Advertisement

Pathwise Solutions for Fully Nonlinear First- and Second-Order Partial Differential Equations with Multiplicative Rough Time Dependence

  • Panagiotis E. SouganidisEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2253)

Abstract

The notes are an overview of the theory of pathwise weak solutions of two classes of scalar fully nonlinear first- and second-order degenerate parabolic partial differential equations with multiplicative rough time dependence, a special case being Brownian. These are Hamilton-Jacobi, Hamilton-Jacobi-Isaacs-Bellman and quasilinear divergence form equations including multidimensional scalar conservation laws. If the time dependence is “regular”, the weak solutions are respectively the viscosity and entropy/kinetic solutions. The main results are the well-posedness and qualitative properties of the solutions. Some concrete applications are also discussed.

Notes

Acknowledgements

I would like to thank Ben Seeger for his help in preparing these notes.

This work was partially supported by the National Science Foundation Grants DMS-1266383 and DMS-1600129, the Office for Naval Research grant N000141712095 and the Air Force Office for Scientific Research grant FA9550-18-1-0494.

References

  1. 1.
    M. Alfaro, D. Antonopoulou, G. Karali, H. Matano, Generation of fine transition layers and their dynamics for the stochastic Allen–Cahn equation (2018). e-prints arXiv:1812.03804Google Scholar
  2. 2.
    M. Bardi, I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, in Systems & Control: Foundations & Applications (Birkhäuser Boston, Inc., Boston, 1997). With appendices by M. Falcone and P. Soravia. https://doi.org/10.1007/978-0-8176-4755-1 zbMATHCrossRefGoogle Scholar
  3. 3.
    M. Bardi, M.G. Crandall, L.C. Evans, H.M. Soner, P.E. Souganidis, Viscosity Solutions and Applications. Lecture Notes in Mathematics, vol. 1660 (Springer/Centro Internazionale Matematico Estivo (C.I.M.E.), Berlin/Florence, 1997). Lectures given at the 2nd C.I.M.E. Session held in Montecatini Terme, June 12–20, 1995, Edited by I. Capuzzo Dolcetta and P. L. Lions, Fondazione C.I.M.E.. [C.I.M.E. Foundation].  https://doi.org/10.1007/BFb0094293
  4. 4.
    G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 17 (Springer, Paris, 1994)Google Scholar
  5. 5.
    G. Barles, P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4(3), 271–283 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    G. Barles, P.E. Souganidis, A new approach to front propagation problems: theory and applications. Arch. Ration. Mech. Anal. 141(3), 237–296 (1998). https://doi.org/10.1007/s002050050077 MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    G. Barles, H.M. Soner, P.E. Souganidis, Front propagation and phase field theory. SIAM J. Control Optim. 31(2), 439–469 (1993). https://doi.org/10.1137/0331021 MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    P. Billingsley, Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics (Wiley, New York, 1999). https://doi.org/10.1002/9780470316962 zbMATHGoogle Scholar
  9. 9.
    R. Buckdahn, J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II. Stoch. Process. Appl. 93(2), 205–228 (2001). https://doi.org/10.1016/S0304-4149(00)00092-2 MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    R. Buckdahn, J. Ma, Pathwise stochastic control problems and stochastic HJB equations. SIAM J. Control Optim. 45(6), 2224–2256 (2007, electronic). https://doi.org/10.1137/S036301290444335X MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    G.Q. Chen, Q. Ding, K.H. Karlsen, On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204(3), 707–743 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    I.D. Chueshov, P.A. Vuillermot, On the large-time dynamics of a class of parabolic equations subjected to homogeneous white noise: Stratonovitch’s case. C. R. Acad. Sci. Paris Sér. I Math. 323(1), 29–33 (1996)MathSciNetzbMATHGoogle Scholar
  13. 13.
    I.D. Chueshov, P.A. Vuillermot, On the large-time dynamics of a class of random parabolic equations. C. R. Acad. Sci. Paris Sér. I Math. 322(12), 1181–1186 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    M.G. Crandall, P.L. Lions, Hamilton-Jacobi equations in infinite dimensions. I. Uniqueness of viscosity solutions. J. Funct. Anal. 62(3), 379–396 (1985). https://doi.org/10.1016/0022-1236(85)90011-4 zbMATHCrossRefGoogle Scholar
  15. 15.
    M.G. Crandall, P.L. Lions, Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions. J. Funct. Anal. 65(3), 368–405 (1986). https://doi.org/10.1016/0022-1236(86)90026-1 zbMATHCrossRefGoogle Scholar
