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Stochastic Partial Differential Equations

  • Tadahisa Funaki
Chapter
Part of the SpringerBriefs in Probability and Mathematical Statistics book series (SBPMS )

Abstract

So far, we have discussed discrete interface models. Taking their (mesoscopic) continuum limit, as a time evolution of interfaces or some other related physical order parameters, one would expect to obtain stochastic partial differential equations (SPDEs), which are partial differential equations having stochastic terms such as a space-time Gaussian white noise. In connection with the problem of random interfaces, we will especially discuss a stochastic Allen-Cahn equation, sometimes called the time-dependent Ginzburg-Landau (TDGL) equation or a dynamic P(ϕ)-model, in Chap.  4, and the Kardar-Parisi-Zhang (KPZ) equation in Chap.  5 This chapter explains, taking the TDGL equation as an example, some fundamental facts concerning SPDEs such as white noises, colored noises, definitions and regularities of solutions of the SPDEs we are interested in. We give some other examples of related SPDEs. The SPDEs used in physics are sometimes ill-posed. They appear in a wide variety of fields, not only in physics but also in biology, engineering (e.g., in control theory, filtering), economics (e.g., in mathematical finance), and others.

Keywords

Gibbs Measure Colored Noise Stochastic Partial Differential Equation Hydrodynamic Limit Regularity Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alfaro, M., Antonopoulou, D., Karali, G., Matano, H.: Generation and propagation of fine transition layers for the stochastic Allen-Cahn equation (2016, preprint)Google Scholar
  2. 2.
    Alfaro, M., Garcke, H., Hilhorst, D., Matano, H., Schätzle, R.: Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation. Proc. R. Soc. Edinb. Sect. A 140, 673–706 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alfaro, M., Hilhorst, D., Matano, H.: The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system. J. Differ. Equs. 245, 505–565 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Alfaro, M., Matano, H.: On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discret. Contin. Dyn. Syst. Ser. B 17, 1639–1649 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Alikakos, N.D., Bates, P.W., Chen, X.: Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Ration. Mech. Anal. 128, 165–205 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Allen, S.M., Cahn, J.W.: A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 27, 1085–1095 (1979)CrossRefGoogle Scholar
  7. 7.
    Almgren, F., Taylor, J.E., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31, 387–438 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Alt, H.W., Caffarelli, L.A.:. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325, 105–144 (1981)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Alt, H.W., Caffarelli, L.A., Friedman, A.: Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282, 431–461 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Commun. Pure Appl. Math. 64, 466–537 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Andjel, E.D.: Invariant measures for the zero range processes. Ann. Probab. 10, 525–547 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Angenent, S.B.: Some recent results on mean curvature flow. In: Recent Advances in Partial Differential Equations. RAM Recherches en mathématiques appliquées, vol. 30, pp. 1–18. Masson, Paris (1994)Google Scholar
  13. 13.
    Antonopoulou, D., Blömker, D., Karali, G.: Front motion in the one-dimensional stochastic Cahn-Hilliard equation. SIAM J. Math. Anal. 44, 3242–3280 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Antonopoulou, D., Karali, G., Kossioris, G.: Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discret. Contin. Dyn. Syst. Ser. A 30, 1037–1054 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Baccelli, F., Karpelevich, F.I., Kelbert, M.Ya., Puhalskii, A.A., Rybko, A.N., Suhov, Yu.M.: A mean-field limit for a class of queueing networks. J. Stat. Phys. 66, 803–825 (1992)Google Scholar
  17. 17.
    M. Balázs, Quastel, J., Seppäläinen, T.: Fluctuation exponent of the KPZ/stochastic Burgers equation. J. Am. Math. Soc. 24, 683–708 (2011)Google Scholar
  18. 18.
    Barles, G., Souganidis, P.E.: A new approach to front propagation problems: theory and applications. Arch. Ration. Mech. Anal. 141, 237–296 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Bellettini, G.: Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations. Lecture Notes, vol. 12. Scuola Normale, Superiore di Pisa (2013)Google Scholar
  20. 20.
    Beltoft, D., Boutillier, C., Enriquez, N.: Random young diagrams in a rectangular box. Mosc. Math. J. 12, 719–745 (2012)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Bertini, L., Brassesco, S., Buttà, P., Presutti, E.: Front fluctuations in one dimensional stochastic phase field equations. Ann. Henri Poincaré 3, 29–86 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Bertini, L., Brassesco, S., Buttà, P.: Soft and hard wall in a stochastic reaction diffusion equation. Arch. Ration. Mech. Anal. 190, 307–345 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Bertini, L., Brassesco, S., Buttà, P.: Front fluctuations for the stochastic Cahn-Hilliard equation. Braz. J. Probab. Stat. 29, 336–371 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Bertini, L., Buttà, P., Pisante, A.: Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound. arXiv:1604.02064Google Scholar
  25. 25.
