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The Kolmogorov Operator Associated to a Burgers SPDE in Spaces of Continuous Functions

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Abstract

We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.

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References

  1. Bogachev, V.I., Röckner, M.: Elliptic equations for measures on infinite-dimensional spaces and applications. Probab. Theory Relat. Fields 120(4), 445–496 (2001)

    Article  MATH  Google Scholar 

  2. Cerrai, S: A Hille-Yosida theorem for weakly continuous semigroups. Semigroup Forum 49(3), 349–367 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers’ equation. NoDEA, Nonlinear Differ. Equ. Appl. 1(4), 389–402 (1994)

    Article  MATH  Google Scholar 

  4. Da Prato, G., Debussche, A.: Differentiability of the transition semigroup of the stochastic burgers equation, and application to the corresponding Hamilton-Jacobi equation. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) (1998); Mat. Appl. 9(4), 267–277 (1999)

  5. Da Prato, G., Debussche, A.: Dynamic programming for the stochastic Burgers equation. Ann. Math. Pures Appl. (4) 178, 143–174 (2000)

    Article  MATH  Google Scholar 

  6. Da Prato, G., Debussche, A.: Dynamic programming for the stochastic Navier-Stokes equations. M2AN Math. Model. Numer. Anal. 34(2), 459–475 (2000) (special issue for R. Temam’s 60th birthday)

    Article  MATH  MathSciNet  Google Scholar 

  7. Da Prato, G., Debussche, A.: Ergodicity for the 3D stochastic Navier-Stokes equations. J. Math. Pures Appl. (9) 82(8), 877–947 (2003)

    MATH  MathSciNet  Google Scholar 

  8. Da Prato, G., Debussche, A.: m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise. Potential Anal. 26(1), 31–55 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Da Prato, G., Tubaro, L.: Some results about dissipativity of Kolmogorov operators. Czechoslovak Math. J. 51(126)(4), 685–699 (2001)

    Article  Google Scholar 

  10. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. In: Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  11. Da Prato, G., Zabczyk, J.: Ergodicity for infinite-dimensional systems. In: London Mathematical Society Lecture Note Series, vol. 229. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  12. Da Prato, G., Zabczyk, J.: Differentiability of the Feynman-Kac semigroup and a control application. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8(3), 183–188 (1997)

    MATH  MathSciNet  Google Scholar 

  13. Da Prato, G., Zabczyk, J.: Second order partial differential equations in Hilbert spaces. In: London Mathematical Society Lecture Note Series, vol. 293. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  14. Da Prato, G.: Kolmogorov equations for Stochastic PDEs. In: Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser, Boston (2004)

    Google Scholar 

  15. Debussche, A., Odasso, C.: Markov solutions for the 3D stochastic Navier-Stokes equations with state dependent noise. J. Evol. Equ. 6(2), 305–324 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Goldys, B., Kocan, M.: Diffusion semigroups in spaces of continuous functions with mixed topology. J. Differ. Equ. 173(1), 17–39 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Henry, D.: Geometric theory of semilinear parabolic equations. In: Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)

    Google Scholar 

  18. Lant, T., Thieme, H.R.: Markov transition functions and semigroups of measures. Semigroup Forum 74(3), 337–369 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Manca, L.: On a class of stochastic semilinear PDEs. Stoch. Anal. Appl. 24(2), 399–426 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Manca, L.: Kolmogorov equations for measures. J. Evol. Equ. 8, 231–262 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Manca, L.: On dynamic programming approach for the 3D-Navier-Stokes equations. Appl. Math. Optim. 57(3), 329–348 (2008)

    Google Scholar 

  22. Manca, L.: Kolmogorov operators in spaces of continuous functions and equations for measures. Ph.D. Thesis (2008)

  23. Manca, L.: Fokker-Planck equation for Kolmogorov operators with unbounded coefficients. Stoch. Anal. Appl. 27(4), 747–769 (2009). http://www.informaworld.com/smpp/content∼db=all∼content=a912617112

    Google Scholar 

  24. Priola, E.: On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math. 136(3), 271–295 (1999)

    MATH  MathSciNet  Google Scholar 

  25. Röckner, M., Sobol, Z.: A new approach to Kolmogorov equations in infinite dimensions and applications to stochastic generalized Burgers equations. C. R. Math. Acad. Sci. Paris 338(12), 945–949 (2004)

    MATH  MathSciNet  Google Scholar 

  26. Röckner, M., Sobol, Z.: Kolmogorov equations in infinite dimensions: well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations. Ann. Probab. 34(2), 663–727 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rothe, F.: Global solutions of reaction-diffusion systems. In: Lecture Notes in Mathematics, vol. 1072. Springer, Berlin (1984)

    Google Scholar 

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  1. Dipartimento di Matematica P. e A., Università di Padova, Via Trieste 63, 35121, Padova, Italy

    Luigi Manca

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  1. Luigi Manca
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Correspondence to Luigi Manca.

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Manca, L. The Kolmogorov Operator Associated to a Burgers SPDE in Spaces of Continuous Functions. Potential Anal 32, 67–99 (2010). https://doi.org/10.1007/s11118-009-9146-4

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  • DOI: https://doi.org/10.1007/s11118-009-9146-4

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