  16. 16.
    M.G. Crandall, P.L. Lions, P.E. Souganidis, Maximal solutions and universal bounds for some partial differential equations of evolution. Arch. Ration. Mech. Anal. 105(2), 163–190 (1989). https://doi.org/10.1007/BF00250835 MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992). https://doi.org/10.1090/S0273-0979-1992-00266-5 MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    G. Da Prato, M. Iannelli, L. Tubaro, Some results on linear stochastic differential equations in Hilbert spaces. Stochastics 6(2), 105–116 (1981/1982)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 325, 4th edn. (Springer, Berlin, 2016). https://doi.org/10.1007/978-3-662-49451-6 zbMATHCrossRefGoogle Scholar
  20. 20.
    A. Debussche, J. Vovelle, Long-time behavior in scalar conservation laws. Differ. Integr. Equ. 22(3-4), 225–238 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    A. Debussche, J. Vovelle, Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259(4), 1014–1042 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    A. Debussche, J. Vovelle, Invariant measure of scalar first-order conservation laws with stochastic forcing. Probab. Theory Relat. Fields 163(3–4), 575–611 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    N. Dirr, S. Luckhaus, M. Novaga, A stochastic selection principle in case of fattening for curvature flow. Calc. Var. Partial Differ. Equ. 13(4), 405–425 (2001). https://doi.org/10.1007/s005260100080 MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. R. Soc. Edinb. Sect. A 111(3–4), 359–375 (1989). https://doi.org/10.1017/S0308210500018631 MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    L.C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinb. Sect. A 120(3–4), 245–265 (1992). https://doi.org/10.1017/S0308210500032121 MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    L.C. Evans, P.E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J. 33(5), 773–797 (1984).  https://doi.org/10.1512/iumj.1984.33.33040 MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    L.C. Evans, H.M. Soner, P.E. Souganidis, Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45(9), 1097–1123 (1992).  https://doi.org/10.1002/cpa.3160450903 MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    J. Feng, D. Nualart, Stochastic scalar conservation laws. J. Funct. Anal. 255(2), 313–373 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    W.H. Fleming, H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Stochastic Modelling and Applied Probability, vol. 25, 2nd edn. (Springer, New York, 2006)Google Scholar
  30. 30.
    P.K. Friz, P. Gassiat, P.L. Lions, P.E. Souganidis, Eikonal equations and pathwise solutions to fully non-linear SPDEs. Stoch. Partial Differ. Equ. Anal. Comput. 5(2), 256–277 (2017). https://doi.org/10.1007/s40072-016-0087-9 MathSciNetzbMATHGoogle Scholar
  31. 31.
    T. Funaki, The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields 102(2), 221–288 (1995). https://doi.org/10.1007/BF01213390 MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    T. Funaki, Singular limit for stochastic reaction-diffusion equation and generation of random interfaces. Acta Math. Sin. (Engl. Ser.) 15(3), 407–438 (1999). https://doi.org/10.1007/BF02650735 MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    P. Gassiat, A stochastic Hamilton-Jacobi equation with infinite speed of propagation. C. R. Math. Acad. Sci. Paris 355(3), 296–298 (2017). https://doi.org/10.1016/j.crma.2017.01.021 MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    P. Gassiat, B. Gess, Regularization by noise for stochastic Hamilton-Jacobi equations. Probab. Theory Relat. Fields 173(3–4), 1063–1098 (2019). https://doi.org/10.1007/s00440-018-0848-7 MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    P. Gassiat, B. Gess, P.L. Lions, P.E. Souganidis, Speed of propagation for Hamilton-Jacobi equations with multiplicative rough time dependence and convex Hamiltonians (2019). ArXiv:1805.08477 [math.PR]Google Scholar
  36. 36.
    P. Gassiat, P.L. Lions, P.E. Souganidis, in preparationGoogle Scholar
  37. 37.
    M. Gerencsér, I. Gyöngy, N. Krylov, On the solvability of degenerate stochastic partial differential equations in Sobolev spaces. Stoch. Partial Differ. Equ. Anal. Comput. 3(1), 52–83 (2015). https://doi.org/10.1007/s40072-014-0042-6 MathSciNetzbMATHGoogle Scholar
  38. 38.