    Bertini, L., Cancrini, N.: The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78, 1377–1401 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Large deviation approach to non equilibrium processes in stochastic lattice gases. Bull. Braz. Math. Soc. 37, 611–643 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Bertini, L., Landim, C., Mourragui, M.: Dynamical large deviations for the boundary driven weakly asymmetric exclusion process. Ann. Probab. 37, 2357–2403 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Bodineau, T.: The Wulff construction in three and more dimensions. Commun. Math. Phys. 207, 197–229 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Bodineau, T., Ioffe, D., Velenik, Y.: Rigorous probabilistic analysis of equilibrium crystal shapes, probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41, 1033–1098 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Bolthausen, E., Chiyonobu, T., Funaki, T.: Scaling limits for weakly pinned Gaussian random fields under the presence of two possible candidates. J. Math. Soc. Jpn. 67, 1359–1412 (2015) (special issue for Kiyosi Itô)Google Scholar
  32. 32.
    Bolthausen, E., Funaki, T., Otobe, T.: Concentration under scaling limits for weakly pinned Gaussian random walks. Probab. Theory Relat. Fields 143, 441–480 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Bolthausen, E., Ioffe, D.: Harmonic crystal on the wall: a microscopic approach. Commun. Math. Phys. 187, 523–566 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Borodin, A., Corwin, I., Ferrari, P., Veto, B.: Height fluctuations for the stationary KPZ equation. Math. Phys. Anal. Geom. 18 (Art. 20), 95 (2015)Google Scholar
  35. 35.
    Bounebache, S.K.: A random string with reflection in a convex domain. Stoch. Anal. Appl. 29, 523–549 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Bounebache, S.K., Zambotti, L.: A skew stochastic heat equation. J. Theor. Probab. 27, 168–201 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Boutillier, C.: Pattern densities in non-frozen planar dimer models. Commun. Math. Phys. 271, 55–91 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Brakke, K.A.: The Motion of a Surface by its Mean Curvature. Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)Google Scholar
  39. 39.
    Brassesco, S., Buttà, P.: Interface fluctuations for the D = 1 stochastic Ginzburg-Landau equation with nonsymmetric reaction term. J. Stat. Phys. 93, 1111–1142 (1998)Google Scholar
  40. 40.
    Brassesco, S., De Masi, A., Presutti, E.: Brownian fluctuations of the instanton in the d = 1 Ginzburg-Landau equation with noise. Ann. Inst. H. Poincaré Probab. Statist. 31, 81–118 (1995)Google Scholar
  41. 41.
    Cahn, J.W., Elliott, C.M., Novick-Cohen, A.: The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature Eur. J. Appl. Math. 7, 287–301 (1996)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Caputo, P., Martinelli, F., Simenhaus, F., Toninelli, F.: “Zero” temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion. Commun. Pure Appl. Math. 64, 778–831 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Caputo, P., Martinelli, F., Toninelli, F.: Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach. Commun. Math. Phys. 311, 157–189 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Carr, J., Pego, R.L.: Metastable patterns in solutions of u t = ε 2 u xxf(u). Commun. Pure Appl. Math. 42, 523–576 (1989)Google Scholar
  45. 45.
    Cerf, R., Kenyon, R.: The low-temperature expansion of the Wulff crystal in the 3D Ising model. Commun. Math. Phys. 222, 147–179 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Cerf, R., Pisztora, A.: On the Wulff crystal in the Ising model. Ann. Probab. 28, 947–1017 (2000)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Chandra, A., Weber, H.: Stochastic PDEs, regularity structures, and interacting particle systems. arXiv:1508.03616Google Scholar
  48. 48.
    Chang, C.C., Yau, H.-T.: Fluctuations of one dimensional Ginzburg-Landau models in nonequilibrium. Commun. Math. Phys. 145, 209–239 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Chen, X.: Generation and propagation of interfaces for reaction-diffusion equations. J. Differ. Equs. 96, 116–141 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Chen, X.: Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Commun. Part. Differ. Equs. 19, 1371–1395 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Chen, X.: Generation, propagation, and annihilation of metastable patterns. J. Differ. Equs. 206, 399–437 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Chen, X., Hilhorst, D., Logak, E.: Mass conserving Allen-Cahn equation and volume preserving mean curvature flow. Interfaces Free Bound. 12, 527–549 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Chen, Y.G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33, 749–786 (1991)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Corwin, I., Tsai, L.-C.: KPZ equation limit of higher-spin exclusion processes. Ann. Probab. (2016, published online)Google Scholar
  55. 55.
