## The Annals of Applied Probability

### Iterated Brownian motion in an open set

R. Dante DeBlassie

#### 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

https://projecteuclid.org/euclid.aoap/1089736295

Digital Object Identifier
doi:10.1214/105051604000000404

Mathematical Reviews number (MathSciNet)
MR2071433

Zentralblatt MATH identifier
1051.60082

#### 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

#### 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.