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Skew Disperson and Continuity of Local Time

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Abstract

Results are provided that highlight the effect of interfacial discontinuities in the diffusion coefficient on the behavior of certain basic functionals of the diffusion, such as local times and occupation times, extending previous results in (Appuhamillage et al., Ann Appl Probab 21:183–214, 2011; Water Resour Res 46:W07511, 2009) on the behavior of first passage times. The main goal is to obtain a characterization of large scale parameters and behavior by an analysis at the fine scale of stochastic particle motions. In particular, considering particle concentration modeled by a diffusion equation with piecewise constant diffusion coefficient, it is shown that the continuity of a natural modification of local time is the individual (stochastic) particle scale equivalent to continuity of flux at the scale of the (macroscopic) particle concentrations. Consequences of this involve the determination of a skewness transmission probability in the presence of an interface, as well as corollaries concerning interfacial effects on occupation time of the associated stochastic particles.

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Notes

  1. While we emphasize “natural”choices from the point of view of modeling (and units), there are very sound and important reasons for the standard mathematical definitions. In particular, no suggestion to change the mathematical definition is intended. Indeed, as the proof of Theorem 2.1 demonstrates, the notion of semimartingale local time and occupation time and their relationship is extremely powerful in singling out the special value of \(\alpha (\lambda )\) for given interface parameter \(\lambda \) and dispersion coefficients \(D^\pm \).

References

  1. Appuhamillage, T.: The skew diffusion with drift: A new class of stochastic processes with applications to parabolic equations with piecewise smooth coefficients. PhD Thesis, Oregon State University, Corvallis (2011).

  2. Appuhamillage, T., Bokil, V., Thomann, E., Waymire, E., Wood, B.: Occupation and local times for Skew Brownian Motion with applications to dispersion across an interface. Ann. Appl. Probab. 21(1), 183–214 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Appuhamillage, T., Bokil, V., Thomann, E., Waymire, E., Wood, B.: Solute transport across an interface: a Fickian theory for skewness in breakthrough curve. Water Resour. Res. 46, W07511 (2009). doi:10.1029/2009WR008258

    ADS  Google Scholar 

  4. Appuhamillage, T.A., Sheldon, D.: First passage time of skew Brownian motion. J. Appl. Probab. 49(3), 686–696 (2012)

    Article  MathSciNet  Google Scholar 

  5. Ben Arous, G., Ramirez, A.F.: Asymptotic survival probabilities in the random saturation process. Ann. Probab. 4(28), 1470–1527 (2000)

    MathSciNet  Google Scholar 

  6. Berkowitz, B., Cortis, A., Dror, I., Scher, H.: Laboratory experiments on dispersive transport across interfaces: the role of flow direction. Water Resour. Res. 45, W02201 (2009). doi:10.1029/2008WR007342

    ADS  Google Scholar 

  7. Bhattacharya, R.N., Waymire, E.: Stochastic processes with applications. Wiley, New York (1990). Reprinted in the SIAM Classics in Applied Mathematics book series, 2009

    Google Scholar 

  8. Cherny, A.S., Shiryaev, A.N., Yor, M.: Limit behavior of the “horizontal-vertical”random walk and some extensions of the Donsker-Prokhorov invariance principle. Th. Probab. Appl. 47(3), 498–516 (2002)

    MathSciNet  Google Scholar 

  9. Cosner, C.C., Cantrell, R.S.: Spatial Ecology via Reaction-Diffusion Equations. Wiley, New York (2003)

    MATH  Google Scholar 

  10. Eisenbaum, N., Kaspi, H.: On the continuity of local times of Borel right Markov processes. Ann. Probab. 35(3), 915–934 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Friedlin, M., Sheu, J.S.: Diffusion processes on graphs: stochastics differential equations, large deviation principle. Probab. Theory Relat. Fields 116, 181–220 (2000)

    Article  Google Scholar 

  12. Guo, F., Jokipii, J.R., Kota, J.: Particle acceleration by collisionless shocks containing large-scale magnetic-field variations. Astrophys. J. 725(1), 128–133 (2010)

    Article  ADS  Google Scholar 

  13. Harrison, J.M., Shepp, L.A.: On skew Brownian motion. Ann. Probab. 9, 309–313 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hill, A.E.: Leakage of Barotropic Slope currents onto the continental shelf’. J. Phys. Oceanogr. 25, 1617–1621 (1995)

    Article  ADS  Google Scholar 

  15. Hoteit, H., Mose, R., Younes, A., Lehmann, F., Ackerer, Ph: Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and random walk methods. Math. Geol. 34(4), 435–456 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Itô, K., McKean, H.P.: Brownian motions on a half line, Illinois. J. Math. 7, 181–231 (1963)

    MATH  Google Scholar 

  17. Itô, K., McKean, H.P.: Diffusion processes and their sample paths. Springer-Verlag, New York (1965)

    Book  MATH  Google Scholar 

  18. Kuo, R.K.H., Irwin, N.C., Greenkorn, R.A., Cushman, J.H.: Experimental investigation of mixing in aperiodic heterogeneous porous media: Comparison with stochastic transport theory. Transport in Porous Media 37, 169–182 (1999)

    Article  Google Scholar 

  19. Le Gall, J.-F.: One-dimensional stochastic differential equations involving the local times of the unknown process. Stochastic analysis and applications (Swansea, 1983). Lecture Notes in Math, vol. 1095, pp. 51–82. Springer, Berlin (1984)

