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Sticky Brownian motion and its numerical solution. (English) Zbl 1444.60048

Summary: Sticky Brownian motion is the simplest example of a diffusion process that can spend finite time both in the interior of a domain and on its boundary. It arises in various applications in fields such as biology, materials science, and finance. This article spotlights the unusual behavior of sticky Brownian motions from the perspective of applied mathematics, and provides tools to efficiently simulate them. We show that a sticky Brownian motion arises naturally for a particle diffusing on \(\mathbb{R}_+\) with a strong, short-ranged potential energy near the origin. This is a limit that accurately models mesoscale particles, those with diameters \(\approx 100\,\mathrm{nm}\)–\(10\,\)\micro m, which form the building blocks for many common materials. We introduce a simple and intuitive sticky random walk to simulate sticky Brownian motion, which also gives insight into its unusual properties. In parameter regimes of practical interest, we show that this sticky random walk is two to five orders of magnitude faster than alternative methods to simulate a sticky Brownian motion. We outline possible steps to extend this method toward simulating multidimensional sticky diffusions.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
60J60 Diffusion processes
60J65 Brownian motion
35K05 Heat equation
35K20 Initial-boundary value problems for second-order parabolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] M. Amir, Sticky Brownian motion as the strong limit of a sequence of random walks, Stochastic Process. Appl., 39 (1991), pp. 221-237. · Zbl 0744.60097
[2] D. E. Apushkinskaya and A. I. Nazarov, A survey of results on nonlinear Venttsel problems, Appl. Math., 45 (2000), pp. 69-80. · Zbl 1058.35118
[3] D. E. Apushkinskaya and A. I. Nazarov, The Venttsel problem for nonlinear elliptic equations, J. Math. Sci., 101 (2000), pp. 2861-2880. · Zbl 1074.35531
[4] S. Asakura and F. Oosawa, On interaction between two bodies immersed in a solution of macromolecules, J. Chem. Phys., 22 (1954), pp. 1255-1256.
[5] G. Barraquand and M. Rychnovsky, Large Deviations for Sticky Brownian Motions, preprint, https://arxiv.org/abs/1905.10280, 2019. · Zbl 1462.60035
[6] R. J. Baxter, Percus-Yevick equation for hard spheres with surface adhesion, J. Chem. Phys., 49 (1968), pp. 2770-2774.
[7] M. Bossy, E. Gobet, and D. Talay, Symmetrized Euler scheme for an efficient approximation of reflected diffusions, J. Appl. Probab., 41 (2004), pp. 877-889. · Zbl 1076.65009
[8] N. Bou-Rabee, SPECTRWM: Spectral random walk method for the numerical solution of stochastic partial differential equations, SIAM Rev., 60 (2018), pp. 386-406, https://doi.org/10.1137/16M1089034. · Zbl 1407.60084
[9] N. Bou-Rabee and E. Vanden-Eijnden, Continuous-time random walks for the numerical solution of stochastic differential equations, Mem. Amer. Math. Soc., 256 (2018). · Zbl 1437.65002
[10] À. Calsina and J. Z. Farkas, Steady states in a structured epidemic model with Wentzell boundary condition, J. Evol. Equ., 12 (2012), pp. 495-512. · Zbl 1254.35223
[11] F. Calvo, J. P. K. Doye, and D. J. Wales, Energy landscapes of colloidal clusters: Thermodynamics and rearrangement mechanisms, Nanoscale, 4 (2012), pp. 1085-1100.
[12] M. T. Casey, R. T. Scarlett, W. B. Rogers, I. Jenkins, T. Sinno, and J. C. Crocker, Driving diffusionless transformations in colloidal crystals using DNA handshaking, Nature Commun., 3 (2012), art. 1209.
[13] G. Ciccotti, T. Lelièvre, and E. Vanden-Eijnden, Projection of diffusions on submanifolds: Application to mean force computation, Comm. Pure Appl. Math., 61 (2007), pp. 371-408. · Zbl 1185.82050
[14] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Stoch. Model. Appl. Probab. 38, B. Rozovskii and G. Grimmett, eds., Springer-Verlag, New York, 2010. · Zbl 1177.60035
[15] M. D. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc., 6 (1951), pp. 1-12. · Zbl 0042.37602
[16] J. L. Doob, Topics in the theory of Markoff chains, Trans. Amer. Math. Soc., 52 (1942), pp. 37-64. · Zbl 0063.09001
[17] J. L. Doob, Markoff chains-denumerable case, Trans. Amer. Math. Soc., 58 (1945), pp. 455-473. · Zbl 0063.01146
[18] J. Doye, D. J. Wales, and R. S. Berry, The effect of the range of the potential on the structures of clusters, J. Chem. Phys., 103 (1995), pp. 4234-4249.
