## Electronic Communications in Probability

### When do skew-products exist?

#### Abstract

The classical skew-product decomposition of planar Brownian motionrepresents the process in polar coordinates as an autonomously Markovian radial part and an angular part that is an independent Brownian motion on the unit circle time-changed according to the radial part. Theorem 4 of L09 gives a broad generalization of this fact to a setting where there is a diffusion on a manifold $X$ with a distribution that is equivariant under the smooth action ofa Lie group $K$. Under appropriate conditions, there is a decomposition into an autonomously Markovian "radial" part that lives on the space of orbits of $K$ and an "angular" part that is an independent Brownian motion on the homogeneous space $K/M$, where $M$ is the isotropy subgroup of a point of $x$, that is time-changed with a time-change that is adapted to the filtration of the radial part. We present two apparent counterexamples to Theorem 4 of L09. In the first counterexample the angular part is not a time-change of any Brownian motionon $K/M$, whereas in the second counterexample the angular part is the time-change of a Brownian motion on $K/M$ but this Brownian motion is not independent of the radial part. In both of these examples $K/M$ has dimension $1$.  The statement and proof of Theorem 4 in L09 remain valid when $K/M$ has dimension greater than $1$. Our examples raise the question of what conditions lead to the usual sort of skew-product decomposition when $K/M$ has dimension $1$ and what conditions lead to there being no decomposition at all or one in which the angular part is a time-changed Brownian motion but this Brownian motion is not independent of the radial part.

#### Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 54, 14 pp.

Dates
Accepted: 27 July 2015
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465320981

Digital Object Identifier
doi:10.1214/ECP.v20-4040

Mathematical Reviews number (MathSciNet)
MR3384112

Zentralblatt MATH identifier
1322.58022

Rights

#### Citation

Evans, Steven; Hening, Alexandru; Wayman, Eric. When do skew-products exist?. Electron. Commun. Probab. 20 (2015), paper no. 54, 14 pp. doi:10.1214/ECP.v20-4040. https://projecteuclid.org/euclid.ecp/1465320981

#### References

• Baudoin, Fabrice; Nualart, David. Notes on the two-dimensional fractional Brownian motion. Ann. Probab. 34 (2006), no. 1, 159–180.
• Chybiryakov, Oleksandr. Skew-product representations of multidimensional Dunkl Markov processes. Ann. Inst. Henri Poincare Probab. Stat. 44 (2008), no. 4, 593–611.
• Elworthy, K. David; Le Jan, Yves; Li, Xue-Mei. The geometry of filtering. Frontiers in Mathematics. Birkhauser Verlag, Basel, 2010. xii+169 pp. ISBN: 978-3-0346-0175-7
• Etheridge, Alison; March, Peter. A note on superprocesses. Probab. Theory Related Fields 89 (1991), no. 2, 141–147.
• Galmarino, Alberto R. Representation of an isotropic diffusion as a skew product. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 1962/1963 359–378.
• Hiraba, Seiji. Jump-type Fleming-Viot processes. Adv. in Appl. Probab. 32 (2000), no. 1, 140–158.
• Horn, Roger A.; Johnson, Charles R. Matrix analysis. Second edition. Cambridge University Press, Cambridge, 2013. xviii+643 pp. ISBN: 978-0-521-54823-6
• Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
• Lazaro-Cami, Joan-Andreu; Ortega, Juan-Pablo. Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations. Stoch. Dyn. 9 (2009), no. 1, 1–46.
• Liao, Ming. A decomposition of Markov processes via group actions. J. Theoret. Probab. 22 (2009), no. 1, 164–185.
• Perkins, Edwin A. Conditional Dawson-Watanabe processes and Fleming-Viot processes. Seminar on Stochastic Processes, 1991 (Los Angeles, CA, 1991), 143–156, Progr. Probab., 29, Birkhauser Boston, Boston, MA, 1992.
• Pauwels, E. J.; Rogers, L. C. G. Skew-product decompositions of Brownian motions. Geometry of random motion (Ithaca, N.Y., 1987), 237–262, Contemp. Math., 73, Amer. Math. Soc., Providence, RI, 1988.
• L. C. G. Rogers and D. Williams, Diffusions, Markov processes, and martingales. Vol. 2: Itô calculus, vol. 2, Cambridge University Press, Cambridge, 2000.
• Taylor, J. C. Skew products, regular conditional probabilities and stochastic differential equations: a technical remark. Seminaire de Probabilites, XXVI, 113–126, Lecture Notes in Math., 1526, Springer, Berlin, 1992.