The Noise Made by a Poisson Snake

Jon Warren (University of Warwick)


The purpose of this article is to study a coalescing flow of sticky Brownian motions. Sticky Brownian motion arises as a weak solution of a stochastic differential equation, and the study of the flow reveals the nature of the extra randomness that must be added to the driving Brownian motion. This can be represented in terms of Poissonian marking of the trees associated with the excursions of Brownian motion. We also study the noise, in the sense of Tsirelson, generated by the flow. It is shown that this noise is not generated by any Brownian motion, even though it is predictable.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-21

Publication Date: June 24, 2002

DOI: 10.1214/EJP.v7-120


  1. Aldous, D.J. (1993), The Continuum random tree III, Annals of Probability 21:1, 248-289. Math. Review 94c:60015
  2. Aldous, D.J. (1994), Recursive self-similarity for random trees, random triangulations and Brownian excursion, Annals of Probability 22:2, 527-545. Math. Review 95i:60007
  3. Adous, D.J. and Pitman, J. (1998) The standard additive coalescent, Annals of Probability, 26:4, 1703-1726. Math.Review 2000d:60121
  4. Dhersin, J.S. and Serlet, L. (2000), A stochastic calculus approach for the Brownian snake, Canadian Journal of Mathematics, 52:1, 92-118. Math. Review 2001f:60075
  5. Feldman, J. (1971), Decomposable processes and continuous products of probability spaces, Journal of Functional Analysis, 8, 1-51. Math. Review 44#7617
  6. Feller, W. (1957), Generalized second order differential operators and their lateral conditions, Illinois Journal of Mathematics, 1, 459-504. Math. Review 19,1052c
  7. Knight, F.B. (1981), Essentials of Brownian motion and diffusion, Mathematical Surveys 18, American Mathematical Society. Math. Review 82m:60098
  8. Le Gall, J.F. (1993), A class of path-valued Markov processes and its applications to superprocesses, Probability Theory and Related Fields, 95:1, 25-46. Math. Review 94f:60093
  9. Revuz, D. and Yor, M.. (1999), Continuous martingales and Brownian motion, Springer, Berlin. Math. Review 2000h:60050
  10. Tsirelson, B. (1998), Within and beyond the reach of Brownian innovation, Documenta Mathematica, extra volume ICM 1998 III, 311-320. Math. Review 99i:60082
  11. Tsirelson, B. (1998), Unitary Brownian motions are linearizable, arXiv:math.PR/9806112
  12. Tsirelson, B.(2002), Scaling limit, noise, stability, Lecture Notes in Mathematics, Springer, Berlin. (To appear)
  13. Tsirelson, B. (1997), Triple points: From non-Brownian filtrations to harmonic measures, Geoemetric and Functional Analysis, 7 1096-1142. Math.Review 2001g:60199
  14. Tsirelson, B. and Vershik, A.M. (1998), Examples of non-linear continuous tensor products of measure spaces and non-Fock factorizations, Reviews in Mathematical Physics, 10:1, 81-145. Math. Review 99c:60085
  15. Warren, J. (1997), Branching Processes, the Ray-Knight theorem and sticky Brownian motion, Seminaire de Probabilites XXXI, Lecture Notes in Mathematics 1655, Springer, Berlin, 1-15. Math. Review 99c:60183
  16. Warren, J. (1999), On the joining of sticky Brownian motion, Seminaire de Probabilites XXXIII, Lecture Notes in Mathematics 1709, Springer, Berlin, 257-266. Math. Review 2001c:60132
  17. Watanabe, S. (1999), The existence of a multiple spider martingale in the natural filtration of a certain diffusion in the plane, Seminaire de Probabilites XXXIII, Lecture Notes in Mathematics 1709, Springer, Berlin, 277-290. Math. Review 2001h:60081
  18. Watanabe, S. (1999), Killing operations in superdiffusions by Brownian snakes, Trends in probabilty and related analysis, Taipai, World Science Publishing, 177-190. Math. Review 2002c:60129
  19. Watanabe, S. (2001), Stochastic flows in duality and noises. Preprint.
  20. Yor, M.. (1997), Some remarks on the joint law of Brownian motion and its supremum, Seminaire de Probabilites XXXI, Lecture Notes in Mathematics 1655, Springer, Berlin, 306-314. Math. Review 99b:60140

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.