Occupation and local times for skew Brownian motion with applications to dispersion across an interface
Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, Edward Waymire, and Brian Wood
Source: Ann. Appl. Probab. Volume 21, Number 1
(2011), 183-214.
Abstract
Advective skew dispersion is a natural Markov process defined by a diffusion with drift across an interface of jump discontinuity in a piecewise constant diffusion coefficient. In the absence of drift, this process may be represented as a function of α-skew Brownian motion for a uniquely determined value of α=α∗; see Ramirez et al. [Multiscale Model. Simul. 5 (2006) 786–801]. In the present paper, the analysis is extended to the case of nonzero drift. A determination of the (joint) distributions of key functionals of standard skew Brownian motion together with some associated probabilistic semigroup and local time theory is given for these purposes. An application to the dispersion of a solute concentration across an interface is provided that explains certain symmetries and asymmetries in recently reported laboratory experiments conducted at Lawrence–Livermore Berkeley Labs by Berkowitz et al. [Water Resour. Res. 45 (2009) W02201].
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Keywords: Skew Brownian motion; advection-diffusion; local time; occupation time; elastic skew Brownian motion; stochastic order; first passage time
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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1292598031
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References
Appuhamillage, T., Bokil, V. A., Thomann, E., Waymire, E. and Wood, B. (2009). Solute transport across an interface: A Fickian theory for skewness in breakthrough curves. Water Resour. Res. DOI: 10.1029/2009WR008258.
Appuhamillage, T. and Sheldon, D. (2010). Ranked excursion heights and first passage time of skew Brownian motion. Available at http://arxiv.org/abs/1008.2989.
Barlow, M., Burdzy, K., Kaspi, H. and Mandelbaum, A. (2000). Variably skewed Brownian motion. Electron. Comm. Probab. 5 57–66 (electronic).
Mathematical Reviews (MathSciNet): MR1752008
Zentralblatt MATH: 0949.60090
Barlow, M., Burdzy, K., Kaspi, H. and Mandelbaum, A. (2001). Coalescence of skew Brownian motions. In Séminaire de Probabilités, XXXV. Lecture Notes in Math. 1755 202–205. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1837288
Zentralblatt MATH: 0977.60074
Barlow, M., Pitman, J. and Yor, M. (1989). On Walsh’s Brownian motions. In Séminaire de Probabilités, XXIII. Lecture Notes in Math. 1372 275–293. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1022893
Berkowitz, B., Cortis, A., Dror, I. and Scher, H. (2008). Laboratory experiments on dispersive transport across interfaces: The role of flow direction. EOS, Transactions of the American Geophysical Union 89 Fall Meeting Supplement, Abstract H23H-05.
Berkowitz, B., Cortis, A., Dror, I. and Scher, H. (2009). Laboratory experiments on dispersive transport across interfaces: The role of flow direction. Water Resour. Res. 45 W02201. DOI: 10.1029/2008WR007342.
Bhattacharya, R. and Waymire, E. (2009). Stochastic Processes with Applications. Classics in Applied Mathematics 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
Burdzy, K. and Chen, Z.-Q. (2001). Local time flow related to skew Brownian motion. Ann. Probab. 29 1693–1715.
Mathematical Reviews (MathSciNet): MR1880238
Zentralblatt MATH: 1037.60057
Digital Object Identifier: doi:10.1214/aop/1015345768
Project Euclid: euclid.aop/1015345768
Decamps, M., Goovaerts, M. and Schoutens, W. (2006). Asymmetric skew Bessel processes and their applications to finance. J. Comput. Appl. Math. 186 130–147.
Mathematical Reviews (MathSciNet): MR2190302
Zentralblatt MATH: 1087.91022
Digital Object Identifier: doi:10.1016/j.cam.2005.03.067
Freidlin, M. and Sheu, S.-J. (2000). Diffusion processes on graphs: Stochastic differential equations, large deviation principle. Probab. Theory Related Fields 116 181–220. DOI: 10.1007/PL00008726.
Mathematical Reviews (MathSciNet): MR1743769
Zentralblatt MATH: 0957.60088
Digital Object Identifier: doi:10.1007/PL00008726
Harrison, J. M. and Shepp, L. A. (1981). On skew Brownian motion. Ann. Probab. 9 309–313.
Mathematical Reviews (MathSciNet): MR606993
Zentralblatt MATH: 0462.60076
Digital Object Identifier: doi:10.1214/aop/1176994472
Project Euclid: euclid.aop/1176994472
Hoteit, H., Mose, R., Younes, A., Lehmann, F. and Ackerer, P. (2002). Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods. Math. Geol. 34 435–456.
