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Nonlinear Stochastic Partial Differential Equations

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Abbreviations

Brownian motion:

A real Brownian motion B = (B(t)) t≥0 on (Ω, ℱ, ℙ) is a continuous real stochastic process in [0, + ∞) such that (1) B(0) = 0 and for any 0 ≤ s < t, B(t) − B(s) is a real Gaussian random variable with mean 0 and covariance t − s, and (2) if 0 < t 1 < ⋯ < t n , the random variables, B(t 1), B(t 2) − B(t 1), …, B(t n ) − B(t n−1) are independent.

Continuous stochastic process:

Let (Ω, ℱ, ℙ) be a probability space. A continuous stochastic process (with values in H) is a family of (H-valued) random variables (X(t) = X(t, ω)) t≥0 (ω∈Ω) such that X(⋅, ω) is continuous for ℙ – almost all ω∈Ω.

Cylindrical Wiener process:

A cylindrical Wiener process in a Hilbert space H is a process of the form

$$ W(t)={\displaystyle \sum_{k=1}^{\infty }{e}_k{\beta}_k(t),}\kern1em t\ge 0, $$

where (e k ) is a complete orthonormal system in H and (β k ) a sequence of mutually independent standard Brownian motions in a probability space (Ω, ℱ, ℙ).

Evolution equations:

Let H be a Hilbert (or Banach) space, T > 0, and f a mapping from [0, T] × H into H. An equation of the form

$$ {u}^{\prime }(t)=f\left(t,u(t)\right),\kern1em t\in \left[0,T\right],\kern2em \left(*\right) $$

is called an abstract evolution equation. If H is finite-dimensional, we call ( * ) an ordinary evolution equation, whereas if H is infinite-dimensional and f is a differential operator, we call ( * ) a partial differential evolution equation.

Noise:

Let W(t) be a cylindrical Wiener process in a Hilbert space H. A noise is an expression of the form BW(t), where B∈L(H). If B = I (the identity operator in H), the noise is called white.

Stochastic dynamical system:

This is a system governed by a partial differential evolution equation perturbed by noise.

Bibliography

  • Albeverio S, Cruizeiro AB (1990) Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids. Commun Math Phys 129:431–444

    Article  ADS  MATH  Google Scholar 

  • Albeverio S, Röckner M (1991) Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab Theory Relat Fields 89:347–386

    Article  MATH  Google Scholar 

  • Bally V, Gyongy I, Pardoux E (1994) White noise driven parabolic SPDEs with measurable drift. J Funct Anal 120(2):484–510

    Article  MATH  MathSciNet  Google Scholar 

  • Barbu V, Da Prato G (2002) The stochastic nonlinear damped wave equation. Appl Math Optim 46:125–141

    Article  MATH  MathSciNet  Google Scholar 

  • Barbu V, Bogachev VI, Da Prato G, Röckner M (2006) Weak solution to the stochastic porous medium equations: the degenerate case. J Funct Anal 235(2):430–448

    Article  MATH  MathSciNet  Google Scholar 

  • Barbu V, Da Prato G, Röckner M (2008) Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. preprint S.N.S. Indiana Univ Math J 57(1):187–212

    Article  MATH  MathSciNet  Google Scholar 

  • Bensoussan A, Temam R (1973) Équations stochastiques du type Navier-Stokes. J Funct Anal 13:195–222

    Article  MATH  MathSciNet  Google Scholar 

  • Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures, vol 5, Studies in mathematics and its applications. North-Holland, Amsterdam

    MATH  Google Scholar 

  • Bertini L, Cancrini N, Jona-Lasinio G (1994) The stochastic Burgers equation. Comm Math Phys 165(2):211–232

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Bismut JM (1984) Large deviations and the Malliavin calculus, vol 45, Progress in mathematics. Birkhäuser, Boston

    MATH  Google Scholar 

  • Bogachev VI, Da Prato G, Röckner M (2004) Existence of solutions to weak parabolic equations for measures. Proc Lond Math Soc 3(88):753–774

    Article  Google Scholar 

  • Bogachev VI, Krylov NV, Röckner M (2006) Elliptic equations for measures: regularity and global bounds of densities. J Math Pures Appl 9(6):743–757