    B. Gess, P.E. Souganidis, Scalar conservation laws with multiple rough fluxes. Commun. Math. Sci. 13(6), 1569–1597 (2015).  https://doi.org/10.4310/CMS.2015.v13.n6.a10 MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    B. Gess, P.E. Souganidis, Long-time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws. Commun. Pure Appl. Math. 70(8), 1562–1597 (2017).  https://doi.org/10.1002/cpa.21646 MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    B. Gess, P.E. Souganidis, Stochastic non-isotropic degenerate parabolic-hyperbolic equations. Stoch. Process. Appl. 127(9), 2961–3004 (2017). https://doi.org/10.1016/j.spa.2017.01.005 MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    B. Gess, B. Perthame, P.E. Souganidis, Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes. SIAM J. Numer. Anal. 54(4), 2187–2209 (2016). https://doi.org/10.1137/15M1053670 MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    H. Hoel, K.H. Karlsen, N.H. Risebro, E.B. Storrø sten, Path-dependent convex conservation laws. J. Differ. Equ. 265(6), 2708–2744 (2018). https://doi.org/10.1016/j.jde.2018.04.045 MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    M. Hofmanová, Strong solutions of semilinear stochastic partial differential equations. Nonlinear Differ. Equ. Appl. 20(3), 757–778 (2013). https://doi.org/10.1007/s00030-012-0178-x MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    M. Hofmanová, Scalar conservation laws with rough flux and stochastic forcing. Stoch. Partial Differ. Equ. Anal. Comput. 4(3), 635–690 (2016). https://doi.org/10.1007/s40072-016-0072-3 MathSciNetzbMATHGoogle Scholar
  45. 45.
    H. Huang, H.J. Kushner, Weak convergence and approximations for partial differential equations with stochastic coefficients. Stochastics 15(3), 209–245 (1985). https://doi.org/10.1080/17442508508833357 MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets. Bull. Fac. Sci. Eng. Chuo Univ. 28, 33–77 (1985)MathSciNetzbMATHGoogle Scholar
  47. 47.
    S.N. Kružkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81(123), 228–255 (1970)Google Scholar
  48. 48.
    N.V. Krylov, On L p-theory of stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 27(2), 313–340 (1996). https://doi.org/10.1137/S0036141094263317 MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    N.V. Krylov, On the foundation of the L p-theory of stochastic partial differential equations, in Stochastic Partial Differential Equations and Applications—VII. Lecture Notes in Pure and Applied Mathematics, vol. 245 (Chapman & Hall/CRC, Boca Raton, 2006), pp. 179–191. https://doi.org/10.1201/9781420028720.ch16 Google Scholar
  50. 50.
    N.V. Krylov, M. Röckner, Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131(2), 154–196 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    H. Kunita, Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics, vol. 24 (Cambridge University Press, Cambridge, 1997). Reprint of the 1990 originalGoogle Scholar
  52. 52.
    J.M. Lasry, P.L. Lions, A remark on regularization in Hilbert spaces. Israel J. Math. 55(3), 257–266 (1986). https://doi.org/10.1007/BF02765025 MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    J.M. Lasry, P.L. Lions, Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)zbMATHCrossRefGoogle Scholar
  54. 54.
    J.M. Lasry, P.L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)zbMATHCrossRefGoogle Scholar
  55. 55.
    J.M. Lasry, P.L. Lions, Mean field games. Jpn. J. Math. 2(1), 229–260 (2007). https://doi.org/10.1007/s11537-007-0657-8 MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    A. Lejay, T.J. Lyons, On the importance of the Lévy area for studying the limits of functions of converging stochastic processes. Application to homogenization, in Current Trends in Potential Theory. Theta Series in Advanced Mathematics, vol. 4 (Theta, Bucharest, 2005), pp. 63–84Google Scholar
  57. 57.
    P.L. Lions, Mean field games. College de France courseGoogle Scholar
  58. 58.
    P.L. Lions, Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics, vol. 69 (Pitman (Advanced Publishing Program), Boston, 1982)Google Scholar
  59. 59.
    P.L. Lions, Some properties of the viscosity semigroups for Hamilton-Jacobi equations, in Nonlinear Differential Equations (Granada, 1984). Research Notes in Mathematics, vol. 132 (Pitman, Boston, 1985), pp. 43–63Google Scholar
  60. 60.
