Construction of diffusions on current groups. (English. Russian original) Zbl 1394.58019

Kalinichenko, Artem A.

Sb. Math. 209, No. 1, 71-95 (2018); translation from Mat. Sb. 209, No. 1, 74-99 (2018).

The author considers the group \(C\left( M, G \right)\) of continuous mappings of a smooth connected compact Riemannian manifold \(M\) into a compact Lie group \(G\). This group is sometimes referred to as a current group. The main claim of the paper is a construction of a diffusion process on \(C\left( M, G \right)\) as a solution to a martingale problem. The techniques used include Prokhorov’s theorem and estimates for the modulus of continuity of random processes by using a method called chaining.
Reviewer’s remarks: Besides some minor issues such as using the Peter-Weyl theorem in Section 3.1 to claim that \(G\) can be taken as a matrix Lie group, the main construction seems to be going back to the case when \(M\) is an interval or a half-line starting from p. 75, without ever saying it clearly. One of the main results of the paper, Theorem 2, uses a process labelled by time \(t\) from a finite interval (going back to Equation (3.3)), then \(t\) suddenly becomes an element of \(M\). Not to mention that the term stochastic exponential usually is used for a different object than the one in the 1984 paper by Hakim-Dowek and Lépingle cited in the article.

Reviewer: Maria Gordina (Storrs)

MSC:

58J65	Diffusion processes and stochastic analysis on manifolds
60J60	Diffusion processes
22E67	Loop groups and related constructions, group-theoretic treatment
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

current groups; diffusion; infinite-dimensional stochastic analysis; Brownian motion

References:

[1]	M.-P. Malliavin and P. Malliavin 1990 Integration on loop groups. I. Quasi invariant measures J. Funct. Anal.93 1 207-237 · Zbl 0715.22024
[2]	J. B. Epperson and T. M. Lohrenz 1993 Brownian motion and the heat semigroup on the path space of a compact Lie group Pacific J. Math.161 2 233-253 · Zbl 0788.60017
[3]	P. Malliavin 1990 Hypoellipticity in infinite dimensions Diffusion processes and related problems in analysisEvanston, IL 1989 Progr. Probab. 22, 1 Birkhäuser Boston, Boston, MA 17-31 · Zbl 0723.60059
[4]	Y. Inahama 2003 Convergence of finite dimensional distributions of heat kernel measures on loop groups J. Funct. Anal.198 2 311-340 · Zbl 1018.60060
[5]	R. Léandre 2003 Brownian surfaces with boundary and Deligne cohomology Rep. Math. Phys.52 3 353-362 · Zbl 1052.60063
[6]	M. Cecil and B. K. Driver 2008 Heat kernel measure on loop and path groups Infin. Dimens. Anal. Quantum Probab. Relat. Top.11 2 135-156 · Zbl 1154.58019
[7]	B. K. Driver 1997 Integration by parts and quasi-invariance for heat kernel measures on loop groups J. Funct. Anal.149 2 470-547 · Zbl 0887.58062
[8]	Z. Brzeźniak and A. Carroll 2003 Approximations of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces Séminaire de Probabilités XXXVII Lecture Notes in Math. 1832 Springer, Berlin 251-289 · Zbl 1040.60047
[9]	Y. Inahama and H. Kawabi 2006 Large deviations for heat kernel measures on loop spaces via rough paths J. London Math. Soc. (2)73 3 797-816 · Zbl 1133.60006
[10]	M. Talagrand 2014 Upper and lower bounds for stochastic processes. Modern methods and classical problems Ergeb. Math. Grenzgeb. (3) 60 Springer, Heidelberg xvi+626 pp. · Zbl 1293.60001
[11]	B. K. Driver and M. Röckner 1992 Construction of diffusions on path and loop spaces of compact Riemannian manifolds C. R. Acad. Sci. Paris Sér. I Math.315 5 603-608 · Zbl 0771.58047
[12]	J. R. Norris 1995 Twisted sheets J. Funct. Anal.132 2 273-334 · Zbl 0848.60055
[13]	J. B. Epperson and T. Lohrenz 1994 Diffusions on finite-energy loop spaces Soochow J. Math.20 1 113-136 · Zbl 0804.58058
[14]	O. Kallenberg 2002 Foundations of modern probability Probab. Appl. (N. Y.) Springer, New York 2nd ed., xx+638 pp. · Zbl 0996.60001
[15]	G. Da Prato and J. Zabczyk 2014 Stochastic equations in infinite dimensions Encyclopedia Math. Appl. 152 Cambridge Univ. Press, Cambridge 2nd ed., xviii+493 pp.
[16]	B. I. Mamporia 1986 Wiener processes and stochastic integrals in a Banach space Probab. Math. Statist.7 1 59-75 · Zbl 0617.60007
[17]	G. R. Shorack and J. A. Wellner 1986 Empirical processes with applications to statistics Wiley Ser. Probab. Math. Statist. Probab. Math. Statist. John Wiley & Sons, Inc., New York xxxviii+938 pp. · Zbl 1170.62365
[18]	J. van Neerven, M. Veraar and L. Weis 2015 Stochastic integration in Banach spaces – a survey Stochastic analysis: a series of lectures Progr. Probab. 68 Birkhäuser/Springer, Basel 297-332 · Zbl 1333.60115
[19]	E. Dettweiler 1983 Banach space valued processes with independent increments and stochastic integration Probability in Banach spaces IVOberwolfach 1982 Lecture Notes in Math. 990 Springer, Berlin-New York 54-83 · Zbl 0514.60010
[20]	A. A. Kalinichenko An approach to stochastic integration in general separable Banach spaces Potential Anal. Submitted
[21]	M. Hakim-Dowek and D. Lépingle 1986 L’exponentielle stochastique des groupes de Lie Séminaire de Probabilités XXStrasbourg 1984/85 Lecture Notes in Math. 1204 Springer, Berlin 352-374 · Zbl 0609.60009
[22]	L. Gross 1994 Analysis on loop groups Stochastic analysis and applications in physicsFunchal 1993 NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 449 Kluwer Acad. Publ., Dordrecht 99-118
[23]	J. L. Kelley 1955 General topology D. Van Nostrand Company, Inc., Toronto-New York-London xiv+298 pp. · Zbl 0066.16604
[24]	M. Ondreját 2007 Integral representations of cylindrical local martingales in every separable Banach space Infin. Dimens. Anal. Quantum Probab. Relat. Top.10 3 365-379 · Zbl 1146.60042
[25]	N. Ikeda and S. Watanabe 1981 Stochastic differential equations and diffusion processes North-Holland Math. Library 24 North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo xiv+464 pp. · Zbl 0495.60005