[19] P. Dupuis and M. R. James, Rates of convergence for approximation schemes in optimal control, SIAM J. Control Optim., 36 (1998), pp. 719-741, https://doi.org/10.1137/S0363012994267789. · Zbl 0914.93072
[20] W. E and E. Vanden-Eijnden, Transition-path theory and path-finding algorithms for the study of rare events, Ann. Rev. Phys. Chem., 61 (2010), pp. 391-420.
[21] A. Eberle and R. Zimmer, Sticky couplings of multidimensional diffusions with different drifts, Ann. Inst. H. Poincaré, 55 (2019), pp. 2370-2394. · Zbl 1434.60213
[22] T. C. Elston and C. R. Doering, Numerical and analytical studies of nonequilibrium fluctuation-induced transport processes, J Statist. Phys., 83 (1996), pp. 359-383.
[23] H.-J. Engelbert and G. Peskir, Stochastic differential equations for sticky Brownian motion, Stochastics, 86 (2014), pp. 993-1021. · Zbl 1337.60120
[24] R. Fantoni and P. Sollich, Multicomponent adhesive hard sphere models and short-ranged attractive interactions in colloidal or micellar solutions, Phys. Rev. E, 74 (2006), art. 051407.
[25] A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli, \(C_0\)-semigroups generated by second order differential operators with general Wentzell boundary conditions, Proc. Amer. Math. Soc., 128 (2000), pp. 1981-1989. · Zbl 0947.47034
[26] A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), pp. 1-19. · Zbl 1043.35062
[27] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed., John Wiley, New York, 1971. · Zbl 0219.60003
[28] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), pp. 468-519. · Zbl 0047.09303
[29] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), pp. 1-31. · Zbl 0059.11601
[30] W. Feller, Generalized second order differential operators and their lateral conditions, Illinois J. Math., 1 (1957), pp. 459-504. · Zbl 0077.29102
[31] D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, 2001. · Zbl 0889.65132
[32] A. Gandolfi, A. Gerardi, and F. Marchetti, Association rates of diffusion-controlled reactions in two dimensions, Acta Appl. Math., 4 (1985), pp. 139-155. · Zbl 0562.92007
[33] C. Gardiner, Stochastic Methods: A Handbook for the Natural Sciences, 4th ed., Springer, 2009. · Zbl 1181.60001
[34] D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), pp. 403-434.
[35] D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), pp. 2340-2361.
[36] E. Gobet, Euler schemes and half-space approximation for the simulation of diffusion in a domain, ESAIM Probab. Statist., 5 (2001), pp. 261-297. · Zbl 0998.60081
[37] T. Grafke and E. Vanden-Eijnden, Numerical computation of rare events via large deviation theory, Chaos, 29 (2019), art. 063118. · Zbl 1416.65042
[38] C. Graham, Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection, Probab. Theory Related Fields, 101 (1995), pp. 291-302. · Zbl 0820.60083
[39] J. M. Harrison and A. J. Lemoine, Sticky Brownian motion as the limit of storage processes, J. Appl. Probab., 18 (2016), pp. 216-226. · Zbl 0453.60072
[40] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), pp. 525-546, https://doi.org/10.1137/S0036144500378302. · Zbl 0979.65007
[41] J. Hinch, Perturbation Methods, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0746.34001
[42] M. Holmes-Cerfon, Sticky-sphere clusters, Ann. Rev. Condensed Matter Phys., 8 (2017), pp. 77-98.
[43] M. Holmes-Cerfon, S. J. Gortler, and M. P. Brenner, A geometrical approach to computing free-energy landscapes from short-ranged potentials, Proc. Natl. Acad. Sci. USA, 110 (2013), pp. E5-E14.
[44] C. J. Howitt, Stochastic Flows and Sticky Brownian Motion, Ph.D. thesis, University of Warwick, UK, 2007.
[45] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1989. · Zbl 0684.60040
[46] K. Itô and H. P. McKean Jr., Brownian motions on a half line, Illinois J. Math., 7 (1963), pp. 181-231. · Zbl 0114.33601
[47] Y. Kabanov, M. Kijima, and S. Rinaz, A positive interest rate model with sticky barrier, Quant. Finance, 7 (2007), pp. 269-284. · Zbl 1142.91530
[48] Y. Kallus and M. Holmes-Cerfon, Free energy of singular sticky-sphere clusters, Phys. Rev. E, 95 (2017), pp. 2491-2518.
