Open Access
August 1998 Brownian motion in a Brownian crack
Krzysztof Burdzy, Davar Khoshnevisan
Ann. Appl. Probab. 8(3): 708-748 (August 1998). DOI: 10.1214/aoap/1028903448
Abstract

Let D be the Wiener sausage of width $\varepsilon$ around two-sided Brownian motion. The components of two-dimensional reflected Brownian motion in D converge to one-dimensional Brownian motion and iterated Brownian motion, respectively, as $\varepsilon$ goes to 0.

References

1.

Barlow, M. T. (1990). Random walks and diffusions on fractals. In Proceedings of the International Congress of Mathematicians, Ky oto, Japan 2 1025-1035. Springer, New York.  MR1159287 0751.60073Barlow, M. T. (1990). Random walks and diffusions on fractals. In Proceedings of the International Congress of Mathematicians, Ky oto, Japan 2 1025-1035. Springer, New York.  MR1159287 0751.60073

2.

Barlow, M. T. and Bass, R. F. (1993). Coupling and Harnack inequalities for Sierpi ´nski carpets. Bull. Amer. Math. Soc. 29 208-212.  MR94a:60011 10.1090/S0273-0979-1993-00424-5Barlow, M. T. and Bass, R. F. (1993). Coupling and Harnack inequalities for Sierpi ´nski carpets. Bull. Amer. Math. Soc. 29 208-212.  MR94a:60011 10.1090/S0273-0979-1993-00424-5

3.

Barlow, M. T. and Bass, R. F. (1997). Brownian motion and harmonic analysis on Sierpinski carpets. Preprint. MR1701339Barlow, M. T. and Bass, R. F. (1997). Brownian motion and harmonic analysis on Sierpinski carpets. Preprint. MR1701339

4.

Barlow, M. T. and Perkins, E. A. (1988). Brownian motion on the Sierpi ´nski carpet. Probab. Theory Related Fields 79 543-623. Barlow, M. T. and Perkins, E. A. (1988). Brownian motion on the Sierpi ´nski carpet. Probab. Theory Related Fields 79 543-623.

5.

Bass, R. F. (1987). Lp-inequalities for functionals of Brownian motion. S´eminaire de Probabilit´es XXI. Lecture Notes in Math. 1247 206-217. Springer, Berlin. Bass, R. F. (1987). Lp-inequalities for functionals of Brownian motion. S´eminaire de Probabilit´es XXI. Lecture Notes in Math. 1247 206-217. Springer, Berlin.

6.

Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes 1992 (E. Çinlar, K. L. Chung and M. Sharpe, eds.) 67-87. Birkh¨auser, Boston. Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes 1992 (E. Çinlar, K. L. Chung and M. Sharpe, eds.) 67-87. Birkh¨auser, Boston.

7.

Burdzy, K. (1994). Variation of iterated Brownian motion. In Workshop and Conference on Measure-Valued Processes, Stochastic Partial Differential Equations and Interacting Sy stems 5 35-53. Amer. Math. Soc., Providence, RI.  MR95h:60123 0803.60077Burdzy, K. (1994). Variation of iterated Brownian motion. In Workshop and Conference on Measure-Valued Processes, Stochastic Partial Differential Equations and Interacting Sy stems 5 35-53. Amer. Math. Soc., Providence, RI.  MR95h:60123 0803.60077

8.

Burdzy, K., Toby, E. and Williams, R. J. (1989). On Brownian excursions in Lipschitz domains II. Local asy mptotic distributions. In Seminar on Stochastic Processes 1988 (E. Çinlar, K. L. Chung, R. Getoor and J. Glover eds.) 55-85. Birkh¨auser, Boston. Burdzy, K., Toby, E. and Williams, R. J. (1989). On Brownian excursions in Lipschitz domains II. Local asy mptotic distributions. In Seminar on Stochastic Processes 1988 (E. Çinlar, K. L. Chung, R. Getoor and J. Glover eds.) 55-85. Birkh¨auser, Boston.

