Acta Mathematica Scientia

, Volume 39, Issue 3, pp 874–914 | Cite as

Some Recent Progress on Stochastic Heat Equations

  • Yaozhong HuEmail author


This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution; Feynman-Kac formula for the moments of the solution; and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.

Key words

random field Gaussian noise stochastic partial differential equation (stochastic heat equation) Feynman-Kac formula for the solution Feynman-Kac formula for the moments of the solution chaos expansion hypercontractivity moment bounds Hölder continuity joint Hölder continuity asymptotic behaviour Trotter-Lie formula Skorohod integral 

2010 MR Subject Classification

60G15 60G22 60H05 60H07 60H10 60H15 28C20 35K15 35R60 


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  1. [1]
    Alberts T, Khanin K, Quastel J. The continuum directed random polymer. J Stat Phys, 2014, 1541/2: 305–326MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Amir G, Corwin I, Quastel J. Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Comm Pure Appl Math, 2011, 644: 466–537MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Balan R, Chen L. Parabolic Anderson Model with space-time homogeneous Gaussian noise and rough initial condition. Journal of Theoretical Probability, 2018, 31: 2216–2265MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Balan R, Jolis M, Quer-Sardanyons L. SPDEs with fractional noise in space with index H < 1/2. Electron J Probab, 2015, 20(54): 36 ppGoogle Scholar
  5. [5]
    Balan R, Quer-Sardanyons L, Song J. Hölder continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise. Acta Mathematica Scientia, 2019, 39B3: 717–730. See also arXiv: 1807.05420Google Scholar
  6. [6]
    Bertini L, Cancrini N. The stochastic heat equation: Feynman- Kac formula and intermittence. J Statist Phys, 1995, 785/6: 1377–1401MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Bezerra S, Tindel S, Viens F. Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann Probab, 2008, 365: 1642–1675MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Biagini F, Hu Y, Øksendal B, Zhang T. Stochastic calculus for fractional Brownian motion and applications//Probability and its Applications (New York). London: Springer-Verlag London, Ltd, 2008Google Scholar
  9. [9]
    Carmona R, Lacroix J. Spectral Theory of Random Schrödinger Operators//Probability and its Applications. Boston, MA: Birkhauser Boston, Inc, 1990Google Scholar
  10. [10]
    Carmona R A, Molchanov S A. Parabolic Anderson problem and intermittency. Mem Amer Math Soc, 1994, 108(518): viii+125MathSciNetzbMATHGoogle Scholar
  11. [11]
    Chen L, Dalang R C. Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions. Annals of Probability, 2015, 43: 3006–3051MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Chen L, Dalang R C. Holder-continuity for the nonlinear stochastic heat equation with rough initial conditions. Stoch Partial Differ Equ Anal Comput, 2014, 23: 316–352MathSciNetzbMATHGoogle Scholar
  13. [13]
    Chen L, Hu G, Hu Y, Huang J. Space-time fractional diffusions in Gaussian noisy environment. Stochastics, 2017, 891: 171–206MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Chen L, Hu Y, Kalbasi K, Nualart D. Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise. Probab Theory Related Fields, 2018, 1711/2: 431–457MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Chen L, Hu Y, Nualart D. Two-point correlation function and Feynman-Kac formula for the stochastic heat equation. Potential Anal, 2017, 464: 779–797MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Chen L, Hu Y, Nualart D. Regularity and strict positivity of densities for the nonlinear stochastic heat equation. Memoirs of American Mathematical Society, 2018 (to appear). See also arXiv:1611.03909Google Scholar
  17. [17]
    Chen L, Hu Y, Nualart D. Nonlinear stochastic time-fractional slow and fast diffusion equations on ∝d. Revised for Stochastic Processes and ApplGoogle Scholar
  18. [18]
    Chen L, Huang J. Comparison principle for stochastic heat equation on Rd. Annals of Probability, 2018, to appearGoogle Scholar
  19. [19]
    Chen L, Kim K. Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency. Acta Mathematica Scientia, 2019, 39B3: 645–668Google Scholar
  20. [20]
