Stochastic Partial Differential Equations Driven by Purely Spatial Noise

We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Even the simplest equations driven by this noise often do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron–Martin version of the Wiener chaos decomposition is an effective tool to study both stationary and evolution equations driven by space-only noise. The paper presents results about solvability of such equations in weighted Wiener chaos spaces and studies the long-time behavior of the solutions of evolution equations with space-only noise.

[1]  T. G. Theting,et al.  Solving wick-stochastic boundary value problems using a finite element method , 2000 .

[2]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[3]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[4]  K. Gawȩdzki,et al.  University in turbulence: An exactly solvable model , 1995, chao-dyn/9504002.

[5]  Stig Larsson,et al.  Introduction to stochastic partial differential equations , 2008 .

[6]  Kiyosi Itô.  Multiple Wiener Integral , 1951 .

[7]  Etienne Pardoux,et al.  Stochastic partial differential equations, a review , 1993 .

[8]  Weighted Stochastic Sobolev Spaces and Bilinear Spde's Driven by Space-time White Noise Proposed Running Head: Weighted Stochastic Sobolev Spaces , 2007 .

[9]  Massimo Vergassola,et al.  Phase transition in the passive scalar advection , 1998 .

[10]  Gjermund Våge,et al.  Variational methods for PDEs aplied to stochastic partial differential equations , 1998 .

[11]  Boris Rozovskii,et al.  Passive scalar equation in a turbulent incompressible Gaussian velocity field , 2004 .

[12]  Hui-Hsiung Kuo.  White Noise Distribution Theory (Probability and Stochastics Series) , 1996 .

[13]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

[14]  D. Nualart,et al.  Quasilinear stochastic elliptic equations with reflection , 1995 .

[15]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[16]  É. Pardoux,et al.  Équations aux dérivées partielles stochastiques non linéaires monotones : étude de solutions fortes de type Ito , 1975 .

[17]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

[18]  D. Nualart.  Application of Malliavin Calculus to Stochastic Partial Differential Equations , 2009 .

[19]  Yuri Kabanov,et al.  From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift , 2006 .

[20]  B. Rozovskii,et al.  Linear Parabolic Stochastic PDE ' s and Wiener , 2007 .

[21]  L. Streit,et al.  Generalized Functionals in Gaussian Spaces: The Characterization Theorem Revisited☆ , 1996, math/0303054.

[22]  Quasi-Linear Elliptic Stochastic Partial Differential Equation: Markov property , 1992 .

[23]  B. Rozovskii.  Stochastic Evolution Systems , 1990 .

[24]  René Carmona,et al.  Stochastic Partial Differential Equations: Six Perspectives , 1998 .

[25]  B. Rozovskii,et al.  Stochastic Differential Equations: A Wiener Chaos Approach , 2005, math/0504559.

[26]  S. Dierential.  Applications of Malliavin Calculus to Stochastic Partial Dierential Equations , 2008 .

[27]  E. Pardoux,et al.  Équations aux dérivées partielles stochastiques de type monotone , 1975 .

[28]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[29]  The spaces of trial and generalized functions of infinite number of variables , 1978 .

[30]  B. Rozovskii,et al.  Wiener chaos solutions of linear stochastic evolution equations , 2005, math/0504558.

[31]  B. Rozovskii,et al.  Stochastic evolution equations , 1981 .

[32]  R. Carmona,et al.  Parabolic Anderson Problem and Intermittency , 1994 .

[33]  An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations , 1999, math/9903042.

[34]  Yaozhong Hu,et al.  Chaos Expansion of Heat Equations with White Noise Potentials , 2002 .

[35]  Hui-Hsiung Kuo,et al.  White noise distribution theory , 1996 .

[36]  Gerd Grubb,et al.  PROBLÉMES AUX LIMITES NON HOMOGÉNES ET APPLICATIONS , 1969 .

[37]  N. Krylov.  Introduction to the theory of diffusion processes , 1994 .

[38]  M. Sanz-Solé,et al.  A Lattice Scheme for Stochastic Partial Differential Equations of Elliptic Type in Dimension d ≥ 4 , 2005 .

[39]  Thomas Y. Hou,et al.  Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics , 2006, J. Comput. Phys..