    Corwin, I., Shen, H., Tsai, L.-C.: ASEP(q, j) converges to the KPZ equation. arXiv:1602.01908Google Scholar
  56. 56.
    Da Prato, G., Debussche, A., Tubaro, L.: A modified Kardar-Parisi-Zhang model. Electron. Commun. Probab. 12, 442–453 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge/New York (1992)zbMATHCrossRefGoogle Scholar
  58. 58.
    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes, vol. 229. Cambridge University Press, Cambridge/New York (1996)Google Scholar
  59. 59.
    Debussche, A., Zambotti, L.: Conservative stochastic Cahn-Hilliard equation with reflection. Ann. Probab. 35, 1706–1739 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    De Masi, A., Presutti, E., Scacciatelli, E.: The weakly asymmetric simple exclusion process. Ann. Inst. H. Poincaré Probab. Statist. 25, 1–38 (1989)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Dembo, A., Vershik, A., Zeitouni, O.: Large deviations for integer partitions. Markov Process. Related Fields 6, 147–179 (2000)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Applications of Mathematics, vol. 38. Springer, New York (1998)Google Scholar
  63. 63.
    Deuschel, J.-D., Giacomin, G., Ioffe, D.: Large deviations and concentration properties for ∇φ interface models. Probab. Theory Relat. Fields 117, 49–111 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Dirr, N., Luckhaus, S., Novaga, M.: A stochastic selection principle in case of fattening for curvature flow. Calc. Var. Part. Differ. Equs. 13, 405–425 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Dobrushin, R.L., Kotecký, R., Shlosman, S.: Wulff Construction: A Global Shape from Local Interaction. AMS Translation Series, vol. 104. American Mathematical Society, Providence (1992)Google Scholar
  66. 66.
    Duits, M.: Gaussian free field in an interlacing particle system with two jump rates. Commun. Pure Appl. Math. 66, 600–643 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. 130, 453–471 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Elliott, C.M., Garcke, H.: Existence results for diffusive surface motion laws. Adv. Math. Sci. Appl. 7, 467–490 (1997)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Elworthy, K.D., Truman, A., Zhao, H.Z., Gaines, J.G.: Approximate travelling waves for generalized KPP equations and classical mechanics. Proc. R. Soc. Lond. A 446, 529–554 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Es-Sarhir, A., von Renesse, M., Stannat, W.: Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow. Nonlinear Differ. Equ. Appl. 19, 663–675 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Es-Sarhir, A., von Renesse, M.: Ergodicity of stochastic curve shortening flow in the plane. SIAM J. Math. Anal. 44, 224–244 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Escher, J., Simonett, G.: The volume preserving mean curvature flow near spheres. Proc. Am. Math. Soc. 126, 2789–2796 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45, 1097–1123 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Differ. Geom. 33, 635–681 (1991)MathSciNetzbMATHGoogle Scholar
  75. 75.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. II. Trans. Am. Math. Soc. 330, 321–332 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Eyink, G., Lebowitz, J.L., Spohn, H.: Lattice gas models in contact with stochastic reservoirs: local equilibrium and relaxation to the steady state. Commun. Math. Phys. 140, 119–131 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    Fatkullin, I., Kovačič, G., Vanden-Eijnden, E.: Reduced dynamics of stochastically perturbed gradient flows. Commun. Math. Sci. 8, 439–461 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Ferrari, P.L., Sasamoto, T., Spohn, H.: Coupled Kardar-Parisi-Zhang equations in one dimension. J. Stat. Phys. 153, 377–399 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Ferrari, P.L., Spohn, H.: Step fluctuations for a faceted crystal. J. Stat. Phys. 113, 1–46 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Fife, P.C., Hsiao, L.: The generation and propagation of internal layers. Nonlinear Anal. 12, 19–41 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Fife, P.C., McLeod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal. 65, 335–361 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Freidlin, M.: Functional Integration and Partial Differential Equations. Princeton University Press, Princeton (1985)zbMATHCrossRefGoogle Scholar
  84. 84.
    Freidlin, M.: Semi-linear PDE’s and limit theorems for large deviations. In: Hennequin (ed.) Lectures on Probability Theory and Statistics: Ecole d’Eté de Probabilités de Saint-Flour XX – 1990. Lecture Notes in Mathematics, vol. 1527, pp. 2–109. Springer (1992)Google Scholar
  85. 85.