    Chen X. Quenched asymptotics for Brownian motion in generalized Gaussian potential. Ann Probab, 2014, 422: 576–622MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Chen X. Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise. Ann Probab, 2016, 442: 1535–1598MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Chen X. Moment asymptotics for parabolic Anderson equation with fractional time-space noise: in Skorokhod regime. Ann Inst Henri Poincar Probab Stat, 2017, 532: 819–841MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Chen X, Hu Y, Nualart D, Tindel S. Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise. Electron J Probab, 2017, 22(65): 38 ppGoogle Scholar
  24. [24]
    Chen X, Hu Y, Song J, Xing F. Exponential asymptotics for time-space Hamiltonians. Ann Inst Henri Poincar Probab Stat, 2015, 514: 1529–1561MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Chen X, Phan T V. Free energy in a mean field of Brownian particles. PreprintGoogle Scholar
  26. [26]
    Chernoff P R. Note on product formulas for operator semigroups. J Funct Anal, 1968, 2: 238–242MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Chernoff P R. Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators//Memoirs of the American Mathematical Society, No. 140. Providence, RI: American Mathematical Society, 1974Google Scholar
  28. [28]
    Conus D, Joseph M, Khoshnevisan D, Shiu S -Y. Initial measures for the stochastic heat equation. Ann Inst Henri Poincar Probab Stat, 2014, 50(1): 136–153MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Dalang R. Extending Martingale Measure Stochastic Integral with Applications to Spatially Homogeneous S.P.D.E’s. Electron J Probab, 1999, 4(6)Google Scholar
  30. [30]
    Duncan T E, Hu Y, Pasik-Duncan B. Stochastic calculus for fractional Brownian motion. I. Theory. SIAM J Control Optim, 2000, 382: 582–612MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Gärtner J, Molchanov S A. Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm Math Phys, 1990, 1323: 613–655MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Gärtner J, Molchanov S A. Parabolic problems for the Anderson model. II. Second-order asymptotics and structure of high peaks. Probab Theory Related Fields, 1998, 1111: 17–55MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Hairer M. Solving the KPZ equation. Ann of Math, 2013, 1782: 559–664MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Hille E, Phillips R S. Functional analysis and semi-groups. Third printing of the revised edition of 1957//American Mathematical Society Colloquium Publications, Vol XXXI. Providence, RI: American Mathematical Society, 1974Google Scholar
  35. [35]
    Hu Y. Integral transformations and anticipative calculus for fractional Brownian motions. Mem Amer Math Soc, 2005, 175(825)Google Scholar
  36. [36]
    Hu Y. Analysis on Gaussian space. Singapore: World Scientific, 2017Google Scholar
  37. [37]
    Hu Y. Heat equation with fractional white noise potentials. Appl Math Optim, 2001, 43: 221–243MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    Hu Y. Chaos expansion of heat equations with white noise potentials. Potential Anal, 2002, 161: 45–66MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    Hu Y. A class of SPDE driven by fractional white noise Leipzig. Stochastic processes, physics and geometry: new interplays, II. 1999: 317–325; CMS Conf Proc, 29. Providence, RI: Amer Math Soc, 2000Google Scholar
  40. [40]
    Hu Y. Schrödinger equation with Gaussian potential (To appear)Google Scholar
  41. [41]
    Hu Y, Huang J, Nualart D, Tindel S. Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron J Probab, 2015, 20(55)Google Scholar
  42. [42]
    Hu Y, Huang J, Le K, Nualart D, Tindel S. Stochastic heat equation with rough dependence in space. Ann Probab, 2017, 45B6: 4561–4616MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    Hu Y, Huang J, Le K, Nualart D, Tindel S. Parabolic Anderson model with rough dependence in space (To appear in Abel Proceedings)Google Scholar
  44. [44]
    Hu Y, Lê K. A multiparameter Garsia-Rodemich-Rumsey inequality and some applications. Stochastic Process Appl, 2013, 1239: 3359–3377MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    Hu Y, Lê K. Nonlinear Young integrals and differential systems in Hölder media. Trans Amer Math Soc, 2017, 3693: 1935–2002MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    Hu Y, Lê K. Joint Holder continuity of parabolic Anderson model. Acta Mathematics Scientia, 2019, 39B3: 764–780Google Scholar
