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2013 | 11 | 11 | 1994-2004
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On mild solutions of gradient systems in Hilbert spaces

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We consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.
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Bibliografia
  • [1] Ball J.M., Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc., 1977, 63(2), 370–373
  • [2] Chojnowska-Michalik A., Transition Semigroups for Stochastic Semilinear Equations on Hilbert Spaces, Dissertationes Math. (Rozprawy Mat.), 396, Polish Academy of Sciences, Warsaw, 2001
  • [3] Da Prato G., Kolmogorov Equations for Stochastic PDEs, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2004 http://dx.doi.org/10.1007/978-3-0348-7909-5
  • [4] Da Prato G., An Introduction to Infinite-Dimensional Analysis, Universitext, Springer, Berlin, 2006
  • [5] Da Prato G., Tubaro L., Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization, Probab. Theory Related Fields, 2000, 118(1), 131–145
  • [6] Da Prato G., Zabczyk J., Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., 44, Cambridge University Press, Cambridge, 1992 http://dx.doi.org/10.1017/CBO9780511666223
  • [7] Da Prato G., Zabczyk J., Second Order Partial Differential Equations in Hilbert Spaces, London Math. Soc. Lecture Note Ser., 293, Cambridge University Press, Cambridge, 2002 http://dx.doi.org/10.1017/CBO9780511543210
  • [8] Fuhrman M., Tessitore G., Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 2002, 30(3), 1397–1465 http://dx.doi.org/10.1214/aop/1029867132
  • [9] Oharu S., Takahashi T., Characterization of nonlinear semigroups associated with semilinear evolution equations, Trans. Amer. Math. Soc., 1989, 311(2), 593–619 http://dx.doi.org/10.1090/S0002-9947-1989-0978369-9
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bwmeta1.element.doi-10_2478_s11533-013-0304-y
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