[49] S. Karlin and H. M. Taylor, A second course in stochastic processes, Academic Press, New York, London, 1981. · Zbl 0469.60001
[50] J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Springer, New York, 1996. · Zbl 0846.34001
[51] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, New York, 1999. · Zbl 1216.60052
[52] H. J. Kushner, Probability limit theorems and the convergence of finite difference approximations of partial differential equations, J. Math. Anal. Appl., 32 (1970), pp. 77-103. · Zbl 0205.43801
[53] H. J. Kushner, Finite difference methods for the weak solutions of the Kolmogorov equations for the density of both diffusion and conditional diffusion processes, J. Math. Anal. Appl., 53 (1976), pp. 251-265. · Zbl 0329.65059
[54] H. J. Kushner, Probabilistic methods for finite difference approximations to degenerate elliptic and parabolic equations with Neumann and Dirichlet boundary conditions, J. Math. Anal. Appl., 53 (1976), pp. 644-668. · Zbl 0329.65055
[55] H. J. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977. · Zbl 0547.93076
[56] H. J. Kushner, Heavy Traffic Analysis of Controlled Queueing and Communication Networks, Appl. Math. (N.Y.) 47, Springer Science & Business Media, New York, 2001. · Zbl 0988.90004
[57] H. J. Kushner, Numerical approximations for stochastic differential games, SIAM J. Control Optim., 41 (2002), pp. 457-486, https://doi.org/10.1137/S0363012901389457. · Zbl 1014.60035
[58] H. J. Kushner, Numerical approximations for stochastic differential games: The ergodic case, SIAM J. Control Optim., 42 (2004), pp. 1911-1933, https://doi.org/10.1137/S0036301290140034. · Zbl 1151.91352
[59] H. J. Kushner, Numerical approximations for stochastic systems with delays in the state and control, Stochastics, 78 (2006), pp. 343-376. · Zbl 1109.93046
[60] H. J. Kushner, Numerical methods for controls for nonlinear stochastic systems with delays and jumps: Applications to admission control, Stochastics, 83 (2011), pp. 277-310. · Zbl 1217.93187
[61] H. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd ed., Springer, New York, 2001. · Zbl 0968.93005
[62] H. J. Kushner and C.-F. Yu, Probability methods for the convergence of finite difference approximations to partial differential equations, J. Math. Anal. Appl., 43 (1973), pp. 603-625. · Zbl 0271.35029
[63] H. J. Kushner and C.-F. Yu, The approximate calculation of invariant measures of diffusions via finite difference approximations to degenerate elliptic partial differential equations, J. Math. Anal. Appl., 51 (1975), pp. 359-367. · Zbl 0313.65083
[64] J. C. Latorre, P. Metzner, C. Hartmann, and C. Schütte, A structure-preserving numerical discretization of reversible diffusions, Commun. Math. Sci., 9 (2011), pp. 1051-1072. · Zbl 1284.60153
[65] T. Lelièvre, M. Rousset, and G. Stoltz, Langevin dynamics with constraints and computation of free energy differences, Math. Comp., 81 (2012), pp. 2071-2125. · Zbl 1274.82034
[66] F. A. Longstaff, Multiple equilibria and term structure models, J. Financial Econom., 32 (1992), pp. 333-344.
[67] P. J. Lu and D. A. Weitz, Colloidal particles: Crystals, glasses, and gels, Ann. Rev. Condensed Matter Phys., 4 (2013), pp. 217-233.
[68] Y. Luo and N. S. Trudinger, Linear second order elliptic equations with Venttsel boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), pp. 193-207. · Zbl 0771.35014
[69] R. J. Macfarlane, B. Lee, M. R. Jones, N. Harris, G. C. Schatz, and C. A. Mirkin, Nanoparticle superlattice engineering with DNA, Science, 334 (2011), pp. 204-208.
[70] A. Malins, S. R. Williams, J. Eggers, H. Tanaka, and C. P. Royall, Geometric frustration in small colloidal clusters, J. Phys. Condensed Matter, 21 (2009), art. 425103.
[71] V. N. Manoharan, Colloidal matter: Packing, geometry, and entropy, Science, 349 (2015), art. 1253751. · Zbl 1355.82031
[72] G. Meng, N. Arkus, M. P. Brenner, and V. N. Manoharan, The free-energy landscape of clusters of attractive hard spheres, Science, 327 (2010), pp. 560-563.