    Article  Google Scholar 

  • Brzezniak Z, Capinski M, Flandoli F (1991) Stochastic partial differential equations and turbulence. Math Models Methods Appl Sci 1(1):41–59

    Article  MATH  MathSciNet  Google Scholar 

  • Capinski M, Cutland N (1994) Statistical solutions of stochastic Navier-Stokes equations. Indiana Univ Math J 43(3):927–940

    Article  MATH  MathSciNet  Google Scholar 

  • Cerrai S (2001) Second order PDE’s in finite and infinite dimensions: a probabilistic approach, vol 1762, Lecture notes in mathematics. Springer, Berlin

    Book  Google Scholar 

  • Cerrai S (2003) Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab Theory Relat Fields 125:271–304

    Article  MATH  MathSciNet  Google Scholar 

  • Cerrai S, Freidlin M (2006) On the Smoluchowski-Kramers approximation for a system with an infinite number of degrees of freedom. Probab Theory Relat Fields 135(3):363–394

    Article  MATH  MathSciNet  Google Scholar 

  • Crauel H, Debussche A, Flandoli F (1997) Random attractors. J Dynam Differen Equ 9:307–341

    Article  MATH  MathSciNet  Google Scholar 

  • Da Prato G (2004) Kolmogorov equations for stochastic PDEs. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  • Da Prato G, Debussche A (1996) Stochastic Cahn-Hilliard equation. Nonlinear Anal 26(2):241–263

    Article  MATH  MathSciNet  Google Scholar 

  • Da Prato G, Debussche A (2002) 2D Navier-Stokes equations driven by a space-time white noise. J Funct Anal 196(1):180–210

    Article  MATH  MathSciNet  Google Scholar 

  • Da Prato G, Debussche A (2003a) Strong solutions to the stochastic quantization equations. Ann Probab 31(4):1900–1916

    Article  MATH  MathSciNet  Google Scholar 

  • Da Prato G, Debussche A (2003b) Ergodicity for the 3D stochastic Navier-Stokes equations. J Math Pures Appl 82:877–947

    Article  MATH  MathSciNet  Google Scholar 

  • Da Prato G, Tubaro L (2000) A new method to prove self-adjointness of some infinite dimensional Dirichlet operator. Probab Theory Relat Fields 118(1):131–145

    Article  MATH  Google Scholar 

  • Da Prato G, Zabczyk J (1992) Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Da Prato G, Zabczyk J (1996) Ergodicity for infinite dimensional systems, vol 229, London Mathematical society lecture notes. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Da Prato G, Zabczyk J (2002) Second order partial differential equations in Hilbert spaces, vol 293, London Mathematical Society lecture notes. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Da Prato G, Debussche A, Temam R (1994) Stochastic Burgers equation. Nonlinear Differ Equ Appl 4:389–402

    Article  Google Scholar 

  • Da Prato G, Röckner M, Rozovskii BL, Wang FY (2006) Strong solutions of stochastic generalized porous media equations: existence, uniqueness and ergodicity. Comm Partial Differ Equ 31(1–3):277–291

    Article  MATH  Google Scholar 

  • Dalang R, Frangos N (1998) The stochastic wave equation in two spatial dimensions. Ann Probab 26(1):187–212

    Article  MATH  MathSciNet  Google Scholar 

  • Dawson DA (1975) Stochastic evolution equations and related measures processes. J Multivariate Anal 5:1–52

    Article  MATH  MathSciNet  Google Scholar 

  • de Bouard A, Debussche A (1999) A stochastic nonlinear Schrödinger equation with multiplicative noise. Comm Math Phys 205(1):161–181

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • de Bouard A, Debussche A (2002) On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation. Probab Theory Relat Fields 123(1):76–79

    Article  MATH  Google Scholar 

  • de Bouard A, Debussche A (2003) The stochastic nonlinear Schrödinger equation in H 1. Stoch Anal Appl 21(1):97–126

    Article  MATH  Google Scholar 

  • de Bouard A, Debussche A (2005) Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise. Ann Probab 33(3):1078–1110

    Article  MATH  MathSciNet  Google Scholar 

  • de Bouard A, Debussche A, Tsutsumi Y (2004–2005) Periodic solutions of the Korteweg-de Vries equation driven by white noise (electronic). SIAM J Math Anal 36(3):815–855

