Quasi-linear stochastic partial differential equations with irregular coefficients: Malliavin regularity of the solutions

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• 作者：Torstein Nilssen
• 关键词：SPDEs ; Malliavin calculus ; Irregular coefficients ; Regularizing effect of noise ; Local time calculus
• 刊名：Stochastic Partial Differential Equations: Analysis and Computations
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：3
• 期：3
• 页码：339-359
• 全文大小：463 KB
• 参考文献：1.Da Prato, G., Flandoli, F., Priola, E., R枚ckner, M.: Strong uniqueness for stochastic evolution equations in Hilbert spaces with unbounded measurable drift. Ann. Probab. 41(5), 3306鈥?344 (2013)MathSciNet View Article MATH
  2.Fernique, X.: Regularite des trajectoires des fonctions aleatoires gaussiennes. Ecole d鈥橢te de Probabilites de Saint-Flour IV-1974. Lecture Notes in Mathematics, vol. 480, pp. 1鈥?6. Springer, Berlin (1975)
  3.Flandoli, F., Nilssen, T., Proske, F.: Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces. University of Oslo series. https://鈥媤ww.鈥媎uo.鈥媢io.鈥媙o/鈥媓andle/鈥?0852/鈥?8087
  4.Geman, D., Horowitz, J.: Occupation densities. Ann. Probab. 8(1), 1鈥?7 (1980)MathSciNet View Article
  5.Gy枚ngy, I., Pardoux, E.: On quasi-linear stochastic partial differential equations. Probab. Theory Relat. Fields 94, 413鈥?25 (1993)View Article MATH
  6.Li, W., Wei, A.: A Gaussian inequality for expected absolute products. J. Theor. Probab. 25(1), 92鈥?9 (2012)MathSciNet View Article MATH
  7.Menoukeu-Pamen, O., Meyer-Brandis, T., Nilssen, T., Proske, F., Zhang, T.: A variational approach to the construction and Malliavin differentiability of strong solutions of SDE鈥檚. Math. Ann. 357(2), 761鈥?99 (2013)MathSciNet View Article MATH
  8.Nualart, D.: The Malliavin Calculus and Related Topics. Springer, Berlin (1995)View Article MATH
  9.Veretennikov, A.Y.: On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24, 354鈥?66 (1979)MathSciNet View Article
  10.Xiao, Y.: Sample Path Properties of Anisotropic Gaussian Random Fields. A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1962, pp. 145鈥?12. Springer, Berlin (2009)View Article
  11.Zvonkin, A.K.: A transformation of the state space of a diffusion process that removes the drift. Math. USSR (Sbornik) 22, 129鈥?49 (1974)MathSciNet View Article
  
• 作者单位：Torstein Nilssen (1)
  
  1. Department of Mathematics, University of Oslo, Moltke Moes vei 35, Blindern, P.O. Box 1053, 0316, Oslo, Norway
  
• 刊物主题：Probability Theory and Stochastic Processes; Partial Differential Equations; Statistical Theory and Methods; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Numerical Analysis;
• 出版者：Springer US
• ISSN：2194-041X

文摘

We study quasi-linear stochastic partial differential equations with discontinuous drift coefficients. Existence and uniqueness of a solution is already known under weaker conditions on the drift, but we are interested in the regularity of the solution in terms of Malliavin calculus. We prove that when the drift is bounded and measurable the solution is directional Malliavin differentiable.