[73] P. Metzner, Transition Path Theory for Markov Processes, Ph.D. thesis, Free University Berlin, 2007. · Zbl 1185.60004
[74] P. Metzner, C. Schütte, and E. Vanden-Eijnden, Transition path theory for Markov jump processes, Multiscale Model. Simul., 7 (2009), pp. 1192-1219, https://doi.org/10.1137/070699500. · Zbl 1185.60086
[75] T. D. T. Nguyen, Sticky Brownian Motions and a Probabilistic Solution to a Two-Point Boundary Value Problem, preprint, https://arxiv.org/abs/1810.06199, 2018.
[76] T. D. T. Nguyen, Fick law and sticky Brownian motions, J. Statist. Phys., 174 (2019), pp. 494-518. · Zbl 1459.82188
[77] M. G. Noro and D. Frenkel, Extended corresponding-states behavior for particles with variable range attractions, J. Chem. Phys., 113 (2000), pp. 2941-2944.
[78] B. Oksendal, Stochastic Differential Equations, 6th ed., Springer, New York, 2005.
[79] R. W. Perry, M. C. Holmes-Cerfon, M. P. Brenner, and V. N. Manoharan, Two-dimensional clusters of colloidal spheres: Ground states, excited states, and structural rearrangements, Phys. Rev. Lett., 114 (2015), art. 228301.
[80] G. Peskir, A probabilistic solution to the Stroock-Williams equation, Ann. Probab., 42 (2014), pp. 2196-2206. · Zbl 1320.60127
[81] G. Peskir, On Boundary Behaviour of One-Dimensional Diffusions: From Brown to Feller and Beyond, Springer, Cham, 2015.
[82] F. Platten, N. E. Valadez-Pérez, R. Castan͂eda-Priego, and S. U. Egelhaaf, Extended law of corresponding states for protein solutions, J. Chem. Phys., 142 (2015), art. 174905.
[83] P. E. Protter, Stochastic Integration and Differential Equations, Stoch. Model. Appl. Probab. 21, Springer, Berlin, 2004. · Zbl 1041.60005
[84] W. B. Rogers, W. M. Shih, and V. N. Manoharan, Using DNA to program the self-assembly of colloidal nanoparticles and microparticles, Nature Rev. Materials, 1 (2016), art. 10760.
[85] J. Ryckaert, G. Ciccotti, and H. Berendsen, Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of \(n\)-alkanes, J. Comput. Phys., 23 (1977), pp. 327-341.
[86] G. Stell, Sticky spheres and related systems, J. Statist. Phys., 63 (1991), pp. 1203-1221.
[87] D. W. Stroock and S. Varadhan, Diffusion processes with boundary conditions, Comm. Pure Appl. Math., 24 (1971), pp. 147-225. · Zbl 0227.76131
[88] A. D. Venttsel, On boundary conditions for multidimensional diffusion processes, Theory Probab. Appl., 4 (1959), pp. 164-177. · Zbl 0089.13404
[89] D. J. Wales, Energy landscapes of clusters bound by short-ranged potentials, ChemPhysChem, 11 (2010), pp. 2491-2494.
[90] H. Wang, C. S. Peskin, and T. C. Elston, A robust numerical algorithm for studying biomolecular transport processes, J. Theoret. Biol., 221 (2003), pp. 491-511. · Zbl 1464.92119
[91] Y. Wang, Y. Wang, X. Zheng, É. Ducrot, J. S. Yodh, M. Weck, and D. J. Pine, Crystallization of DNA-coated colloids, Nature Commun., 6 (2015), art. 7253.
[92] J. Warren, Branching processes, the Ray-Knight theorem, and sticky Brownian motion, in Séminaire de Probabilités XXXI, Springer, Berlin, 1997, pp. 1-15. · Zbl 0884.60081
[93] E. Zappa, M. Holmes-Cerfon, and J. Goodman, Monte Carlo on manifolds: Sampling densities and integrating functions, Comm. Pure Appl. Math., 71 (2018), pp. 2609-2647. · Zbl 06993640
[94] Y. Zeng and Y. Luo, Linear parabolic equations with Venttsel initial boundary conditions, Bull. Austral. Math. Soc., 50 (1994), pp. 465-479. · Zbl 0813.35024
[95] Z. Zeravcic, V. N. Manoharan, and M. P. Brenner, Size limits of self-assembled colloidal structures made using specific interactions, Proc. Natl. Acad. Sci. USA, 111 (2014), pp. 15918-15923.
[96] Y. Zhang, A. McMullen, L.-L. Pontani, X. He, R. Sha, N. C. Seeman, J. Brujic, and P. M. Chaikin, Sequential self-assembly of DNA functionalized droplets, Nature Commun., 8 (2017), art. 21.
[97] R. Zimmer, Couplings and Kantorovich Contractions with Explicit Rates for Diffusions, Ph.D. thesis, University of Bonn, 2017.
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