    P.L. Lions, Axiomatic derivation of image processing models. Math. Models Methods Appl. Sci. 4(4), 467–475 (1994). https://doi.org/10.1142/S0218202594000261 MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    P.L. Lions, B. Perthame, Remarks on Hamilton-Jacobi equations with measurable time-dependent Hamiltonians. Nonlinear Anal. 11(5), 613–621 (1987). https://doi.org/10.1016/0362-546X(87)90076-9 MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    P.L. Lions, B. Perthame, E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems. Commun. Math. Phys. 163(2), 415–431 (1994). http://projecteuclid.org/euclid.cmp/1104270470 MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    P.L. Lions, G. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations (1996). PreprintGoogle Scholar
  64. 64.
    P.L. Lions, B. Perthame, P.E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun. Pure Appl. Math. 49(6), 599–638 (1996). https://doi.org/10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5 MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    P.L. Lions, B. Perthame, P.E. Souganidis, Scalar conservation laws with rough (stochastic) fluxes. Stoch. Partial Differ. Equ. Anal. Comput. 1(4), 664–686 (2013)MathSciNetzbMATHGoogle Scholar
  66. 66.
    P.L. Lions, B. Perthame, P.E. Souganidis, Stochastic averaging lemmas for kinetic equations, in Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2011–2012, Sémin. Équ. Dériv. Partielles, pp. Exp. No. XXVI, 17. (École Polytech., Palaiseau, 2013)Google Scholar
  67. 67.
    P.L. Lions, B. Perthame, P.E. Souganidis, Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case. Stoch. Partial Differ. Equ. Anal. Comput. 2(4), 517–538 (2014). https://doi.org/10.1007/s40072-014-0038-2 MathSciNetzbMATHGoogle Scholar
  68. 68.
    P.L. Lions, B. Seeger, P.E. Souganidis, in preparationGoogle Scholar
  69. 69.
    P.L. Lions, P.E. Souganidis, The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation. PreprintGoogle Scholar
  70. 70.
    P.L. Lions, P.E. Souganidis, Ill-posedness of fronts moving with space-time white noise (in preparation)Google Scholar
  71. 71.
    P.L. Lions, P.E. Souganidis, Pathwise solutions for nonlinear partial differential equations with rough signals (in preparation)Google Scholar
  72. 72.
    P.L. Lions, P.E. Souganidis, Well posedness of pathwise solutions of fully nonlinear pde with multiple rough signals (in preparation)Google Scholar
  73. 73.
    P.L. Lions, P.E. Souganidis, Well posedness of pathwise solutions of Hamilton-Jacobi equations with convex Hamiltonians (in preparation)Google Scholar
  74. 74.
    P.L. Lions, P.E. Souganidis, Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 326(9), 1085–1092 (1998). https://doi.org/10.1016/S0764-4442(98)80067-0 MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    P.L. Lions, P.E. Souganidis, Fully nonlinear stochastic partial differential equations: non-smooth equations and applications. C. R. Acad. Sci. Paris Sér. I Math. 327(8), 735–741 (1998). https://doi.org/10.1016/S0764-4442(98)80161-4 MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    P.L. Lions, P.E. Souganidis, Équations aux dérivées partielles stochastiques nonlinéaires et solutions de viscosité, in Seminaire: Équations aux Dérivées Partielles, 1998–1999, Sémin. Équ. Dériv. Partielles, pp. Exp. No. I, 15 (École Polytech., Palaiseau, 1999)Google Scholar
  77. 77.
    P.L. Lions, P.E. Souganidis, Fully nonlinear stochastic pde with semilinear stochastic dependence. C. R. Acad. Sci. Paris Sér. I Math. 331(8), 617–624 (2000). https://doi.org/10.1016/S0764-4442(00)00583-8 MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    P.L. Lions, P.E. Souganidis, Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 331(10), 783–790 (2000). https://doi.org/10.1016/S0764-4442(00)01597-4 MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    P.L. Lions, P.E. Souganidis, Viscosity solutions of fully nonlinear stochastic partial differential equations. Sūrikaisekikenkyūsho Kōkyūroku 1287, 58–65 (2002). Viscosity solutions of differential equations and related topics (Japanese) (Kyoto, 2001)Google Scholar
  80. 80.
    P.L. Lions, P. Souganidis, New regularity results and long time behavior of pathwise (stochastic) Hamilton-Jacobi equations (2018). PreprintGoogle Scholar
  81. 81.
    T.J. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14(2), 215–310 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    T.J. Lyons, Z. Qian, Flow equations on spaces of rough paths. J. Funct. Anal. 149(1), 135–159 (1997).  https://doi.org/10.1006/jfan.1996.3088 MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    T.J. Lyons, Z. Qian, System Control and Rough Paths. Oxford Mathematical Monographs (Oxford University Press, Oxford, 2002)zbMATHCrossRefGoogle Scholar
