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Analytic expressions of the solutions of advection-diffusion problems in one dimension with discontinuous coefficients. (English) Zbl 07106896
MSC:
65C05 Monte Carlo methods
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35R05 PDEs with low regular coefficients and/or low regular data
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[1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th ed., Dover, New York, 1970.
[2] T. Appuhamillage, V. Bokil, E. Thomann, E. Waymire, and B. Wood, Occupation and local times for skew Brownian motion with applications to dispersion across an interface, Ann. Appl. Probab., 21 (2011), pp. 183–214, https://doi.org/10.1214/10-AAP691. · Zbl 1226.60113
[3] T. Appuhamillage, V. Bokil, E. Thomann, E. Waymire, and B. Wood, Corrections: Occupation and local times for skew Brownian motion with applications to dispersion across an interface, Ann. Appl. Probab., 21 (2011), pp. 2050–2051, https://doi.org/10.1214/11-AAP775. · Zbl 1226.60113
[4] D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. (N.S.), 73 (1967), pp. 890–896, https://doi.org/10.1090/S0002-9904-1967-11830-5.
[5] M. Bechtold, J. Vanderborght, O. Ippisch, and H. Vereecken, Efficient random walk particle tracking algorithm for advective-dispersive transport in media with discontinuous dispersion coefficients and water contents, Water Res. Res., 47 (2011), W10526, https://doi.org/10.1029/2010WR010267.
[6] S. Berezin and O. Zayats, Skew Brownian Motion with dry friction: The Pugachev-Sveshnikov equation approach, Mater. Phys. Mech., 41 (2019), pp. 103–110.
[7] A. N. Borodin and P. Salminen, Handbook of Brownian Motion – Facts and Formulae, 2nd ed., Probab. Appl., Birkhäuser, Basel, 2002.
[8] M. Bossy, N. Champagnat, S. Maire, and D. Talay, Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics, ESAIM Math. Model. Numer. Anal., 44 (2010), pp. 997–1048, https://doi.org/10.1051/m2an/2010050. · Zbl 1204.82020
[9] R. Cantrell and C. Cosner, Diffusion models for population dynamics incorporating individual behavior at boundaries: Applications to refuge design, Theor. Popul. Biol., 55 (1999), pp. 189–207, https://doi.org/10.1006/tpbi.1998.1397. · Zbl 0958.92028
[10] Z.-Q. Chen and M. Zili, One-dimensional heat equation with discontinuous conductance, Sci. China Math., 58 (2015), pp. 97–108, https://doi.org/10.1007/s11425-014-4912-1.
[11] M. Decamps, A. De Schepper, and M. Goovaerts, Applications of \(\delta\)-function perturbation to the pricing of derivative securities, Phys. A, 342 (2004), pp. 677–692, https://doi.org/10.1016/j.physa.2004.05.035.
[12] F. Delay, P. Ackerer, and C. Danquigny, Simulating solute transport in porous or fractured formations using random walks particle tracking: A review, Vadose Zone J., 4 (2005), pp. 360–379.
[13] D. Dereudre, S. Mazzonetto, and S. Roelly, An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers, Monte Carlo Methods Appl., 22 (2016), pp. 1–23, https://doi.org/10.1515/mcma-2016-0100. · Zbl 1335.60151
[14] D. Dereudre, S. Mazzonetto, and S. Roelly, Exact simulation of Brownian diffusions with drift admitting jumps, SIAM J. Sci. Comput., 39 (2017), pp. A711–A740, https://doi.org/10.1137/16M107699X. · Zbl 1370.60113
[15] J. Eckhardt and G. Teschl, Sturm-Liouville operators with measure-valued coefficients, J. Anal. Math., 120 (2013), pp. 151–224. · Zbl 1315.34001
[16] P. Étoré, On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients, Electron. J. Probab., 11 (2006), pp. 249–275, https://doi.org/10.1214/EJP.v11-311. · Zbl 1112.60061
[17] P. Étoré and A. Lejay, A Donsker theorem to simulate one-dimensional processes with measurable coefficients, ESAIM Probab. Stat., 11 (2007), pp. 301–326, https://doi.org/10.1051/ps:2007021. · Zbl 1181.60123
[18] P. Étoré and M. Martinez, Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process, Monte Carlo Methods Appl., 19 (2013), pp. 41–71. · Zbl 1269.65007
[19] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), pp. 1–31. · Zbl 0059.11601
[20] W. Feller, The general diffusion operator and positivity preserving semi-groups in one dimension, Ann. of Math. (2), 60 (1954), pp. 417–436. · Zbl 0057.09805
[21] W. Feller, On second order differential operators, Ann. of Math. (2), 61 (1955), pp. 90–105. · Zbl 0064.11301
[22] W. Feller, Generalized second order differential operators and their lateral conditions, Illinois J. Math., 1 (1957), pp. 459–504. · Zbl 0077.29102
[23] W. Feller, On the intrinsic form for second order differential operator, Illinois J. Math., 2 (1959), pp. 1–18.