    Article  Google Scholar 

  • Debussche A, Printems J (2001) Effect of a localized random forcing term on the Korteweg-de Vries equation. J Comput Anal Appl 3(3):183–206

    MATH  MathSciNet  Google Scholar 

  • Debussche A, Zambotti L (2007) Conservative stochastic Cahn-Hilliard equation with reflection. Ann Probab 35(5):1706–1739

    Article  MATH  MathSciNet  Google Scholar 

  • Eckmann JP, Ruelle D (1985) Rev Mod Phys 53:643–653

    Article  ADS  Google Scholar 

  • Elworthy KD (1992) Stochastic flows on Riemannian manifolds. In: Pinsky MA, Wihstutz V (eds) Diffusion processes and related problems in analysis, vol II (Charlotte, NC, 1990), vol 27, Progr probab. Birkhäuser, Boston, pp 37–72

    Chapter  Google Scholar 

  • EW, Mattingly JC, Sinai YG (2001) Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Commun Math Phys 224:83–106

    Article  ADS  Google Scholar 

  • Faris WG, Jona-Lasinio G (1982) Large fluctuations for a nonlinear heat equation with noise. J Phys A 15:3025–3055

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Flandoli F (1994) Dissipativity and invariant measures for stochastic Navier-Stokes equations. Nonlinear Differ Equ Appl 1:403–423

    Article  MATH  MathSciNet  Google Scholar 

  • Flandoli F (1997) Irreducibilty of the 3D stochastic Navier-Stokes equation. J Funct Anal 149:160–177

    Article  MATH  MathSciNet  Google Scholar 

  • Flandoli F, Gątarek D (1995) Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab Theory Relat Fields 102:367–391

    Article  MATH  Google Scholar 

  • Flandoli F, Maslowski B (1995) Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Commun Math Phys 171:119–141

    Article  ADS  MathSciNet  Google Scholar 

  • Flandoli F, Romito M (2002) Partial regularity for the stochastic Navier-Stokes equations. Trans Am Math Soc 354(6):2207–2241

    Article  MATH  MathSciNet  Google Scholar 

  • Flandoli F, Romito M (2006) Markov selections and their regularity for the three-dimensional stochastic Navier-Stokes equations. C R Math Acad Sci Paris 343(1):47–50

    Article  MATH  MathSciNet  Google Scholar 

  • Fleming WH (1975) A selection-migration model in population genetics. J Math Biol 2(3):219–233

    Article  MATH  MathSciNet  Google Scholar 

  • Freidlin M (1996) Markov processes and differential equations: asymptotic problems, Lectures in mathematics ETH Zürich. Birkhäuser, Basel

    Book  MATH  Google Scholar 

  • Fuhrman M, Tessitore G (2002) Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control. Ann Probab 30(3):1397–1465

    Article  MATH  MathSciNet  Google Scholar 

  • Funaki T (1991) The reversible measures of multi-dimensional Ginzburg-Landau type continuum model. Osaka J Math 28(3):463–494

    MATH  MathSciNet  Google Scholar 

  • Funaki T, Olla S (2001) Fluctuations for ∇ϕ interface model on a wall. Stoch Process Appl 94(1):1–27

    Article  MATH  MathSciNet  Google Scholar 

  • Funaki T, Spohn H (1997) Motion by mean curvature from the Ginzburg-Landau ∇ϕ interface model. Comm Math Phys 185(1):1–36

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Hairer M, Mattingly JC (2006) Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann Math 3:993–1032

    Article  MathSciNet  Google Scholar 

  • Jona-Lasinio G, Mitter PK (1985) On the stochastic quantization of field theory. Commun Math Phys 101(3):409–436

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Krylov NV (1999) An analytic approach to SPDEs. In: Stochastic partial differential equations: six perspectives, vol 64, Mathematical surveys and monographs. American Mathematical Society, Providence, pp 185–242

    Chapter  Google Scholar 

  • Krylov NV, Rozovskii BL (1981) Stochastic evolution equations, Translated from Itogi Naukii Tekhniki, Seriya Sovremennye Problemy Matematiki 14:71–146. J Soviet Math 14:1233–1277

    Article  Google Scholar 

  • Kuksin S, Shirikyan A (2001) A coupling approach to randomly forced PDE’s I. Commun Math Phys 221:351–366

