Invariance for stochastic reaction-diffusion equations
Piermarco Cannarsa Giuseppe Da Prato
Evolution Equations & Control Theory 2012, 1(1): 43-56 doi: 10.3934/eect.2012.1.43
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 $L^2(\mathcal O)$.
keywords: invariance of space domains. reaction-diffusion equations Stochastic partial differential equations
Transition semigroups corresponding to Lipschitz dissipative systems
Giuseppe Da Prato
Discrete & Continuous Dynamical Systems - A 2004, 10(1&2): 177-192 doi: 10.3934/dcds.2004.10.177
We consider a semilinear differential stochastic equation in a Hilbert space $H$ with a dissipative and Lipschitz nonlinearity. We study the corresponding transition semigroup in a space $L^2(H,\nu)$ where $\nu$ is an invariant measure.
keywords: dissipative systems transition semigroups Stochastic differential equations. Lipschitz nonlinearities Markov semigroups
On a class of elliptic and parabolic equations in convex domains without boundary conditions
Giuseppe Da Prato Alessandra Lunardi
Discrete & Continuous Dynamical Systems - A 2008, 22(4): 933-953 doi: 10.3934/dcds.2008.22.933
We consider the operator $\A u = \frac{1}{2} \Delta u - \langle DU, Du\right$, where $U $ is a convex real function defined in a convex open set $\O \subset \R^N$ and $\lim_{|x|\to \infty} U(x) = \lim_{ x \to \partial \O} U(x)$ $ =$ $ +\infty$. We study the realization of $\A $ in the spaces $C_{b}(\overline{\O})$, $C_{b}(\O)$ and $B_{b}(\O)$, and prove several properties of the associated Markov semigroup. In contrast with the case of bounded coefficients, elliptic equations and parabolic Cauchy problems such as (3) and (4) below are uniquely solvable in reasonable classes of functions, without imposing any boundary condition. We prove that the associated semigroup coincides with the transition semigroup of a stochastic variational inequality on $C_{b}(\overline{\O})$.
keywords: Markov semigroups stochastic variational inequalities unbounded coefficients
Some results for pathwise uniqueness in Hilbert spaces
Giuseppe Da Prato Franco Flandoli
Communications on Pure & Applied Analysis 2014, 13(5): 1789-1797 doi: 10.3934/cpaa.2014.13.1789
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
Schauder estimates for some perturbation of an infinite dimensional Ornstein--Uhlenbeck operator
Giuseppe Da Prato
Discrete & Continuous Dynamical Systems - S 2013, 6(3): 637-647 doi: 10.3934/dcdss.2013.6.637
We prove some Shauder estimates for an elliptic equation in Hilbert spaces.
keywords: elliptic equations with infinitely many variables. Schauder estimates Ornstein-Uhlenbeck operators
Asymptotic behavior of stochastic PDEs with random coefficients
Giuseppe Da Prato Arnaud Debussche
Discrete & Continuous Dynamical Systems - A 2010, 27(4): 1553-1570 doi: 10.3934/dcds.2010.27.1553
We study the long time behavior of the solution of a stochastic PDEs with random coefficients assuming that randomness arises in a different independent scale. We apply the obtained results to $2D$- Navier-Stokes equations.
keywords: Poincaré recurrences multifractal analysis. Dimension theory
Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces
Giuseppe Da Prato Alessandra Lunardi
Discrete & Continuous Dynamical Systems - B 2006, 6(4): 751-760 doi: 10.3934/dcdsb.2006.6.751
We consider a degenerate elliptic Kolmogorov--type operator arising from second order stochastic differential equations in $\mathbb R^{n}$ perturbed by noise. We study a realization of such an operator in $L^2$ spaces with respect to an explicit invariant measure, and we prove that it is $m$-dissipative.
keywords: Kolmogorov operators Second order stochastic equations invariant measures. degenerate elliptic operators

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