[24] E. R. Fernholz, T. Ichiba, and I. Karatzas, Two Brownian particles with rank-based characteristics and skew-elastic collisions, Stochastic Process. Appl., 123 (2013), pp. 2999–3026. · Zbl 1296.60148
[25] A. Gairat and V. Shcherbakov, Density of skew Brownian motion and its functionals with application in finance, Math. Finance, 26 (2016), pp. 1069–1088, https://doi.org/10.1111/mafi.12120. · Zbl 1411.91555
[26] E. M. Garon and J. V. Lambers, Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle, Comput. Appl. Math., 37 (2018), pp. 2566–2587, https://doi.org/10.1007/s40314-017-0465-6. · Zbl 1404.65194
[27] B. Gaveau, M. Okada, and T. Okada, Second order differential operators and Dirichlet integrals with singular coefficients, I. Functional calculus of one-dimensional operators, Tohoku Math. J. (2), 39 (1987), pp. 465–504. · Zbl 0653.35034
[28] U. Gräwe, E. Deleersnijder, S. H. A. M. Shah, and A. W. Heemink, Why the Euler scheme in particle tracking is not enough: The shallow-sea pycnocline test case, Ocean Dynam., 62 (2012), pp. 501–514, https://doi.org/10.1007/s10236-012-0523-y.
[29] Y. Güldü, On discontinuous Dirac operator with eigenparameter dependent boundary and two transmission conditions, Bound. Value Probl. (2016), 135, https://doi.org/10.1186/s13661-016-0639-y. · Zbl 1342.34027
[30] H. Hoteit, R. Mose, A. Younes, F. Lehmann, and P. Ackerer, Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods, Math. Geol., 34 (2002), pp. 435–456. · Zbl 1107.76401
[31] K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, 2nd ed., Springer, Berlin, 1974.
[32] E. M. LaBolle, G. E. Fogg, and A. F. B. Tompson, Random-walk simulation of transport in heterogeneous porous media: Local mass-conservation problem and implementation methods, Water Res. Res., 32 (1996), pp. 583–593, https://doi.org/10.1029/95WR03528.
[33] O. A. Ladyženskaja, V. J. Rivkind, and N. N. Ural‘ceva, Equations aux dérivées partielles de type elliptique, Monogr. Univ. Math. 31, Dunod, Paris, 1968.
[34] O. A. Ladyženskaja, V. J. Rivkind, and N. N. Ural‘ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 33, American Mathematical Society, Providence, RI, 1968.
[35] O. A. Ladyženskaja, V. J. Rivkind, and N. N. Ural‘ceva, Classical solvability of diffraction problems for equations of elliptic and parabolic types, Dokl. Akad. Nauk SSSR, 158 (1964), pp. 513–515.
[36] J. Langebrake, L. Riotte-Lambert, C. W. Osenberg, and P. De Leenheer, Differential movement and movement bias models for marine protected areas, J. Math. Biol., 64 (2012), pp. 667–696. · Zbl 1262.34049
[37] H. Langer and W. Schenk, Knotting of one-dimensional Feller processes, Math. Nachr., 113 (1983), pp. 151–161. · Zbl 0532.60066
[38] J.-F. Le Gall, One-dimensional stochastic differential equations involving the local times of the unknown process, in Stochastic Analysis, Lecture Notes in Math. 1095, Springer, Berlin, 1984, pp. 51–82.
[39] A. Lejay, On the constructions of the skew Brownian motion, Probab. Surv., 3 (2006), pp. 413–466. · Zbl 1189.60145
[40] A. Lejay, Simulation of a stochastic process in a discontinuous layered medium, Electron. Commun. Probab., 16 (2011), pp. 764–774. · Zbl 1243.60062
[41] A. Lejay, L. Lenôtre, and G. Pichot, An exponential timestepping algorithm for diffusion with discontinuous coefficients, J. Comput. Phys., 396 (2019), pp. 888–904.
