## Journals

- Advances in Mathematics of Communications
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### Open Access Journals

EECT

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 $L^2(\mathcal O)$.

DCDS

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.

DCDS

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 $C_{b}(\overline{\O})$.

CPAA

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.

DCDS-S

We prove some Shauder estimates for an elliptic equation in Hilbert spaces.

DCDS

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.

DCDS-B

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.

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