    Freiman, G., Vershik, A., Yakubovich, Y.: A local limit theorem for random strict partitions. Theory Probab. Appl. 44, 453–468 (2000)Google Scholar
  86. 86.
    Friedrichs, K.: Über ein Minimumproblem für Potentialströmungen mit freiem Rande. Math. Ann. 109, 60–82 (1934)MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    Fritz, J.: On the diffusive nature of entropy flow in infinite systems: remarks to a paper by Guo-Papanicolau-Varadhan. Commun. Math. Phys. 133, 331–352 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Friz, P.K., Hairer, M.: A Course on Rough Paths. With an Introduction to Regularity Structures. Universitext. Springer, Cham (2014)zbMATHCrossRefGoogle Scholar
  89. 89.
    Funaki, T.: Random motion of strings and related stochastic evolution equations. Nagoya Math. J. 89, 129–193 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Funaki, T.: Derivation of the hydrodynamical equation for one-dimensional Ginzburg-Landau model. Probab. Theory Relat. Fields 82, 39–93 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Funaki, T.: The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian. Probab. Theory Relat. Fields 90, 519–562 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Funaki, T.: The reversible measures of multi-dimensional Ginzburg-Landau type continuum model. Osaka J. Math. 28, 463–494 (1991)MathSciNetzbMATHGoogle Scholar
  93. 93.
    Funaki, T.: Regularity properties for stochastic partial differential equations of parabolic type. Osaka J. Math. 28, 495–516 (1991)MathSciNetzbMATHGoogle Scholar
  94. 94.
    Funaki, T.: A stochastic partial differential equation with values in a manifold. J. Funct. Anal. 109, 257–288 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Funaki, T.: Low temperature limit and separation of phases for Ginzburg-Landau stochastic equation. In: Kunita and Kuo (eds.) Stochastic Analysis on Infinite Dimensional Spaces, Proceedings of the U.S.-Japan Bilateral Seminar at Baton Rouge. Pitman Research Notes in Mathematical Series, vol. 310, pp. 88–98. Longman, Essex (1994)Google Scholar
  96. 96.
    Funaki, T.: The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields 102, 221–288 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Funaki, T.: Singular limit for reaction-diffusion equation with self-similar Gaussian noise. In: Elworthy, Kusuoka, Shigekawa (eds.) Proceedings of Taniguchi Symposium, New Trends in Stochastic Analysis, pp. 132–152. World Scientific (1997)Google Scholar
  98. 98.
    Funaki, T.: Singular limit for stochastic reaction-diffusion equation and generation of random interfaces. Acta Math. Sinica Engl. Ser. 15, 407–438 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Funaki, T.: Hydrodynamic limit for ∇ϕ interface model on a wall. Probab. Theory Relat. Fields 126, 155–183 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    Funaki, T.: Stochastic models for phase separation and evolution equations of interfaces. Sugaku Expositions 16, 97–116 (2003)MathSciNetzbMATHGoogle Scholar
  101. 101.
    Funaki, T.: Zero temperature limit for interacting Brownian particles, I. Motion of a single body. Ann. Probab. 32, 1201–1227 (2004)MathSciNetzbMATHGoogle Scholar
  102. 102.
    Funaki, T.: Zero temperature limit for interacting Brownian particles, II. Coagulation in one dimension. Ann. Probab. 32, 1228–1246 (2004)MathSciNetzbMATHGoogle Scholar
  103. 103.
    Funaki, T.: Stochastic interface models. In: Picard, J. (ed.) Lectures on Probability Theory and Statistics, Ecole d’Eté de Probabilités de Saint-Flour XXXIII – 2003. Lecture Notes in Mathematics, vol. 1869, pp. 103–274. Springer, Berlin (2005)Google Scholar
  104. 104.
    Funaki, T.: Stochastic Differential Equations (in Japanese) Iwanami, 1997, 203p (2005). xviii+187 pagesGoogle Scholar
  105. 105.
    Funaki, T.: Stochastic analysis on large scale interacting systems. In: Selected Papers on Probability and Statistics. Am. Math. Soc. Trans. Ser. 2 227, 49–73 (2009)MathSciNetGoogle Scholar
  106. 106.
    Funaki, T.: Equivalence of ensembles under inhomogeneous conditioning and its applications to random Young diagrams. J. Stat. Phys. 154, 588–609 (2014) (special issue for Herbert Spohn)Google Scholar
  107. 107.
    Funaki, T.: Infinitesimal invariance for the coupled KPZ equations, Memoriam Marc Yor – Séminaire de Probabilités XLVII. Lecture Notes in Mathematics, vol. 2137, pp. 37–47. Springer (2015)Google Scholar
  108. 108.
