×

Lenses in skew Brownian flow. (English) Zbl 1071.60073

Lenses in a skew Brownian flow are investigated. In this stochastic flow individual particles follow skew Brownian flow with each one of these processes is driven by the same Brownian motion, i.e. \(X^{s,x}_t=x + B_t - B_s + \beta L^{s,x}_t ,\) where \(B\) is a Brownian motion, \(L\) is the local time of \(X\). The authors present qualitative and distributional results on lenses and bifurcation times (ordinary, semi-flat, anticipated). This paper is the continuation of papers on skew Brownian flows [see M. Barlow, the authors and A. Mandelbaum, in: Séminaire de probabilités XXXV. Lect. Notes Math. 1755, 202–205 (2001; Zbl 0977.60074); K. Burdzy and Z.-Q. Chen, Ann. Probab. 29, 1693–1715 (2001; Zbl 1037.60057)].

MSC:

60J65 Brownian motion
60J55 Local time and additive functionals
60G17 Sample path properties
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Barlow, M., Burdzy, K., Kaspi, H. and Mandelbaum, A. (2001). Coalescence of skew Brownian motions. Séminaire de Probabilités XXXV . Lecture Notes in Math. 1755 202–206. Springer, Berlin. · Zbl 0977.60074
[2] Blumenthal, R. M. (1992). Excursions of Markov Processes . Birkhäuser, Boston. · Zbl 0983.60504
[3] Burdzy, K. (1987). Multidimensional Brownian Excursions and Potential Theory . Longman, Harlow. · Zbl 0691.60066
[4] Burdzy, K. and Chen, Z. Q. (2001). Local time flow related to skew Brownian motion. Ann. Probab. 29 1693–1715. · Zbl 1037.60057
[5] Harrison, J. M. and Shepp, L. A. (1981). On skew Brownian motion. Ann. Probab . 9 309–313. JSTOR: · Zbl 0462.60076
[6] Itô, K. and McKean, H. P. (1965). Diffusion Processes and Their Sample Paths . Springer, New York. · Zbl 0127.09503
[7] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus , 2nd ed. Springer, New York. · Zbl 0734.60060
[8] Knight, F. B. (1981). Essentials of Brownian Motion and Diffusion . Amer. Math. Soc., Providence, RI. · Zbl 0458.60002
[9] Le Jan, Y. and Raimond, O. (2002). Integration of Brownian vector fields. Ann. Probab. 30 826–873. · Zbl 1037.60061
[10] Le Jan, Y. and Raimond, O. (2002). The noise of a Brownian sticky flow is black. Available at arxiv.org/abs/math.PR/0212269.
[11] Le Jan, Y. and Raimond, O. (2002). Sticky flows on the circle. Available at arxiv.org/abs/math.PR/0211387.
[12] Le Jan, Y. and Raimond, O. (2002). Flows, coalescence and noise. Available at arxiv.org/abs/math.PR/0203221. · Zbl 1065.60066
[13] Maisonneuve, B. (1975). Exit systems. Ann. Probab. 3 399–411. · Zbl 0311.60047
[14] Nagasawa, M. (1964). Time reversions of Markov processes. Nagoya Math. J. 24 177–204. · Zbl 0133.10702
[15] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion . Springer, Berlin. · Zbl 0731.60002
[16] Rogers, L. C. G. and Williams, D. (1987). Diffusions , Markov Processes , and Martingales 2 . Wiley, New York. · Zbl 0627.60001
[17] Sharpe, M. J. (1980). Some transformations of diffusions by time reversal. Ann. Probab. 8 1157–1162. JSTOR: · Zbl 0465.60066
[18] Walsh, J. B. (1978). A diffusion with discontinuous local time. Temps Locaux Asterisque 52 – 53 37–45. · Zbl 0385.60063
[19] Watanabe, S. (1987). Construction of semimartingales from pieces by the method of excursion point processes. Ann. Inst. H. Poincaré Probab. Statist. 23 297–320. · Zbl 0622.60050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.