Mathematical Reviews (MathSciNet): MR1951790
Zentralblatt MATH: 1107.76401
Digital Object Identifier: doi:10.1023/A:1015083111971
Itô, K. and McKean, H. P. Jr. (1963). Brownian motions on a half line. Illinois J. Math. 7 181–231.
Mathematical Reviews (MathSciNet): MR154338
Zentralblatt MATH: 0114.33601
Project Euclid: euclid.ijm/1255644633
Itô, K. and McKean, H. P. Jr. (1996). Diffusion Processes and Their Sample Paths. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR345224
Karatzas, I. and Shreve, S. E. (1984). Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control. Ann. Probab. 12 819–828.
Mathematical Reviews (MathSciNet): MR744236
Zentralblatt MATH: 0544.60069
Digital Object Identifier: doi:10.1214/aop/1176993230
Project Euclid: euclid.aop/1176993230
Kuo, R., Irwin, N., Greenkorn, R. and Cushman, J. (1999). Experimental investigation of mixing in aperiodic heterogeneous porous media: Comparison with stochastic transport theory. Transport in Porous Media 37 169–182.
LaBolle, E., Quastel, J. and Fogg, G. (1998). Diffusion theory for transport in porous media: Transition-probability densities of diffusion processes corresponding to advection-dispersion equations. Water Resour. Res. 34 1685–1693.
LaBolle, E., Quastel, J., Fogg, G. and Gravner, J. (2000). Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients. Water Resour. Res. 36 651–662.
Le Gall, J. F. (1982). Temps locaux et equations differentielles stochastiques. Ph.D. thesis, Th’ese de troisi’eme cycle, Univ. Pierre et Marie Curie (Paris VI), Paris.
Le Gall, J. F. (1984). One-dimensional stochastic differential equations involving the local times of the unknown process. In Stochastic Analysis and Applications (Swansea, 1983). Lecture Notes in Math. 1095 51–82. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR777514
Zentralblatt MATH: 0551.60059
Digital Object Identifier: doi:10.1007/BFb0099122
Lejay, A. and Martinez, M. (2006). A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Probab. 16 107–139.
Mathematical Reviews (MathSciNet): MR2209338
Zentralblatt MATH: 1094.60056
Digital Object Identifier: doi:10.1214/105051605000000656
Project Euclid: euclid.aoap/1141654283
Ouknine, Y. (1990). Le “Skew-Brownian motion” et les processus qui en dérivent. Teor. Veroyatnost. i Primenen. 35 173–179.
Mathematical Reviews (MathSciNet): MR1050069
Portenko, N. I. (1990). Generalized Diffusion Processes. Translations of Mathematical Monographs 83. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR1104660
Ramirez, J. (2007). Skew Brownian motion and branching processes applied to diffusion-advection in heterogenous media and fluid flow. Ph.D. thesis, Oregon State Univ.
Mathematical Reviews (MathSciNet): MR2710544
Ramirez, J. (2010). Multi-skewed Brownian motion and diffusion. Trans. Amer. Math. Soc. To appear.
Ramirez, J. M., Thomann, E. A., Waymire, E. C., Haggerty, R. and Wood, B. (2006). A generalized Taylor–Aris formula and skew diffusion. Multiscale Model. Simul. 5 786–801 (electronic).
Mathematical Reviews (MathSciNet): MR2257235
Ramirez, J., Thomann, E., Waymire, E., Chastanet, J. and Wood, J. (2008). A note on the theoretical foundations of particle tracking methods in heterogeneous porous media. Water Resour. Res. 44 W01501. DOI: 10.1029/2007WR005914.
Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1083357
Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales. Volume 2: Itô Calculus. Wiley, New York.
Mathematical Reviews (MathSciNet): MR921238
Zentralblatt MATH: 0977.60005
Trotter, H. F. (1958). A property of Brownian motion paths. Illinois J. Math. 2 425–433.
Mathematical Reviews (MathSciNet): MR96311
Zentralblatt MATH: 0117.35502
Project Euclid: euclid.ijm/1255454547
Uffink, G. J. M. (1985). A random walk method for the simulation of macrodispersion in a stratified aquifer. In Relation of Groundwater Quantity and Quality. IAHS Publication 146 103–114. IAHS Press, Walingford, UK.
Walsh, J. B. (1978). A diffusion with a discontinuous local time. Asterisque 52–53 37–45.
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