  47. [47]
    Hu Y, Liu Y, Tindel S. On the necessary and sufficient conditions to solve a heat equation with general Additive Gaussian noise. Acta Mathematics Scientia, 2019, 39B(3): 669–690Google Scholar
  48. [48]
    Hu Y, Lu F, Nualart D. Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2. Ann Probab, 2012, 403: 1041–1068MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    Hu Y, Nualart D. Stochastic heat equation driven by fractional noise and local time. Probab Theory Related Fields, 2009, 1431/2: 285–228MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    Hu Y, Nualart D, Song J. Feynman-Kac formula for heat equation driven by fractional white noise. Ann. Probab, 2011, 391: 291–326MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    Hu Y, Nualart D, Zhang T. Large deviations for stochastic heat equation with rough dependence in space. Bernoulli, 2018, 241: 354–385MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    Hu Y, Øksendal B, Zhang T. General fractional multiparameter white noise theory and stochastic partial differential equations. Comm Partial Differential Equations, 2004, 29(1/2): 123MathSciNetzbMATHGoogle Scholar
  53. [53]
    Hu Y, Yan J A. Wick calculus for nonlinear Gaussian functionals. Acta Math Appl Sin Engl Ser, 2009, 253: 399–414MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    Ikeda N, Watanabe S. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. Amsterdam: North-Holland Publishing Co; Tokyo: Kodansha, Ltd, 1989zbMATHGoogle Scholar
  55. [55]
    Johnson G W, Lapidus M L. The Feynman integral and Feynman’s operational calculus. Oxford Mathematical Monographs. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press, 2000Google Scholar
  56. [56]
    Kato T. Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups//Gohberg I, Kac M. Topics in Functional Analysis. London: Academic Press, 1978Google Scholar
  57. [57]
    Kilbas A A, Srivastava H M, Trujillo J J. Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Amsterdam: Elsevier Science BV, 2006Google Scholar
  58. [58]
    Khoshnevisan D. Analysis of stochastic partial differential equations//CBMS Regional Conference Series in Mathematics, 119. Published for the Conference Board of the Mathematical Sciences, Washington, DC. Providence, RI: the American Mathematical Society, 2014: viii+116 ppGoogle Scholar
  59. [59]
    König W. The parabolic Anderson model. Random walk in random potential. Pathways in Mathematics. Birkhauser/Springer, [Cham], 2016Google Scholar
  60. [60]
    Memin J, Mishura Y, Valkeila E. Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist Probab Lett, 2001, 51: 197–206MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    Mueller C. Long-time existence for the heat equation with a noise term. Prob. Theory Rel Fields, 1991, 9: 505–517MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    Nualart D. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Berlin: Springer-Verlag, 2006Google Scholar
  63. [63]
    Peszat S, Zabczyk J. Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process Appl, 1997, 722: 187–204MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    Rhandi A. Dyson-Phillips expansion and unbounded perturbations of linear C0-semigroups. J Comput Appl Math, 1992, 44: 339–349MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    Trotter H F. On the product of semi-groups of operators. Proc Amer Math Soc, 1959, 10: 545–551MathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    Vuillermot P -A. A generalization of Chernoff’s product formula for time-dependent operators. J Funct Anal, 2010, 259(11): 2923–2938MathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    Vuillermot P -A, Wreszinski W F, Zagrebnov V A. A general Trotter-Kato formula for a class of evolution operators. J Funct Anal, 2009, 257: 2246–2290MathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    Walsh B. An introduction to Stochastic Partial Differential Equations//Lecture Notes in Mathematics 1180. Springer-Verlag, 1986: 265–439Google Scholar
  69. [69]
    Yosida K. Functional analysis. Reprint of the sixth edition (1980)//Classics in Mathematics. Berlin: Springer-Verlag, 1995Google Scholar

Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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