9.

Chen, Z.-Q. (1992). Pseudo Jordan domains and reflecting Brownian motions. Probab. Theory Related Fields 94 271-280.  0767.60079 MR1191110 10.1007/BF01192446Chen, Z.-Q. (1992). Pseudo Jordan domains and reflecting Brownian motions. Probab. Theory Related Fields 94 271-280.  0767.60079 MR1191110 10.1007/BF01192446

10.

Chudnovsky, A. and Kunin, B. (1987). A probabilistic model of brittle crack formation. J. Appl. Phy s. 62 4124-4129. Chudnovsky, A. and Kunin, B. (1987). A probabilistic model of brittle crack formation. J. Appl. Phy s. 62 4124-4129.

11.

Fitzsimmons, P. J. (1989). Time changes of sy mmetric Markov processes and a Fey nman-Kac formula. J. Theoret. Probab. 2 485-501.  MR91h:60076 0683.60052 10.1007/BF01051880Fitzsimmons, P. J. (1989). Time changes of sy mmetric Markov processes and a Fey nman-Kac formula. J. Theoret. Probab. 2 485-501.  MR91h:60076 0683.60052 10.1007/BF01051880

12.

Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Sy mmetric Markov Processes. de Gruy ter, New York.  MR96f:60126 0838.31001Fukushima, M., Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Sy mmetric Markov Processes. de Gruy ter, New York.  MR96f:60126 0838.31001

13.

Goldstein, S. (1987). Random walks and diffusion on fractals. In Percolation Theory and Ergodic Theory of Infinite Particle Sy stems (H. Kesten, ed.) 121-129. Springer, New York.  MR894545Goldstein, S. (1987). Random walks and diffusion on fractals. In Percolation Theory and Ergodic Theory of Infinite Particle Sy stems (H. Kesten, ed.) 121-129. Springer, New York.  MR894545

14.

Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.  MR89c:60096Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.  MR89c:60096

15.

Khoshnevisan, D. and Lewis, T. M. (1997). Stochastic calculus for Brownian motion on a Brownian fracture. Ann. Appl. Probab. To appear.  MR1722276 0956.60054 10.1214/aoap/1029962807 euclid.aoap/1029962807 Khoshnevisan, D. and Lewis, T. M. (1997). Stochastic calculus for Brownian motion on a Brownian fracture. Ann. Appl. Probab. To appear.  MR1722276 0956.60054 10.1214/aoap/1029962807 euclid.aoap/1029962807

16.

Kunin, B. and Gorelik, M. (1991). On representation of fracture profiles by fractional integrals of a Wiener process. J. Appl. Phy s. 70 7651-7653. Kunin, B. and Gorelik, M. (1991). On representation of fracture profiles by fractional integrals of a Wiener process. J. Appl. Phy s. 70 7651-7653.

17.

Kusuoka, S. (1985). A diffusion process on a fractal. In Probabilistic Methods in Mathematical physics. Proceedings of the Taniguchi International Sy mposium (K. It o and N. Ikeda, eds.) 251-274. Academic Press, New York.  MR933827 0645.60081Kusuoka, S. (1985). A diffusion process on a fractal. In Probabilistic Methods in Mathematical physics. Proceedings of the Taniguchi International Sy mposium (K. It o and N. Ikeda, eds.) 251-274. Academic Press, New York.  MR933827 0645.60081

18.

Silverstein, M. L. (1974). Sy mmetric Markov Processes. Lecture Notes in Math. 426. Springer, New York. MR52:6891Silverstein, M. L. (1974). Sy mmetric Markov Processes. Lecture Notes in Math. 426. Springer, New York. MR52:6891
Copyright © 1998 Institute of Mathematical Statistics
Krzysztof Burdzy and Davar Khoshnevisan "Brownian motion in a Brownian crack," The Annals of Applied Probability 8(3), 708-748, (August 1998). https://doi.org/10.1214/aoap/1028903448
Published: August 1998
Vol.8 • No. 3 • August 1998
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