The Annals of Applied Probability

Iterated Brownian motion in an open set

R. Dante DeBlassie

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Abstract

Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If τ is the first exit time of iterated Brownian motion from the solid, then P(τ>t) can be viewed as a measurement of the amount of contaminant left in the crack at time t. We determine the large time asymptotics of P(τ>t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator $\frac{1}{8}\Delta^{2}-\frac{\partial}{\partial t}$.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 3 (2004), 1529-1558.

Dates
First available in Project Euclid: 13 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1089736295

Digital Object Identifier
doi:10.1214/105051604000000404

Mathematical Reviews number (MathSciNet)
MR2071433

Zentralblatt MATH identifier
1051.60082

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60K99: None of the above, but in this section

Keywords
Iterated Brownian motion exit time

Citation

DeBlassie, R. Dante. Iterated Brownian motion in an open set. Ann. Appl. Probab. 14 (2004), no. 3, 1529--1558. doi:10.1214/105051604000000404. https://projecteuclid.org/euclid.aoap/1089736295


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References

  • Accetta, G. and Orsingher, E. (1997). Asymptotic expansion of fundamental solutions of higher order heat equations. Random Oper. Stochastic Equations 5 217–226.
  • Allouba, H. and Zheng, W. (2001). Brownian-time processes: The PDE connection and the half-derivative generator. Ann. Probab. 29 1780–1795.
  • Allouba, H. (2002). Brownian-time processes: The PDE connection II and the corresponding Feyman–Kac formula. Trans. Amer. Math. Soc. 354 4627–4637.
  • Arcones, M. A. (1995). On the law of the iterated logarithm for Gaussian processes. J. Theoret. Probab. 8 877–903.
  • Bañuelos, R. and Smits, R. G. (1997). Brownian motion in cones. Probab. Theory Related Fields 108 299–319.
  • Beghin, L., Hochberg, K. J. and Orsingher, E. (2000). Conditional maximal distributions of processes related to higher-order heat-type equations. Stochastic Process. Appl. 85 209–223.
  • Beghin, L., Orsingher, E. and Rogozina, T. (2001). Joint distributions of the maximum and the process for higher-order diffusions. Stochastic Process. Appl. 94 71–93.
  • Bertoin, J. (1996). Iterated Brownian motion and stable ($1/4$) subordinator. Statist. Probab. Lett. 27 111–114.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion–-Facts and Formulae. Birkhäuser, Basel.
  • Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes (E. Çinlar, K. L. Chung and M. J. Sharpe, eds.) 67–87. Birkhäuser, Boston.
  • Burdzy, K. (1994). Variations of iterated Brownian motion. In Workshop and Conference on Measure-Valued Processes, Stochastic Partial Differential Equations and Interacting Particle Systems (D. A. Dawson, ed.) 35–53. Amer. Math. Soc., Providence, RI.
  • Burdzy, K. and Khoshnevisan, D. (1995). The level sets of iterated Brownian motion. Séminaire de Probabilités XXIX. Lecture Notes in Math. 1613 231–236. Springer, Berlin.
  • Burdzy, K. and Khoshnevisan, D. (1998). Brownian motion in a Brownian crack. Ann. Appl. Probab. 8 708–748.
  • Burkholder, D. L. (1977). Exit times of Brownian motion, harmonic majorization and Hardy spaces. Adv. Math. 26 182–205.
  • Chavel, I. (1984). Eigenvalues in Riemannian Geometry. Academic Press, New York.
  • Csáki, E., Csörgő, M., Földes, A. and Révész, P. (1995). Global Strassen type theorems for iterated Brownian motion. Stochastic Process. Appl. 59 321–341.
  • Csáki, E., Csörgő, M., Földes, A. and Révész, P. (1996). The local time of iterated Brownian motion. J. Theoret. Probab. 9 717–743.
  • Davies, E. B. (1995). Spectral Theory and Differential Operators. Cambridge Univ. Press.
  • DeBlassie, R. D. (1987). Exit times from cones in $\mathbb{R}^n$ of Brownian motion. Probab. Theory Related Fields 74 1–29.