    Funaki, T., Hoshino, M.: A coupled KPZ equation and its two types of approximations (2016, preprint)Google Scholar
  109. 109.
    Funaki, T., Nishikawa, T.: Large deviations for the Ginzburg-Landau ∇ϕ interface model. Probab. Theory Related Fields 120, 535–568 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    Funaki, T., Olla, S.: Fluctuations for ∇ϕ interface model on a wall. Stoch. Proc. Appl. 94, 1–27 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    Funaki, T., Otobe, T.: Scaling limits for weakly pinned random walks with two large deviation minimizers. J. Math. Soc. Jpn. 62, 1005–1041 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    Funaki, T., Quastel, J.: KPZ equation, its renormalization and invariant measures. Stoch. Partial Differ. Equ. Anal. Comput. 3, 159–220 (2015)MathSciNetzbMATHGoogle Scholar
  113. 113.
    Funaki, T., Sakagawa, H.: Large deviations for ∇φ interface model and derivation of free boundary problems. In: Funaki, T., Osada, H. (eds.) Stochastic Analysis on Large Scale Interacting Systems. Advanced Studies in Pure Mathematics, vol. 39, pp. 173–211. Mathematical Society of Japan, Tokyo (2004)Google Scholar
  114. 114.
    Funaki, T., Sasada, M.: Hydrodynamic limit for an evolutional model of two-dimensional Young diagrams. Commun. Math. Phys. 299, 335–363 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    Funaki, T., Sasada, M., Sauer, M., Xie, B.: Fluctuations in an evolutional model of two-dimensional Young diagrams. Stoch. Proc. Appl. 123, 1229–1275 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Funaki, T., Spohn, H.: Motion by mean curvature from the Ginzburg-Landau ∇ϕ interface model. Commun. Math. Phys. 185, 1–36 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    Funaki, T., Xie, B.: A stochastic heat equation with the distributions of Lévy processes as its invariant measures. Stoch. Proc. Appl. 119, 307–326 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  118. 118.
    Funaki, T., Yokoyama, S.: Sharp interface limit for stochastically perturbed mass conserving Allen-Cahn equation (2016, preprint)Google Scholar
  119. 119.
    Gage, M., Hamilton, R.S.: The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)MathSciNetzbMATHGoogle Scholar
  120. 120.
    Gärtner, J.: Bistable reaction-diffusion equations and excitable media. Math. Nachr. 112, 125–152 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  121. 121.
    Gärtner, J.: Convergence towards Burger’s equation and propagation of chaos for weakly asymmetric exclusion processes. Stoch. Process. Appl. 27, 233–260 (1988)zbMATHCrossRefGoogle Scholar
  122. 122.
    Giga, Y.: Surface Evolution Equations. A Level Set Approach. Monographs in Mathematics, vol. 99. Birkhäuser, Basel/Boston (2006)Google Scholar
  123. 123.
    Giga, Y., Mizoguchi, N.: Existence of periodic solutions for equations of evolving curves. SIAM J. Math. Anal. 27, 5–39 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edn. Springer, New York (1987)Google Scholar
  125. 125.
    Glimm, J., Jaffe, A., Spencer, T.: Phase transitions for φ 2 4 quantum fields. Commun. Math. Phys. 45, 203–216 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  126. 126.
    Glimm, J., Jaffe, A., Spencer, T.: Phase transitions in P(ϕ)2 quantum fields. Bull. Am. Math. Soc. 82, 713–715 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  127. 127.
    Gonçalves, P., Jara, M.: Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212, 597–644 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    Gonçalves, P., Jara, M., Sethuraman, S.: A stochastic Burgers equation from a class of microscopic interactions. Ann. Probab. 43, 286–338 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285–314 (1987)MathSciNetzbMATHGoogle Scholar
  130. 130.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3 (e6), 75 (2015)MathSciNetzbMATHGoogle Scholar
  131. 131.
    Gubinelli, M., Perkowski, N.: KPZ reloaded. arXiv:1508.03877Google Scholar
  132. 132.
    Gubinelli, M., Perkowski, N.: Energy solutions of KPZ are unique. arXiv:1508.07764Google Scholar
  133. 133.
    Gubinelli, M., Perkowski, N.: The Hairer-Quastel universality result in equilibrium. RIMS Kôkyûroku Bessatsu B59 (2016, to appear)Google Scholar
  134. 134.
    Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118, 31–59 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  135. 135.
