# American Institute of Mathematical Sciences

June  2012, 1(1): 43-56. doi: 10.3934/eect.2012.1.43

## Invariance for stochastic reaction-diffusion equations

 1 Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scienti ca 1, I-00133 Roma, Italy 2 Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, I-56125 Pisa, Italy

Received  December 2011 Revised  February 2012 Published  March 2012

Given a stochastic reaction-diffusion equation on a bounded open subset $\mathcal O$ of $\mathbb{R}^n$, we discuss conditions for the invariance of a nonempty closed convex subset $K$ of $L^2(\mathcal O)$ under the corresponding flow. We obtain two general results under the assumption that the fourth power of the distance from $K$ is of class $C^2$, providing, respectively, a necessary and a sufficient condition for invariance. We also study the example where $K$ is the cone of all nonnegative functions in $L^2(\mathcal O)$.
Citation: Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43
##### References:
 [1] S. Agmon, "Lectures on Elliptic Boundary Value Problems," Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr., Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, NJ-Toronto-London, 1965.  Google Scholar [2] P. Cannarsa and G. Da Prato, Stochastic viability for regular closed sets in Hilbert spaces, Rend. Lincei Math. Appl., 22 (2011), 1-10. Google Scholar [3] S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Probab. Theory Related Fields, 125 (2003), 271-304. doi: 10.1007/s00440-002-0230-6.  Google Scholar [4] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  Google Scholar [5] G. Da Prato and J. Zabczyk, "Second Order Partial Differential Equations in Hilbert Spaces," London Mathematical Society Lecture Notes, 293, Cambridge University Press, Cambridge, 2002.  Google Scholar [6] L. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar [7] D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Comm. Partial Differential Equations, 27 (2002), 1283-1299.  Google Scholar [8] S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab., 23 (1995), 157-172. doi: 10.1214/aop/1176988381.  Google Scholar

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##### References:
 [1] S. Agmon, "Lectures on Elliptic Boundary Value Problems," Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr., Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, NJ-Toronto-London, 1965.  Google Scholar [2] P. Cannarsa and G. Da Prato, Stochastic viability for regular closed sets in Hilbert spaces, Rend. Lincei Math. Appl., 22 (2011), 1-10. Google Scholar [3] S. Cerrai, Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Probab. Theory Related Fields, 125 (2003), 271-304. doi: 10.1007/s00440-002-0230-6.  Google Scholar [4] G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992.  Google Scholar [5] G. Da Prato and J. Zabczyk, "Second Order Partial Differential Equations in Hilbert Spaces," London Mathematical Society Lecture Notes, 293, Cambridge University Press, Cambridge, 2002.  Google Scholar [6] L. Evans and R. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar [7] D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Comm. Partial Differential Equations, 27 (2002), 1283-1299.  Google Scholar [8] S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab., 23 (1995), 157-172. doi: 10.1214/aop/1176988381.  Google Scholar
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