Electronic Journal of Probability

Stochastic evolution equations with multiplicative noise

Tijana Levajković, Stevan Pilipović, Dora Seleši, and Milica Žigić

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We study parabolic stochastic partial differential equations(SPDEs), driven by two types of operators: one linear closedoperator generating a $C_0-$semigroup and one linear boundedoperator with Wick-type multiplication, all of them set in theinfinite dimensional space framework of white noise analysis. Weprove existence and uniqueness of solutions for this class of SPDEs.In particular, we also treat the stationary case when thetime-derivative is equal to zero.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 19, 23 pp.

Accepted: 28 February 2015
First available in Project Euclid: 4 June 2016

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Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60H40: White noise theory 60G20: Generalized stochastic processes 60H07: Stochastic calculus of variations and the Malliavin calculus 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 46N30: Applications in probability theory and statistics

generalized stochastic process chaos expansion stochastic evolution equation Wick product white noise $C_0-$semigroup infinitesimal generator stochastic operator

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Levajković, Tijana; Pilipović, Stevan; Seleši, Dora; Žigić, Milica. Stochastic evolution equations with multiplicative noise. Electron. J. Probab. 20 (2015), paper no. 19, 23 pp. doi:10.1214/EJP.v20-3696. https://projecteuclid.org/euclid.ejp/1465067125

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  • Applebaum, David. On the infinitesimal generators of Ornstein-Uhlenbeck processes with jumps in Hilbert space. Potential Anal. 26 (2007), no. 1, 79–100.
  • Bogachev, Vladimir I. Differentiable measures and the Malliavin calculus. Mathematical Surveys and Monographs, 164. American Mathematical Society, Providence, RI, 2010. xvi+488 pp. ISBN: 978-0-8218-4993-4
  • Catuogno, Pedro; Olivera, Christian. On stochastic generalized functions. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (2011), no. 2, 237–260.
  • Engel, Klaus-Jochen; Nagel, Rainer. One-parameter semigroups for linear evolution equations. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. xxii+586 pp. ISBN: 0-387-98463-1
  • Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. phSpringer-Verlag, Berlin, 1998.
  • Hida, Takeyuki; Kuo, Hui-Hsiung; Potthoff, Jürgen; Streit, Ludwig. White noise. An infinite-dimensional calculus. Mathematics and its Applications, 253. Kluwer Academic Publishers Group, Dordrecht, 1993. xiv+516 pp. ISBN: 0-7923-2233-9
  • Holden, Helge; Øksendal, Bernt; Ubøe, Jan; Zhang, Tusheng. Stochastic partial differential equations. A modeling, white noise functional approach. Second edition. Universitext. Springer, New York, 2010. xvi+305 pp. ISBN: 978-0-387-89487-4
  • Hu, Yaozhong. Chaos expansion of heat equations with white noise potentials. Potential Anal. 16 (2002), no. 1, 45–66.
  • Kalpinelli, E. A.; Frangos, N. E.; Yannacopoulos, A. N. A Wiener chaos approach to hyperbolic SPDEs. Stoch. Anal. Appl. 29 (2011), no. 2, 237–258.
  • Levajkovi' c, T., Pilipovi'c, S., Sele si, D.: Fundamental equations with higher order Malliavin operators. phStochastics: An International Journal of Probability and Stochastic Processes, accepted for publication.
  • Levajković, Tijana; Pilipović, Stevan; Selesi, Dora. The stochastic Dirichlet problem driven by the Ornstein-Uhlenbeck operator: approach by the Fredholm alternative for chaos expansions. Stoch. Anal. Appl. 29 (2011), no. 2, 317–331.
  • Lototsky, Sergey V.; Rozovskii, Boris L. Stochastic partial differential equations driven by purely spatial noise. SIAM J. Math. Anal. 41 (2009), no. 4, 1295–1322.
  • Lototsky, Sergey V.; Rozovskii, Boris L. Bilinear stochastic elliptic equations. Stochastic partial differential equations and applications, 207–221, Quad. Mat., 25, Dept. Math., Seconda Univ. Napoli, Caserta, 2010.
  • Alshanskiy, M. A.; Melnikova, I. V. Regularized and generalized solutions of infinite-dimensional stochastic problems. (Russian) Mat. Sb. 202 (2011), no. 11, 3–30; translation in Sb. Math. 202 (2011), no. 11-12, 1565–1592
  • Melnikova, I. V.; Alshanskiy, M. A. Generalized solutions of abstract stochastic problems. Pseudo-differential operators, generalized functions and asymptotics, 341–352, Oper. Theory Adv. Appl., 231, Birkh�user/Springer Basel AG, Basel, 2013.
  • Pazy, A. Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp. ISBN: 0-387-90845-5
  • Pilipović, Stevan; Seleši, Dora. Expansion theorems for generalized random processes, Wick products and applications to stochastic differential equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), no. 1, 79–110.
  • Pilipović, Stevan; Seleši, Dora. On the generalized stochastic Dirichlet problem. I. The stochastic weak maximum principle. Potential Anal. 32 (2010), no. 4, 363–387.
  • Pilipović, Stevan; Seleši, Dora. On the generalized stochastic Dirichlet problem-Part II: solvability, stability and the Colombeau case. Potential Anal. 33 (2010), no. 3, 263–289.
  • Proske, Frank. The stochastic transport equation driven by Lévy white noise. Commun. Math. Sci. 2 (2004), no. 4, 627–641.
  • Seleši, Dora. Fundamental solutions of singular SPDEs. Chaos Solitons Fractals 44 (2011), no. 7, 526–537.