    Gurtin, M.E.: Thermomechanics of Evolving Phase Boundaries in the Plane. Clarendon Press, Oxford (1993)zbMATHGoogle Scholar
  136. 136.
    Hadeler, K.P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251–263 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  137. 137.
    Hairer, M.: Ergodic Theory for Stochastic PDEs (2008). Available online at http://www.hairer.org/notes/Imperial.pdf
  138. 138.
    Hairer, M.: Solving the KPZ equation. Ann. Math. 178, 559–664 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  139. 139.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198, 269–504 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  140. 140.
    Hairer, M., Matetski, K.: Discretisations of rough stochastic PDEs. arXiv:1511.06937Google Scholar
  141. 141.
    Hairer, M., Mattingly, J.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164, 993–1032 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  142. 142.
    Hairer, M., Pardoux, E.: A Wong-Zakai theorem for stochastic PDEs. J. Math. Soc. Jpn. 67, 1551–1604 (2015) (special issue for Kiyosi Itô.)Google Scholar
  143. 143.
    Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ. arXiv:1512.07845Google Scholar
  144. 144.
    Hairer, M., Shen, H.: A central limit theorem for the KPZ equation. arXiv:1507.01237Google Scholar
  145. 145.
    Hofmanová, M., Röger, M., von Renesse, M.: Weak solutions for a stochastic mean curvature flow of two-dimensional graphs. Probab. Theory Relat. Fields (2016, published online)Google Scholar
  146. 146.
    Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–475 (1977)CrossRefGoogle Scholar
  147. 147.
    Hora, A.: A diffusive limit for the profiles of random Young diagrams by way of free probability. Publ. RIMS Kyoto Univ. 51, 691–708 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  148. 148.
    Hoshino, M.: KPZ equation with fractional derivatives of white noise. Stoch. Partial Differ. Equ. Anal. Comput. (2016, published online)Google Scholar
  149. 149.
    Hoshino, M.: Paracontrolled calculus and Funaki-Quastel approximation for KPZ equation. arXiv:1605.02624v2Google Scholar
  150. 150.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)MathSciNetzbMATHGoogle Scholar
  151. 151.
    Huisken, G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987)MathSciNetzbMATHGoogle Scholar
  152. 152.
    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990)MathSciNetzbMATHGoogle Scholar
  153. 153.
    Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)MathSciNetzbMATHGoogle Scholar
  154. 154.
    Ioffe, D. Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Stat. Phys. 74, 411–432 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  155. 155.
    Ioffe, D.: Exact large deviation bounds up to T c for the Ising model in two dimensions. Probab. Theory Relat. Fields 102, 313–330 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  156. 156.
    Ioffe, D., Schonmann, R.H.: Dobrushin-Kotecký-Shlosman theorem up to the critical temperature. Commun. Math. Phys. 199, 117–167 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  157. 157.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  158. 158.
    Kaimanovich, V.A.: Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators. Potential Anal. 1, 61–82 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  159. 159.
    Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)zbMATHCrossRefGoogle Scholar
  160. 160.
    Katsoulakis, M.A., Kossioris, G.T., Lakkis, O.: Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem. Interfaces Free Bound. 9, 1–30 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  161. 161.
    Kawasaki, K.: Non-equilibrium and Phase Transition–Statistical Physics in Mesoscopic Scale, in Japanese. Asakura (2000)Google Scholar
  162. 162.
    Kawasaki, K., Ohta, T.: Kinetic drumhead model of interface I. Prog. Theoret. Phys. 67, 147–163 (1982)CrossRefGoogle Scholar
  163. 163.
    Kawasaki, K., Ohta, T.: Kinetic drumhead models of interface. II. Prog. Theoret. Phys. 68, 129–147 (1982)CrossRefGoogle Scholar
  164. 164.
    Kenyon, R.: Height fluctuations in the honeycomb dimer model. Commun. Math. Phys. 281, 675–709 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  165. 165.
    Kenyon, R.: Lectures on dimers. In: Statistical Mechanics, pp. 191–230. I AS/Park City Mathematics Series, vol. 16. American Mathematical Society, Providence (2009)Google Scholar
  166. 166.
    Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. 163, 1019–1056 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  167. 167.
    Kerov, S.V.: Asymptotic representation theory of the symmetric group and its applications in analysis. Translations of Mathematics Monographs, vol. 219. American Mathematics Society, Providence (2003)Google Scholar
  168. 168.
    Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, New York (1999)zbMATHCrossRefGoogle Scholar
  169. 169.
    Kipnis, C., Olla, S., Varadhan, S.R.S.: Hydrodynamics and large deviation for simple exclusion processes. Commun. Pure Appl. Math. 42, 115–137 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  170. 170.
    Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Springer, Berlin/New York (2012)zbMATHCrossRefGoogle Scholar
  171. 171.
    Komorowski, T., Peszat, S., Szarek, T.: On ergodicity of some Markov processes. Ann. Probab. 38, 1401–1443 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  172. 172.
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge/New York (1990)zbMATHGoogle Scholar
  173. 173.
    Lacoin, H.: The scaling limit of polymer pinning dynamics and a one dimensional Stefan freezing problem. Commun. Math. Phys. 331, 21–66 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  174. 174.
    Landim, C., Yau, H.-T.: Large deviations of interacting particle systems in infinite volume. Commun. Pure Appl. Math. 48, 339–379 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  175. 175.
    Lee, K.: Generation and motion of interfaces in one-dimensional stochastic Allen-Cahn equation. arXiv:1511.05727Google Scholar
  176. 176.
    Lee, K.: Generation of interfaces for multi-dimensional stochastic Allen-Cahn equation with a noise smooth in space. arXiv:1604.06535Google Scholar
  177. 177.
    Lions, P.L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Ser. I Math. 326, 1085–1092 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  178. 178.
    Lions, P.L., Souganidis, P.E.: Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. C. R. Acad. Sci. Paris Ser. I Math. 327, 735–741 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  179. 179.
    Major, P.: Multiple Wiener-Itô Integrals, with Applications to Limit Theorems, 2nd edn. Lecture Notes Mathematics, vol. 849. Springer, Cham (2014)Google Scholar
  180. 180.
    Matano, H., Nakamura, K., Lou, B.: Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit. Netw. Heterog. Media 1, 537–568 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  181. 181.
    Miller, J.: Fluctuations for the Ginzburg-Landau ∇ϕ interface model on a bounded domain. Commun. Math. Phys. 308, 591–639 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  182. 182.
    Mogul’skii, A.A.: Large deviations for trajectories of multi-dimensional random walks. Theory Probab. Appl. 21, 300–315 (1976)MathSciNetCrossRefGoogle Scholar
  183. 183.
    Mourrat, J.-C., Weber, H.: Convergence of the two-dimensional dynamic Ising-Kac model to Φ 2 4. Commun. Pure Appl. Math. (2016, published online)Google Scholar
  184. 184.
    Mourrat,J.-C., Weber, H.: Global well-posedness of the dynamic Φ 4 model in the plane. Ann. Probab. (2016, published online)Google Scholar
  185. 185.
    Mourrat, J.-C., Weber, H.: Global well-posedness of the dynamic Φ 3 4 model on the torus. arXiv:1601.01234Google Scholar
  186. 186.
    de Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347, 1533–1589 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  187. 187.
    Mueller, C.: On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37, 225–245 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  188. 188.
    Naddaf, A., Spencer, T.: On homogenization and scaling limit of some gradient perturbations of a massless free field. Commun. Math. Phys. 183, 55–84 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  189. 189.
    Nagahata, Y.: A remark on equivalence of ensembles for surface diffusion model. RIMS Kôkyûroku Bessatsu B59 (2016, to appear)Google Scholar
  190. 190.
    Nagahata, Y.: Spectral gap for surface diffusion (2015, preprint)Google Scholar
  191. 191.
    Nualart, D., Pardoux, E.: White noise driven quasilinear SPDEs with reflection. Probab. Theory Relat. Fields 93, 77–89 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  192. 192.
    Petrov, V.V.: Sums of Independent Random Variables. Springer, Berlin/New York (1975)zbMATHCrossRefGoogle Scholar
  193. 193.
    Pimpinelli, A., Villain, J.: Physics of Crystal Growth. Cambridge University Press, Cambridge/New York (1998)CrossRefGoogle Scholar
  194. 194.
    Pittel, B.: On a likely shape of the random Ferrers diagram. Adv. Appl. Math. 18, 432–488 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  195. 195.
    Quastel, J.: Introduction to KPZ. In: Current Developments in Mathematics, vol. 2011, pp. 125–194. International Press, Somerville (2012)Google Scholar
  196. 196.
    Röger, M., Weber, H.: Tightness for a stochastic Allen-Cahn equation. Stoch. Partial Differ. Equ. Anal. Comput. 1, 175–203 (2013)MathSciNetzbMATHGoogle Scholar
  197. 197.
    Rybko, A., Shlosman, S., Vladimirov, A.: Spontaneous resonances and the coherent states of the queuing networks. J. Stat. Phys. 134, 67–104 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  198. 198.