[42] A. Lejay and M. Martinez, A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients, Ann. Appl. Probab., 16 (2006), pp. 107–139. · Zbl 1094.60056
[43] A. Lejay and G. Pichot, Simulating diffusion processes in discontinuous media: A numerical scheme with constant time steps, J. Comput. Phys., 231 (2012), pp. 7299–7314. · Zbl 1284.65007
[44] A. Lejay and G. Pichot, Simulating diffusion processes in discontinuous media: Benchmark tests, J. Comput. Phys., 314 (2016), pp. 384–413, https://doi.org/10.1016/j.jcp.2016.03.003. · Zbl 1349.65019
[45] F. T. Lindstrom and F. Oberhettinger, A note on a Laplace transform pair associated with mass transport in porous media and heat transport problems, SIAM J. Appl. Math., 29 (1975), pp. 288–292. · Zbl 0325.44001
[46] A. Lipton, Oscillating Bachelier and Black–Scholes formulas, World Scientific, in Financial Engineering, Hackensack, NJ, 2018, pp. 371–394, https://doi.org/10.1142/10425.
[47] M. Martinez, Interprétations probabilistes d’opérateurs sous forme divergence et analyse de méthodes numériques probabilistes associées, Ph.D. thesis, Université de Provence, Marseille, France, 2004.
[48] H. P. McKean, Jr., Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc., 82 (1956), pp. 519–548, https://doi.org/10.1090/S0002-9947-1956-0087012-3.
[49] T. Okada, Asymptotic behavior of skew conditional heat kernels on graph networks, Canad. J. Math., 45 (1993), pp. 863–878, https://doi.org/10.4153/CJM-1993-049-6. · Zbl 0799.58084
[50] J. Pitman and M. Yor, Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches, Bernoulli, 9 (2003), pp. 1–24, https://doi.org/10.3150/bj/1068129008. · Zbl 1024.60032
[51] P. Puri and P. Kythe, Some inverse Laplace transforms of exponential form, Z. Angew. Math. Phys., 39 (1988), pp. 150–156, https://doi.org/10.1007/BF00945761. · Zbl 0644.44002
[52] J. M. Ramirez, E. A. Thomann, E. C. Waymire, J. Chastanet, and B. D. Wood, A note on the theoretical foundations of particle tracking methods in heterogeneous porous media, Water Resour. Res., 44 (2008), W01501, https://doi.org/10.1029/2007WR005914.
[53] P. Salamon, D. Fernàndez-Garcia, and J. J. Gómez-Hernández, A review and numerical assessment of the random walk particle tracking method, J. Contam. Hydrol., 87 (2006), pp. 277–305, https://doi.org/10.1016/j.jconhyd.2006.05.005.
[54] D. Spivakovskaya, A. W. Heemink, and E. Deleersnijder, Lagrangian modelling of multi-dimensional advection-diffusion with space-varying diffusivities: Theory and idealized test cases, Ocean Dynam., 57 (2007), pp. 189–203.
[55] D. W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, in Séminaire de Probabilités, XXII, Lecture Notes in Math. 1321, Springer, Berlin, 1988, pp. 316–347, https://doi.org/10.1007/BFb0084145.
[56] D. Thomson, W. Physick, and R. Maryon, Treatment of interfaces in random walk dispersion models, J. Appl. Meteorol., 36 (1997), pp. 1284–1295.
[57] G. Uffink, A random walk method for the simulation of macrodispersion in a stratified aquifer, in Relation of Groundwater Quantity and Quality, IAHS Publication 146, International Association of Hydrological Sciences, Wallingford, United Kingdom, 1985, pp. 103–114.
[58] J. B. Walsh, A diffusion with discontinuous local time, in Temps locaux, Vol. 52-53, Société Mathématique de France, Paris, 1978, pp. 37–45.
[59] S. Wang, S. Song, and Y. Wang, Skew Ornstein–Uhlenbeck processes and their financial applications, J. Comput. Appl. Math., 273 (2015), pp. 363–382, https://doi.org/10.1016/j.cam.2014.06.023. · Zbl 1304.60046
[60] E. Zauderer, Partial Differential Equations of Applied Mathematics, 3rd ed., Pure Appl. Math., Wiley, Hoboken, NJ, 2006, https://doi.org/10.1002/9781118033302.
[61] M. Zhang, Calculation of diffusive shock acceleration of charged particles by skew Brownian motion, Astrophys. J., 541 (2000), pp. 428–435.
[62] C. Zheng and G. D. Bennett, Applied Contaminant Transport Modelling, 2nd ed., Wiley-Interscience, New York, 2002.
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