    Google Scholar 

  20. Loukianova, D., Loukianov, O.: Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions. Ann. de Inst. Henri Poincarè - Prob et Stat 44(4), 771–786 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. LaBolle, E.M., Quastel, J., Gravner, J.: Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients. Water Resour. Res. 36(3), 651–662 (2000)

    Article  ADS  Google Scholar 

  22. Lejay, A.: On the constructions of the skew Brownian motion. Probab. Surveys 3(2), 413–466 (2006)

    MATH  MathSciNet  Google Scholar 

  23. Lejay, A., Martinez, M.: A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients. Ann. Appl. Probab. 116(1), 107–139 (2008)

    MathSciNet  Google Scholar 

  24. McKenzie, H.W., Lewis, M.A., Merrill, E.H.: First passage time analysis of animal movement and insights into functional response. Bull. Math. Bio. 71(1), 107–129 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Matano, R.P., Palma, E.D.: On the upwelling of downwelling currents. J. Phys. Ocean. 38(11), 2482–2500 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  26. Newburgh, R., Peidle, J., Rueckner, W.: Einstein, Perrin, and the reality of atoms: 1905 revisited. Am. J. Phys. 74(6), 478–481 (2006)

    Article  Google Scholar 

  27. Nilsen, W., Sayit, H.: No arbitrage in markets with bounces and sinks. Int. Rev. Appld. Financ. Issues Econ. 3(4), 696–699 (2011)

    Google Scholar 

  28. Okada, T.: Asymptotics behavior of skew conditional heat kernels on graph networks. Can. J. Math. 45(4), 863–878 (1993)

    Article  MATH  Google Scholar 

  29. Ouknine, Y.: Skew-Brownian motion and associated processes. Theory Probab. Appl. 35(1), 163–169 (1991)

    Article  MathSciNet  Google Scholar 

  30. Ovaskainen, O.: Habitat-specific movement parameters estimated using mark-recapture data and a diffusion model. Ecology 85(1), 242–257 (2004)

    Article  Google Scholar 

  31. Perrin, J.: Le Mouvement Brownien et la Rèalité Moleculaire. Ann. Chimi. Phys. 18(8), 5–114 (1909)

    Google Scholar 

  32. Ramirez, J., Thomann, E., Waymire, E.: Advection-dispersion across interfaces, Statistical Sciences. Inst. Math. Stat. 28(4), 487–509 (2013). doi:10.1214/13-STS442

    MathSciNet  Google Scholar 

  33. Ramirez, J. (2007): Skew Brownian motion and branching processes applied to advective-diffusion in porous media and fluid flow, PhD Thesis, Oregon State University, Corvallis.

  34. Ramirez, J., Thomann, E., Waymire, E., Haggerty, R., Wood, B.: A generalized Taylor-Aris formula and skew diffusion. SIAM Multiscale Model. Simul. 5(3), 786–801 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. Ramirez, J.M., Thomann, E.A., Waymire, E.C., Chastanet, J., Wood, B.D.: A note on the theoretical foundations of particle tracking methods in heterogeneous porous media. Water Resour. Res. 44, W01501 (2008). doi:10.1029/2007WR005914

    ADS  Google Scholar 

  36. Ramirez, J.M.: Multi-skewed Brownian motion and diffusion in layered media. Proc. Am. Math. Soc. 139, 3739–3752 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  37. Ramirez, J.M.: Population persistence under advection-diffusion in river networks. Math. Biol. 65(5), 919–942 (2011). doi:10.1007/s00285-011-0485-6

    Article  MathSciNet  Google Scholar 

  38. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, New York (1999)

    Book  MATH  Google Scholar 

  39. Rogers, L.C.G., Williams, D.: Diffusions, markov processes and martingales. It\(\hat{0}\) Calculus, vol. II. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  40. Schultz, C.B., Crone, E.E.: Patch size and connectivity thresholds for butterfly habitat restoration. Conserv. Biol. 19(3), 887–896 (2005)

    Article  Google Scholar 

  41. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, New York (1997)

    Google Scholar 

  42. Trotter, H.: A property of Brownian motion paths Ill. J. Math. 2, 425–433 (1958)

    MATH  MathSciNet  Google Scholar 

  43. Walsh, J.: A diffusion with a discontinuous local time. Asterisque 52–53, 37–45 (1978)

    Google Scholar 

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Acknowledgments

This work was partially supported by a Grant DMS-11-22699 from the National Science Foundation. The authors also wish to gratefully acknowledge the comments provided by two referees that both improved the exposition, and enriched the results by noting the use of the Chacon-Ornstein theorem to obtain the limit of ratio occupation times.

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Authors and Affiliations

  1. Department of Mathematics, Oregon State University, Corvallis, OR, USA

    Thilanka A. Appuhamillage, Vrushali A. Bokil, Enrique A. Thomann & Edward C. Waymire

  2. School of Chemical, Biological and Environmental Engineering, Oregon State University, Corvallis, OR, USA

    Brian D. Wood

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  1. Thilanka A. Appuhamillage
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  2. Vrushali A. Bokil
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  3. Enrique A. Thomann
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  4. Edward C. Waymire
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  5. Brian D. Wood
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Correspondence to Edward C. Waymire.

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Appuhamillage, T.A., Bokil, V.A., Thomann, E.A. et al. Skew Disperson and Continuity of Local Time. J Stat Phys 156, 384–394 (2014). https://doi.org/10.1007/s10955-014-1010-2

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