    Sasamoto, T., Spohn, H.: Superdiffusivity of the 1D lattice Kardar-Parisi-Zhang equation. J. Stat. Phys. 137, 917–935 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  199. 199.
    Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834, 523–542 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  200. 200.
    Sasamoto, T., Spohn, H.: One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. Phys. Rev. Lett. 104, 230602 (2010)CrossRefGoogle Scholar
  201. 201.
    Shiga, T.: Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. 46, 415–437 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  202. 202.
    Soner, H.M.: Ginzburg-Landau equation and motion by mean curvature. I. Convergence. J. Geom. Anal. 7, 437–475 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  203. 203.
    Soner, H.M.: Ginzburg-Landau equation and motion by mean curvature. II. Development of the initial interface. J. Geom. Anal. 7, 477–491 (1997)MathSciNetzbMATHGoogle Scholar
  204. 204.
    Souganidis, P.E., Yip, N.K.: Uniqueness of motion by mean curvature perturbed by stochastic noise. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 1–23 (2004)MathSciNetzbMATHGoogle Scholar
  205. 205.
     Spohn, H.: Interface motion in models with stochastic dynamics. J. Stat. Phys. 71, 1081–1132 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  206. 206.
    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191–1227 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  207. 207.
    Spohn, H., Stoltz, G.: Nonlinear fluctuating hydrodynamics in one dimension: the case of two conserved fields. J. Stat. Phys. 160, 861–884 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  208. 208.
    Taylor, J., Cahn, J.W., Handwerker, C.A.: I-geometric models of crystal growth. Acta Metall. Meter. 40, 1443–1474 (1992)CrossRefGoogle Scholar
  209. 209.
    W. van Saarloos, Hohenberg, P.C.: Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations. Phys. D 56, 303–367 (1992)Google Scholar
  210. 210.
    van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003)zbMATHCrossRefGoogle Scholar
  211. 211.
    Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions – II. In: Elworthy and Ikeda (eds.) Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, pp. 75–128. Longman, Essex (1993)Google Scholar
  212. 212.
    Vershik, A.: Statistical mechanics of combinatorial partitions and their limit shapes. Funct. Anal. Appl. 30, 90–105 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  213. 213.
    Vershik, A., Yakubovich, Yu.: The limit shape and fluctuations of random partitions of naturals with fixed number of summands. Mosc. Math. J. 1, 457–468 (2001)MathSciNetzbMATHGoogle Scholar
  214. 214.
    Vershik, A., Yakubovich, Yu.: Fluctuations of the maximal particle energy of the quantum ideal gas and random partitions. Commun. Math. Phys. 261, 759–769 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  215. 215.
    H. Weber, Sharp interface limit for invariant measures of the stochastic Allen-Cahn equation, Commun. Pure Appl. Math., 63 (2010), 1071–1109.MathSciNetzbMATHGoogle Scholar
  216. 216.
    Weber, H.: On the short time asymptotic of the stochastic Allen-Cahn equation. Ann. Inst. H. Poincaré Probab. Statist. 46, 965–975 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  217. 217.
    Weber, S.: The sharp interface limit of the stochastic Allen-Cahn equation. PhD thesis, University of Warwick (2014)Google Scholar
  218. 218.
    Weiss, G.S.: A free boundary problem for non-radial-symmetric quasi-linear elliptic equations. Adv. Math. Sci. Appl. 5, 497–555 (1995)MathSciNetzbMATHGoogle Scholar
  219. 219.
    Wilson, D.B.: Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14, 274–325 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  220. 220.
    Wulff, G.: Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Krystallflächen. Z. Krystallogr. 34, 449–530 (1901)Google Scholar
  221. 221.
    Yakubovich, Yu.: Central limit theorem for random strict partitions. J. Math. Sci. 107, 4296–4304 (2001)MathSciNetCrossRefGoogle Scholar
  222. 222.
    Yip, N.K.: Stochastic motion by mean curvature. Arch. Ration. Mech. Anal. 144, 313–355 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  223. 223.
    Yip, N.K.: Stochastic curvature driven flows. In: Da Prato, G., Tubaro, L. (eds.) Stochastic Partial Differential Equations and Applications. Lecture Notes in Pure and Applied Mathematics, vol. 227, pp. 443–460. Marcel Dekker, New York (2002)Google Scholar
  224. 224.
    Zhu, R., Zhu, X.: Three-dimensional Navier-Stokes equations driven by space-time white noise. J. Differ. Equ. 259, 4443–4508 (2015)MathSciNetzbMATHCrossRefGoogle Scholar

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© The Author(s) 2016

Authors and Affiliations

  • Tadahisa Funaki
    • 1
  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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