zbMATH — the first resource for mathematics

Occupation and local times for skew Brownian motion with applications to dispersion across an interface. (English) Zbl 1226.60113
Loosely speaking a skew Brownian motion is a Brownian motion with two different diffusion coefficients \(D^-\) and \(D^+\) on the negative and positive half-lines, respectively. The paper provides new theoretical results for functionals of skew Brownian motions and its associated semi-group theory. In particular, the trivariate density of position, the symmetric local time at zero and the occupation time on the positive half-line are obtained via a Feynman-Kac formula for an elastic skew Brownian motion. Also, an elastic change of measure is devised to obtain an integral representation of the transition density for a skew Brownian motion with non-zero drift. Besides this, several other distributional results on functionals of skew Brownian motions are given. The paper concludes with a specific application to dispersion across an interface. In an actual physical experiment by B. Berkowitz, A. Cortis, I. Dror and H. Scher [“Laboratory experiments on dispersive transport across interfaces: the role of flow direction”, Water Resour. Res. 45, W02201 (2009), doi:10.1029/2008WR007342], it was experimentally observed that fine to coarse breakthrough is faster than coarse to fine breakthrough. The theoretical results given in the present paper explain this phenomenon within the framework of Fickian flux laws. Here, fine and course media are characterized by their relative dispersion rates \(D^{-}< D^+\).

60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60K40 Other physical applications of random processes
60G44 Martingales with continuous parameter
60J55 Local time and additive functionals
35C15 Integral representations of solutions to PDEs
35R05 PDEs with low regular coefficients and/or low regular data
47D07 Markov semigroups and applications to diffusion processes
Full Text: DOI arXiv
[1] 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: . · Zbl 1226.60113 · dx.doi.org
[2] Appuhamillage, T. and Sheldon, D. (2010). Ranked excursion heights and first passage time of skew Brownian motion. Available at . · Zbl 1266.60140 · arxiv.org
[3] Barlow, M., Burdzy, K., Kaspi, H. and Mandelbaum, A. (2000). Variably skewed Brownian motion. Electron. Comm. Probab. 5 57-66 (electronic). · Zbl 0949.60090 · emis:journals/EJP-ECP/EcpVol5/paper6.abs.html · eudml:120650
[4] 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. · Zbl 0977.60074 · numdam:SPS_2001__35__202_0 · eudml:114062
[5] 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. · Zbl 0747.60072 · numdam:SPS_1989__23__275_0 · eudml:113680
[6] 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.
[7] 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: . · dx.doi.org
[8] Bhattacharya, R. and Waymire, E. (2009). Stochastic Processes with Applications. Classics in Applied Mathematics 61 . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. · Zbl 1171.60333
[9] Burdzy, K. and Chen, Z.-Q. (2001). Local time flow related to skew Brownian motion. Ann. Probab. 29 1693-1715. · Zbl 1037.60057 · doi:10.1214/aop/1015345768
[10] Decamps, M., Goovaerts, M. and Schoutens, W. (2006). Asymmetric skew Bessel processes and their applications to finance. J. Comput. Appl. Math. 186 130-147. · Zbl 1087.91022 · doi:10.1016/j.cam.2005.03.067
[11] 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: . · Zbl 0957.60088 · doi:10.1007/PL00008726 · dx.doi.org
[12] Harrison, J. M. and Shepp, L. A. (1981). On skew Brownian motion. Ann. Probab. 9 309-313. · Zbl 0462.60076 · doi:10.1214/aop/1176994472
[13] 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. · Zbl 1107.76401 · doi:10.1023/A:1015083111971
[14] Itô, K. and McKean, H. P. Jr. (1963). Brownian motions on a half line. Illinois J. Math. 7 181-231. · Zbl 0114.33601
[15] Itô, K. and McKean, H. P. Jr. (1996). Diffusion Processes and Their Sample Paths . Springer, Berlin. · Zbl 0837.60001
[16] 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. · Zbl 0544.60069 · doi:10.1214/aop/1176993230
[17] 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.
[18] 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.
[19] 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.
[20] 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.
[21] 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. · Zbl 0551.60059 · doi:10.1007/BFb0099122
[22] Lejay, A. and Martinez, M. (2006). A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Probab. 16 107-139. · Zbl 1094.60056 · doi:10.1214/105051605000000656
[23] Ouknine, Y. (1990). Le “Skew-Brownian motion” et les processus qui en dérivent. Teor. Veroyatnost. i Primenen. 35 173-179. · Zbl 0696.60080
[24] Portenko, N. I. (1990). Generalized Diffusion Processes. Translations of Mathematical Monographs 83 . Amer. Math. Soc., Providence, RI. · Zbl 0727.60088
[25] 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.
[26] Ramirez, J. (2010). Multi-skewed Brownian motion and diffusion. Trans. Amer. Math. Soc.
[27] 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). · Zbl 1122.60072 · doi:10.1137/050642770
[28] 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: . · dx.doi.org
[29] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Springer, Berlin. · Zbl 0731.60002
[30] Rogers, L. C. G. and Williams, D. (1987). Diffusions , Markov Processes , and Martingales. Volume 2: Itô Calculus . Wiley, New York. · Zbl 0977.60005
[31] Trotter, H. F. (1958). A property of Brownian motion paths. Illinois J. Math. 2 425-433. · Zbl 0117.35502
[32] 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.
[33] Walsh, J. B. (1978). A diffusion with a discontinuous local time. Asterisque 52-53 37-45.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.