## The Annals of Probability

### The heat equation and reflected Brownian motion in time-dependent domains

#### Abstract

The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as "noncylindrical domains,'' and its connections with partial differential equations. Construction is given for RBM in $C^3$-smooth time-dependent domains in the n-dimensional Euclidean space $\R^n$. We present various sample path properties of the process, two-sided estimates for its transition density function, and a probabilistic representation of solutions to some partial differential equations. Furthermore, the one-dimensional case is thoroughly studied, with the assumptions on the smoothness of the boundary drastically relaxed.

#### Article information

Source
Ann. Probab., Volume 32, Number 1B (2004), 775-804.

Dates
First available in Project Euclid: 11 March 2004

https://projecteuclid.org/euclid.aop/1079021464

Digital Object Identifier
doi:10.1214/aop/1079021464

Mathematical Reviews number (MathSciNet)
MR2039943

Zentralblatt MATH identifier
1046.60060

#### Citation

Burdzy, Krzysztof; Chen, Zhen-Qing; Sylvester, John. The heat equation and reflected Brownian motion in time-dependent domains. Ann. Probab. 32 (2004), no. 1B, 775--804. doi:10.1214/aop/1079021464. https://projecteuclid.org/euclid.aop/1079021464

#### References

• Anderson, J. and Pitt, L. (1997). Large time asymptotics for Brownian hitting densities of transient concave curves. J. Theoret. Probab. 10 921--934.
• Bass, R. F. (1997). Diffusions and Elliptic Operators. Springer, New York.
• Bass, R. and Burdzy, K. (1996). A critical case for Brownian slow points. Probab. Theory Related Fields 105 85--108.
• Bass, R. and Burdzy, K. (1999). Stochastic bifurcation models. Ann. Probab. 27 50--108.
• Bass, R., Burdzy, K. and Chen, Z.-Q. (2002). Uniqueness for reflecting Brownian motion in lip domains. Preprint.
• Bass, R. F. and Hsu, P. (1990). The semimartingale structure of reflecting Brownian motion. Proc. Amer. Math. Soc. 108 1007--1010.
• Bass, R. F. and Hsu, P. (1991). Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19 486--508.
• Burdzy, K. and Chen, Z.-Q. (1998). Weak convergence of reflecting Brownian motions. Electron. Comm. Probab. 3 29--33.
• Burdzy, K., Chen, Z.-Q. and Sylvester, J. (2003). The heat equation and reflected Brownian motion in time-dependent domains II: Singularities of solutions. J. Funct. Anal. 204 1--34.
• Burdzy, K., Chen, Z.-Q. and Sylvester, J. (2004). The heat equation in time dependent domains with insulated boundaries. J. Math. Anal. Appl. To appear.
• Burdzy, K. and Kendall, W. (2000). Efficient Markovian couplings, examples and counterexamples. Ann. Appl. Probab. 10 362--409.
• Burdzy, K. and Khoshnevisan, D. (1998). Brownian motion in a Brownian crack. Ann. Appl. Probab. 8 708--748.
• Burdzy, K. and Toby, E. (1995). A Skorohod-type lemma and a decomposition of reflected Brownian motion. Ann. Probab. 23 586--604.
• Chen, Z.-Q. (1993). On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Related Fields 94 281--316.
• Chen, Z.-Q. (1996). Reflecting Brownian motions and a deletion result for Sobolev spaces of order $(1,2)$. Potential Anal. 5 383--401.
• Chen, Z.-Q., Fitzsimmons, P. J. and Williams, R. J. (1993). Reflecting Brownian motions: Quasimartingales and strong Caccioppoli sets. Potential Anal. 2 219--243.
• Costantini, C. (1992). The Skorohod oblique reflection principle in domains with corners and applications to stochastic differential equations. Probab. Theory Related Fields 91 43--70.
• Crank, J. (1984). Free and Moving Boundary Problems. Clarendon Press, Oxford.
• Cranston, M. and Le Jan, Y. (1989). Simultaneous boundary hitting for a two point reflecting Brownian motion. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 234--238. Springer, Berlin.
• Doob, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York.