  84. 84.
    T. Otha, D. Jasnow, K. Kawasaki, Universal scaling in the motion of random interfaces. Phys. Rev. Lett. 49, 1223–1226 (1982)CrossRefGoogle Scholar
  85. 85.
    E. Pardoux, Sur des équations aux dérivées partielles stochastiques monotones. C. R. Acad. Sci. Paris Sér. A-B 275, A101–A103 (1972)zbMATHGoogle Scholar
  86. 86.
    E. Pardoux, Équations aux dérivées partielles stochastiques de type monotone, in Séminaire sur les Équations aux Dérivées Partielles (1974–1975), III, Exp. No. 2 (Collège de France, Paris, 1975), p. 10Google Scholar
  87. 87.
    E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3(2), 127–167 (1979). https://doi.org/10.1080/17442507908833142 MathSciNetzbMATHGoogle Scholar
  88. 88.
    Y. Peres, Points of increase for random walks. Israel J. Math. 95, 341–347 (1996). https://doi.org/10.1007/BF02761045 MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    B. Perthame, Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure. J. Math. Pures Appl. (9) 77(10), 1055–1064 (1998). https://doi.org/10.1016/S0021-7824(99)80003-8 MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    B. Perthame, Kinetic Formulation of Conservation Laws. Oxford Lecture Series in Mathematics and its Applications, vol. 21 (Oxford University Press, Oxford, 2002)Google Scholar
  91. 91.
    B. Perthame, P.E. Souganidis, Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws. Arch. Ration. Mech. Anal. 170(4), 359–370 (2003). https://doi.org/10.1007/s00205-003-0282-5 MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    B. Perthame, E. Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws. Commun. Math. Phys. 136(3), 501–517 (1991). http://projecteuclid.org/euclid.cmp/1104202434 MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    F. Rezakhanlou, J.E. Tarver, Homogenization for stochastic Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 151(4), 277–309 (2000). https://doi.org/10.1007/s002050050198 MathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    B.L. Rozovskiı̆, Stochastic partial differential equations that arise in nonlinear filtering problems. Usp. Mat. Nauk 27(3(165)), 213–214 (1972)Google Scholar
  95. 95.
    B.L. Rozovskiı̆, Stochastic partial differential equations. Mat. Sb. (N.S.) 96(138), 314–341, 344 (1975)Google Scholar
  96. 96.
    B. Seeger, Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations (2018). ArXiv:1802.04740 [math.AP]Google Scholar
  97. 97.
    B. Seeger, Scaling limits and homogenization of stochastic Hamilton-Jacobi equations (in preparation)Google Scholar
  98. 98.
    B. Seeger, Homogenization of pathwise Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 110, 1–31 (2018). https://doi.org/10.1016/j.matpur.2017.07.012 MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    B. Seeger, Perron’s method for pathwise viscosity solutions. Commun. Partial Differ. Equ. 43(6), 998–1018 (2018). https://doi.org/10.1080/03605302.2018.1488262 MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    B. Seeger, Fully nonlinear stochastic partial differential equations, Thesis (Ph.D.), The University of Chicago, 2019Google Scholar
  101. 101.
    D. Serre, Systems of Conservation Laws 1 (Cambridge University Press, Cambridge, 1999). Hyperbolicity, entropies, shock waves. Translated from the 1996 French original by I. N. Sneddon.  https://doi.org/10.1017/CBO9780511612374
  102. 102.
    P.E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differ. Equ. 59(1), 1–43 (1985). https://doi.org/10.1016/0022-0396(85)90136-6 MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    P.E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications. Asymptot. Anal. 20(1), 1–11 (1999)MathSciNetzbMATHGoogle Scholar
  104. 104.
    P.E. Souganidis, N.K. Yip, Uniqueness of motion by mean curvature perturbed by stochastic noise. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(1), 1–23 (2004). https://doi.org/10.1016/S0294-1449(03)00029-5 MathSciNetzbMATHGoogle Scholar
  105. 105.
    H. Watanabe, On the convergence of partial differential equations of parabolic type with rapidly oscillating coefficients to stochastic partial differential equations. Appl. Math. Optim. 20(1), 81–96 (1989). https://doi.org/10.1007/BF01447648 MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    N.K. Yip, Stochastic motion by mean curvature. Arch. Ration. Mech. Anal. 144(4), 313–355 (1998). https://doi.org/10.1007/s002050050120 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

Personalised recommendations