236. · Zbl 1099.58504
[10] K. Bichteler, Stochastic Integration with Jumps, Encyclopedia Math. Appl. 89, Cambridge University, Cambridge, 2002. · Zbl 1002.60001
[11] G. Cicogna and G. Gaeta, Symmetry and Perturbation Theory in Nonlinear Dynamics, Lecture Notes in Phys. 57, Springer, Berlin, 1999. · Zbl 0981.34025
[12] S. Cohen, Some Markov properties of stochastic differential equations with jumps, Séminaire de Probabilités. 29, Lecture Notes in Math. 1613, Springer, Berlin (1995), 181-193. · Zbl 0833.60060
[13] S. Cohen, Géométrie différentielle stochastique avec sauts. I, Stoch. Stoch. Rep. 56 (1996), no. 3-4, 179-203. · Zbl 0911.58044
[14] F. C. De Vecchi, A de Finetti-type theorem for random-rotation-invariant continuous semimartingales, preprint (2017), https://arxiv.org/abs/1712.08374.
[15] F. C. De Vecchi, Lie symmetry analysis and geometrical methods for finite and infinite dimensional stochastic differential equations, PhD thesis, Università degli Studi di Milano, 2018.
[16] F. C. De Vecchi, P. Morando and S. Ugolini, Reduction and reconstruction of stochastic differential equations via symmetries, J. Math. Phys. 57 (2016), no. 12, Article ID 123508. · Zbl 1432.60061
[17] F. C. De Vecchi, P. Morando and S. Ugolini, Symmetries of stochastic differential equations: A geometric approach, J. Math. Phys. 57 (2016), no. 6, Article ID 063504. · Zbl 1382.60081
[18] F. C. De Vecchi, P. Morando and S. Ugolini, A note on symmetries of diffusions within a martingale problem approach, Stoch. Dyn. 19 (2019), no. 2, Article ID 1950011. · Zbl 1435.58004
[19] F. C. De Vecchi, P. Morando and S. Ugolini, Symmetries of stochastic differential equations using Girsanov transformations, J. Phys. A 53 (2020), no. 13, Article ID 135204. · Zbl 1514.60066
[20] P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev. 41 (1999), no. 1, 45-76. · Zbl 0926.60056
[21] K. D. Elworthy, Stochastic Differential Equations on Manifolds, London Math. Soc. Lecture Note Ser. 70, Cambridge University, Cambridge, 1982. · Zbl 0514.58001
[22] M. Émery, Stochastic Calculus in Manifolds, Universitext, Springer, Berlin, 1989. · Zbl 0697.60060
[23] P. Feinsilver, Processes with independent increments on a Lie group, Trans. Amer. Math. Soc. 242 (1978), 73-121. · Zbl 0403.60006
[24] P. K. Friz and A. Shekhar, General rough integration, Lévy rough paths and a Lévy-Kintchine-type formula, Ann. Probab. 45 (2017), no. 4, 2707-2765. · Zbl 1412.60103
[25] P. K. Friz and H. Zhang, Differential equations driven by rough paths with jumps, J. Differential Equations 264 (2018), no. 10, 6226-6301. · Zbl 1432.60098
[26] T. Fujiwara and H. Kunita, Canonical SDE’s based on semimartingales with spatial parameters. I. Stochastic flows of diffeomorphisms, Kyushu J. Math. 53 (1999), no. 2, 265-300. · Zbl 0951.60075
[27] G. Gaeta, Nonlinear Symmetries and Nonlinear Equations, Mathe. Appl. 299, Kluwer Academic, Dordrecht, 1994. · Zbl 0813.58002
[28] G. Gaeta and C. Lunini, Symmetry and integrability for stochastic differential equations, J. Nonlinear Math. Phys. 25 (2018), no. 2, 262-289. · Zbl 1411.35293
[29] R. Gangolli, Isotropic infinitely divisible measures on symmetric spaces, Acta Math. 111 (1964), 213-246. · Zbl 0154.43804
[30] J. Glover, Symmetry groups and translation invariant representations of Markov processes, Ann. Probab. 19 (1991), no. 2, 562-586. · Zbl 0732.60079
[31] J. Glover and J. Mitro, Symmetries and functions of Markov processes, Ann. Probab. 18 (1990), no. 2, 655-668. · Zbl 0714.60060
[32] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, 1974. · Zbl 0361.57001
[33] Q. Han and J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Math. Surveys Monogr. 130, American Mathematical Society, Providence, 2006. · Zbl 1113.53002
[34] D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry, Oxford Texts Appl. Eng. Math. 12, Oxford University, Oxford, 2009. · Zbl 1175.70001
[35] G. A. Hunt, Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293. · Zbl 0073.12402
[36] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, 2nd ed., Grundlehren Math. Wiss. 288, Springer, Berlin, 2003. · Zbl 1018.60002
[37] O. Kallenberg, Probabilistic Symmetries and Invariance Principles, Probab. Appl. (N. Y.), Springer, New York, 2005. · Zbl 1084.60003
[38] N. V. Krylov, Controlled Diffusion Processes, Appl. Math. 14, Springer, New York, 1980. · Zbl 0459.93002
[39] H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, Real and Stochastic Analysis, Trends Math., Birkhäuser, Boston (2004), 305-373. · Zbl 1082.60052
[40] T. G. Kurtz, E. Pardoux and P. Protter, Stratonovich stochastic differential equations driven by general semimartingales, Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 2, 351-377. · Zbl 0823.60046
[41] J.-A. Lázaro-Camí and J.-P. Ortega, Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations, Stoch. Dyn. 9 (2009), no. 1, 1-46. · Zbl 1187.60045
[42] M. Liao, Symmetry groups of Markov processes, Ann. Probab. 20 (1992), no. 2, 563-578. · Zbl 0755.60061
[43] M. Liao, Lévy Processes in Lie Groups, Cambridge Tracts in Math. 162, Cambridge University, Cambridge, 2004. · Zbl 1076.60004
[44] M. Liao, Inhomogeneous Lévy processes in Lie groups and homogeneous spaces, J. Theoret. Probab. 27 (2014), no. 2, 315-357. · Zbl 1302.60075
[45] M. Liao, Markov processes and group actions, Theory Stoch. Process. 21 (2016), no. 2, 29-57. · Zbl 1374.60143
[46] S. I. Marcus, Modeling and approximation of stochastic differential equations driven by semimartingales, Stochastics 4 (1980/81), no. 3, 223-245. · Zbl 0456.60064
[47] T. Misawa, Conserved quantities and symmetry for stochastic dynamical systems, Phys. Lett. A 195 (1994), no. 3-4, 185-189. · Zbl 0941.34512
[48] P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math. 107, Springer, New York, 1993. · Zbl 0785.58003
[49] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406-413. · Zbl 0065.24603
[50] N. Privault, Invariance of Poisson measures under random transformations, Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 4, 947-972. · Zbl 1278.60084
[51] P. Protter, Stochastic Integration and Differential Equations, Appl. Math. (N. Y.) 21, Springer, Berlin, 1990. · Zbl 0694.60047
[52] K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Stud. Adv. Math. 68, Cambridge University, Cambridge, 1999. · Zbl 0973.60001
[53] H. Stephani, Differential Equations, Cambridge Universitys, Cambridge, 1989. · Zbl 0704.34001
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