• Dupuis, P. and Ishii, H. (1993). SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 554--580.
• Durbin, J. (1992). The first-passage density of the Brownian morion process to a curved boundary. With an appendix by D. Williams. J. Appl. Probab. 29 291--304.
• Durrett, R. (1984). Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, CA.
• El Karoui, N. and Karatzas, I. (1991a). A new approach to the Skorohod problem, and its applications. Stochastics Stochastics Rep. 34 57--82.
• El Karoui, N. and Karatzas, I. (1991b). Correction: A new approach to the Skorohod problem, and its applications.'' Stochastics Stochastics Rep. 36 265.
• Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffts, NJ.
• Fukushima, M. (1967). A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math. 4 183--215.
• Fukushima, M. (1999). On semimartingale characterization of functionals of symmetric Markov processes. Electron J. Probab. 4 1--31.
• Fukushima, M. and Tomisaki, M. (1996). Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps. Probab. Theory Related Fields 106 521--557.
• Greenwood, P. and Perkins, E. (1983). A conditioned limit theorem for random walk and Brownian local time on square root boundaries. Ann. Probab. 11 227--261.
• Hofmann, S. and Lewis, J. L. (1996). $L^2$ solvability and representation by caloric layer potentials in time-varying domains. Ann. Math. 144 349--420.
• Hsu, P. (1984). Reflecting Brownian, boundary local time, and the Neumann boundary value problem. Ph.D. dissertation, Stanford Univ.
• Hsu, P. (1987). On the Poisson kernel for the Neumann problem of Schrödinger operators. J. London Math. Soc. 36 370--384.
• Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
• Itô, S. (1957). Fundamental solutions of parabolic differential equations and boundary value problems. Japan. J. Math. 27 55--102.
• Karatzas, I. and Shreve, S. (1994). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
• Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
• Knight, F. (1981). Essentials of Brownian Motion and Diffusion. Amer. Math. Soc., Providence, RI.
• Knight, F. (1999). On the path of an inert object impinged on one side by a Brownian particle. Preprint.
• Lewis, J. L. and Murray, M. A. M. (1995). The method of layer potentials for the heat equation in time-varying domains. Memoir AMS 114 545.
• Lions, P. L. and Sznitman, A. S. (1984). Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511--537.
• Moser, J. (1964). A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 101--134.
• Oshima, Y. (2001). On a construction of diffusion processes on moving domains. Preprint.
• Port, S. and Stone, C. (1978). Brownian Motion and Classical Potential Theory. Academic Press, New York.
• Roberts, G. (1991). A comparison theorem for conditioned Markov processes. J. Appl. Probab. 28 74--83.
• Saisho, Y. (1987). Stochastic differential equations for multi-dimensional domain with reflecting boundary. Probab. Theory Related Fields 74 455--477.
• Sato, K. and Ueno, T. (1965). Multi-dimensional diffusion and the Markov process on the boundary. J. Math. Kyoto Univ. 4-3 529--605.
• Shimura, M. (1985). Excursions in a cone for two-dimensional Brownian motion. J. Math. Kyoto Univ. 25 433--443.
• Soucaliuc, F., Toth, B. and Werner, W. (2000). Reflection and coalescence between independent one-dimensional Brownian paths. Ann. Inst. H. Poincaré Probab. Statist. 36 509--545.
• Stroock, D. W. and Varadhan, S. R. S. (1971). Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 147--225.
• Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, New York.
• Tanaka, M. (1979). Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9 163--177.
• Williams, R. J. and Zheng, W. A. (1990). On reflecting Brownian motion---a weak convergence approach. Ann. Inst. H. Poincaré Probab. Statist. 26 461--488.