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September  2014, 13(5): 1789-1797. doi: 10.3934/cpaa.2014.13.1789

Some results for pathwise uniqueness in Hilbert spaces

Giuseppe Da Prato 1, and  Franco Flandoli 2,

1. 

Scuola Normale Superiore, Piazza dei Cavalieri 6, 56126 Pisa, Italy
2. 

Dipartimento di Matematica Applicata, "U.Dini" Università di Pisa, V.le B. Pisano 26/b, 56126 Pisa

Received  September 2013 Revised  March 2014 Published  June 2014

An abstract evolution equation in Hilbert spaces with Hölder continuous drift is considered. By proceeding as in [3], we transform the equation in another equation with Lipschitz continuous coefficients.Then we prove existence and uniqueness of this equation by a fixed point argument.
Keywords: Pathwise uniqueness, Hölder continuous drift., stochastic PDEs.
Mathematics Subject Classification: 35R60, 60H1.
Citation: Giuseppe Da Prato, Franco Flandoli. Some results for pathwise uniqueness in Hilbert spaces. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1789-1797. doi: 10.3934/cpaa.2014.13.1789
References:
[1]

S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum, 49 (1994), 349-367. doi: 10.1007/BF02573496.  Google Scholar

[2]

G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society, Lecture Notes 293, Cambridge University Press, 2002. doi: 10.1017/CBO9780511543210.  Google Scholar

[3]

G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), 243-267. doi: 10.1016/j.jfa.2009.11.019.  Google Scholar

[4]

F. Flandoli, Random perturbation of PDEs and fluid dynamic models, Lecture Notes in Mathematics, 2015, Springer, Berlin, 2011. doi: 10.1007/978-3-642-18231-0.  Google Scholar

[5]

J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math., 55 (1986), 257-266. doi: 10.1007/BF02765025.  Google Scholar

[6]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, 1968. Google Scholar

[7]

M. Ondreját, Uniqueness for Stochastic Evolution Equations in Banach Spaces, Dissertationes Math. (Rozprawy Mat.), no. 426, 2004. doi: 10.4064/dm426-0-1.  Google Scholar

[8]

M. Röckner, B. Schmuland and X. Zhang, The Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Comm. Mat. Phys., 11 (2008), 247-259. Google Scholar

[9]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.  Google Scholar

show all references

References:
[1]

S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum, 49 (1994), 349-367. doi: 10.1007/BF02573496.  Google Scholar

[2]

G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society, Lecture Notes 293, Cambridge University Press, 2002. doi: 10.1017/CBO9780511543210.  Google Scholar

[3]

G. Da Prato and F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, J. Funct. Anal., 259 (2010), 243-267. doi: 10.1016/j.jfa.2009.11.019.  Google Scholar

[4]

F. Flandoli, Random perturbation of PDEs and fluid dynamic models, Lecture Notes in Mathematics, 2015, Springer, Berlin, 2011. doi: 10.1007/978-3-642-18231-0.  Google Scholar

[5]

J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math., 55 (1986), 257-266. doi: 10.1007/BF02765025.  Google Scholar

[6]

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, 1968. Google Scholar

[7]

M. Ondreját, Uniqueness for Stochastic Evolution Equations in Banach Spaces, Dissertationes Math. (Rozprawy Mat.), no. 426, 2004. doi: 10.4064/dm426-0-1.  Google Scholar

[8]

M. Röckner, B. Schmuland and X. Zhang, The Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Comm. Mat. Phys., 11 (2008), 247-259. Google Scholar

[9]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.  Google Scholar

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