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Liskevich V, Röckner M (1998) Strong uniqueness for a class of infinite dimensional Dirichlet operators and application to stochastic quantization. Ann Scuola Norm Sup Pisa Cl Sci 4(XXVII):69–91

    Google Scholar 

  • Mattingly J (2002) Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Commun Math Phys 230(3):421–462

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Mikulevicius R, Rozovskii B (1998) Martingale problems for stochastic PDE’s. In: Carmona RA, Rozoskii B (eds) Stochastic partial differential equations: six perspectives, vol 64, Mathematical surveys and monograph. American Mathematical Society, Providence, pp 243–325

    Chapter  Google Scholar 

  • Millet A, Morien PL (2001) On a nonlinear stochastic wave equation in the plane: existence and uniqueness of the solution. Ann Appl Probab 11:922–951

    Article  MATH  MathSciNet  Google Scholar 

  • Millet A, Sanz-Solé M (2000) Approximation and support theorem for a wave equation in two space dimensions. Bernoulli 6:887–915

    Article  MATH  MathSciNet  Google Scholar 

  • Musiela M, Rutkowski M (1997) Martingale methods in financial modelling, vol 36, Applications of mathematics (New York). Springer, Berlin

    Book  MATH  Google Scholar 

  • Nualart D, Pardoux E (1992) White noise driven quasilinear SPDEs with reflection. Prob Theory Relat Fields 93:77–89

    Article  MATH  MathSciNet  Google Scholar 

  • Pardoux E (1975) Èquations aux derivées partielles stochastiques nonlinéaires monotones. Thèse, Université Paris XI

    Google Scholar 

  • Pardoux E, Peng SG (1990) Adapted solution of a backward stochastic differential equation. Syst Control Lett 14(1):55–61

    Article  MATH  MathSciNet  Google Scholar 

  • Parisi G, Wu YS (1981) Sci Sin 24:483–490

    MathSciNet  Google Scholar 

  • Peszat S, Zabczyk J (2000) Nonlinear stochastic wave and heat equations. Probab Theory Relat Fields 116(3):421–443

    Article  MATH  MathSciNet  Google Scholar 

  • Petersen K (1983) Ergodic theory. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Prevot C, Röckner M (2007) A concise course on stochastic partial differential equations, Monograph 2006, Lecture notes in mathematics. Springer, Berlin

    Google Scholar 

  • Priola E (1999) On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math 136:271–295

    MATH  MathSciNet  Google Scholar 

  • Ren J, Röckner M, Wang FY (2007) Stochastic generalized porous media and fast diffusions equations. J Diff Equ 238(1):118–152

    Article  ADS  MATH  Google Scholar 

  • Rozovskii BL (2003) Linear theory and applications to non-linear filtering (mathematics and its applications). Kluwer, Dordrecht

    Google Scholar 

  • Sanz-Solé M (2005) Malliavin calculus with applications to stochastic partial differential equations, Fundamental sciences. EPFL Press, Lausanne; distributed by CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  • Simon B (1974) The P(ϕ)2 Euclidean (quantum) field theory. Princeton University Press, Princeton

    Google Scholar 

  • Stannat W (2000) On the validity of the log-Sobolev inequality for symmetric Fleming-Viot operators. Ann Probab 28(2):667–684

    Article  MATH  MathSciNet  Google Scholar 

  • Temam R (1988) Infinite-dimensional dynamical systems in mechanics and physics. Springer, New York

    Book  MATH  Google Scholar 

  • Viot M (1976) Solution faibles d’équations aux dérivées partielles non linéaires. Thèse, Université Pierre et Marie Curie, Paris

    Google Scholar 

  • Walsh JB (1986) An introduction to stochastic partial differential equations. In: Cole d’t de probabilits de Saint-Flour, XIV-1984, vol 1180, Lecture notes in math. Springer, Berlin, pp 265–439

    Chapter  Google Scholar 

  • Zambotti L (2001) A reflected stochastic heat equation as a symmetric dynamic with respect to 3-d Bessel Bridge. J Funct Anal 180(1):195–209

    Article  MATH  MathSciNet  Google Scholar 

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  1. Scuola Normale Superiore, Pisa, Italy

    Giuseppe Da Prato

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    Robert A. Meyers

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Da Prato, G. (2013). Nonlinear Stochastic Partial Differential Equations. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_367-3

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