  • Deheuvels, P. and Mason, D. M. (1992). A functional LIL approach to pointwise Bahadur–Kiefer theorems. In Probability in Banach Spaces (R. M. Dudley, M. G. Hahn and J. Kuelbs, eds.) 255–266. Birkhäuser, Boston.
  • Feller, W. (1971). An Introduction to Probability Theory and its Applications. Wiley, New York.
  • Funaki, T. (1979). A probabilistic construction of the solution of some higher order parabolic differential equations. Proc. Japan Acad. Ser. A Math. Sci. 55 176–179.
  • Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series and Products. Academic Press, New York.
  • Helms, L. L. (1967). Biharmonic functions and Brownian motion. J. Appl. Probab. 4 130–136.
  • Helms, L. L. (1987). Biharmonic functions with prescribed fine normal derivative on the Martin boundary. Acta Math. Hungar. 49 139–143.
  • Hochberg, K. J. and Orsingher, E. (1996). Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations. J. Theoret. Probab. 9 511–532.
  • Hu, Y. (1999). Hausdorff and packing functions of the level sets of iterated Brownian motion. J. Theoret. Probab. 12 313–346.
  • Hu, Y., Pierre-Lotti-Viand, D. and Shi, Z. (1995). Laws of the iterated logarithm for iterated Wiener processes. J. Theoret. Probab. 8 303–319.
  • Hu, Y. and Shi, Z. (1995). The Csörgő–Révész modulus of non-differentiability of iterated Brownian motion. Stochastic Process. Appl. 58 267–279.
  • Khoshnevisan, D. and Lewis, T. M. (1996). The uniform modulus of continuity for iterated Brownian motion. J. Theoret. Probab. 9 317–333.
  • Khoshnevisan, D. and Lewis, T. M. (1996). Chung's law of the iterated logarithm for iterated Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 32 349–359.
  • Khoshnevisan, D. and Lewis, T. M. (1999). Stochastic calculus for Brownian motion in a Brownian fracture. Ann. Appl. Probab. 9 629–667.
  • Krylov, V. Ju. (1960). Some properties of the distribution corresponding to the equation $\frac{\partial u}{\partial t} = (-1)^{q+1} \frac{\partial^{2q}u}{\partial x^{2q}}$. Soviet Math. Dokl. 1 760–763.
  • Mądrecki, A. and Rybaczuk, M. (1990). On a Feyman–Kac type formula. In Stochastic Methods in Experimental Sciences (W. Kasprzak and A. Weron, eds.) 312–321. World Scientific, River Edge, NJ.
  • Mądrecki, A. and Rybaczuk, M. (1993). New Feyman–Kac type formula. Rep. Math. Phys. 32 301–327.
  • Nikitin, Y. and Orsingher, E. (2000). On sojourn distributions of processes related to some higher-order heat-type equations. J. Theoret. Probab. 13 997–1012.
  • Nishioka, K. (1987). A stochastic solution of a higher order parabolic equation. J. Math. Soc. Japan 39 209–231.
  • Nishioka, K. (1996). Monopoles and dipoles in biharmonic pseudo-process. Proc. Japan Acad. Ser. A. Math. Sci. 72 47–50.
  • Nishioka, K. (1997). The first hitting time and place of a half-line by a biharmonic pseudo-process. Japan. J. Math. 23 235–280.
  • Nishioka, K. (2001). Boundary conditions for one-dimensional biharmonic pseudo process. Electron. J. Probab. 6 13.
  • Orsingher, E. (1990). Random motions governed by third-order equations. Adv. in Appl. Probab. 22 915–928.
  • Orsingher, E. (1991). Processes governed by signed measures connected with third-order “heat type” equations. Lithuanian Math. J. 31 220–231.
  • Orsingher, E. and Zhao, X. (1999). Iterated processes and their applications to higher order ODE's. Acta Math. Sinica 15 173–180.
  • Port, S. C. and Stone, C. J. (1978). Brownian Motion and Potential Theory. Academic Press, New York.
  • Shi, Z. (1995). Lower limits of iterated Wiener processes. Statist. Probab. Lett. 23 259–270.
  • Vanderbei, R. J. (1984). Probabilistic solution of the Dirichlet problem for biharmonic functions in discrete space. Ann. Probab. 12 311–324.
  • Xiao, Y. (1998). Local times and related properties of multidimensional iterated Brownian motion. J. Theoret. Probab. 11 383–408.