The Annals of Probability

Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension

Giuseppe Da Prato and Alessandra Lunardi

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Abstract

We consider an elliptic Kolmogorov equation $\lambda u-Ku=f$ in a separable Hilbert space $H$. The Kolmogorov operator $K$ is associated to an infinite dimensional convex gradient system: $dX=(AX-DU(X))\,dt+dW(t)$, where $A$ is a self-adjoint operator in $H$, and $U$ is a convex lower semicontinuous function. Under mild assumptions we prove that for $\lambda>0$ and $f\in L^{2}(H,\nu)$ the weak solution $u$ belongs to the Sobolev space $W^{2,2}(H,\nu)$, where $\nu$ is the log-concave probability measure of the system. Moreover maximal estimates on the gradient of $u$ are proved. The maximal regularity results are used in the study of perturbed nongradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction–diffusion and Cahn–Hilliard stochastic PDEs.

Article information

Source
Ann. Probab., Volume 42, Number 5 (2014), 2113-2160.

Dates
First available in Project Euclid: 29 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1409319474

Digital Object Identifier
doi:10.1214/14-AOP936

Mathematical Reviews number (MathSciNet)
MR3262499

Zentralblatt MATH identifier
1328.35291

Subjects
Primary: 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 37L40: Invariant measures 35B65: Smoothness and regularity of solutions

Keywords
Kolmogorov operators in infinite dimensions maximal Sobolev regularity invariant measures

Citation

Da Prato, Giuseppe; Lunardi, Alessandra. Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension. Ann. Probab. 42 (2014), no. 5, 2113--2160. doi:10.1214/14-AOP936. https://projecteuclid.org/euclid.aop/1409319474


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References

  • [1] Albeverio, S. and Röckner, M. (1991). Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probab. Theory Related Fields 89 347–386.
  • [2] Barbu, V. (2010). Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York.
  • [3] Bogachev, V. I. (1998). Gaussian Measures. Amer. Math. Soc., Providence, RI.
  • [4] Brézis, H. (1973). Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam.
  • [5] Chojnowska-Michalik, A. and Gołdys, B. (1995). Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces. Probab. Theory Related Fields 102 331–356.
  • [6] Davies, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge.
  • [7] Da Prato, G. (2004). Kolmogorov Equations for Stochastic PDEs. Birkhäuser, Basel.
  • [8] Da Prato, G. (2006). An Introduction to Infinite-Dimensional Analysis. Springer, Berlin.
  • [9] Da Prato, G. and Debussche, A. (1996). Stochastic Cahn–Hilliard equation. Nonlinear Anal. 26 241–263.
  • [10] Da Prato, G., Debussche, A. and Goldys, B. (2002). Some properties of invariant measures of nonsymmetric dissipative stochastic systems. Probab. Theory Related Fields 123 355–380.
  • [11] Da Prato, G., Flandoli, F., Priola, E. and Röckner, M. (2013). Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 3306–3344.
  • [12] Da Prato, G. and Röckner, M. (2002). Singular dissipative stochastic equations in Hilbert spaces. Probab. Theory Related Fields 124 261–303.
  • [13] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [14] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge.
  • [15] Da Prato, G. and Zabczyk, J. (2002). Second Order Partial Differential Equations in Hilbert Spaces. London Mathematical Society Lecture Notes 293. Cambridge Univ. Press, Cambridge.
  • [16] Elezović, N. and Mikelić, A. (1991). On the stochastic Cahn–Hilliard equation. Nonlinear Anal. 16 1169–1200.
  • [17] Haussmann, U. G. and Pardoux, É. (1986). Time reversal of diffusions. Ann. Probab. 14 1188–1205.
  • [18] Jona-Lasinio, G. and Sénéor, R. (1991). On a class of stochastic reaction–diffusion equations in two space dimensions. J. Phys. A 24 4123–4128.
  • [19] Lunardi, A., Metafune, G. and Pallara, D. (2005). Dirichlet boundary conditions for elliptic operators with unbounded drift. Proc. Amer. Math. Soc. 133 2625–2635.
  • [20] Lunardi, A. and Vespri, V. (1998). Optimal $L^{\infty}$ and Schauder estimates for elliptic and parabolic operators with unbounded coefficients. In Reaction Diffusion Systems (Trieste, 1995) (G. Caristi and E. Mitidieri, eds.). Lecture Notes in Pure and Applied Mathematics 194 217–239. Dekker, New York.
  • [21] Phelps, R. R. (1978). Gaussian null sets and differentiability of Lipschitz map on Banach spaces. Pacific J. Math. 77 523–531.
  • [22] Röckner, M. (1999). $L^{p}$-analysis of finite and infinite-dimensional diffusion operators. In Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998) (G. Da Prato, ed.). Lecture Notes in Math. 1715 65–116. Springer, Berlin.
  • [23] Shigekawa, I. (1992). Sobolev spaces over the Wiener space based on an Ornstein–Uhlenbeck operator. J. Math. Kyoto Univ. 32 731–748.