126 research outputs found
Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noise
The paper is concerned with spatial and time regularity of solutions to
linear stochastic evolution equation perturbed by L\'evy white noise "obtained
by subordination of a Gaussian white noise". Sufficient conditions for spatial
continuity are derived. It is also shown that solutions do not have in general
\cadlag modifications. General results are applied to equations with fractional
Laplacian. Applications to Burgers stochastic equations are considered as well.Comment: This is an updated version of the same paper. In fact, it has already
been publishe
Stochastic Reaction-diffusion Equations Driven by Jump Processes
We establish the existence of weak martingale solutions to a class of second
order parabolic stochastic partial differential equations. The equations are
driven by multiplicative jump type noise, with a non-Lipschitz multiplicative
functional. The drift in the equations contains a dissipative nonlinearity of
polynomial growth.Comment: See journal reference for teh final published versio
Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains
We show that the stochastic flow generated by the Stochastic Navier-Stokes
equations in a 2-dimensional Poincar\'e domain has a unique random attractor.
This result complements a recent result by Brze\'zniak and Li [10] who showed
that the flow is asymptotically compact and generalizes a recent result by
Caraballo et al. [12] who proved existence of a unique pullback attractor for
the time-dependent deterministic Navier-Stokes equations in a 2-dimensional
Poincar\'e domain
Stochastic Quasilinear Evolution Equations in UMD Banach Spaces
In this work we prove the existence and uniqueness up to a stopping time for the stochastic counterpart of Tosio Kato\u27s quasilinear evolutions in UMD Banach spaces. These class of evolutions are known to cover a large class of physically important nonlinear partial differential equations. Existence of a unique maximal solution as well as an estimate on the probability of positivity of stopping time is obtained. An example of stochastic Euler and Navier–Stokes equation is also given as an application of abstract theory to concrete models
Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations
In this paper we study the following non-autonomous stochastic evolution
equation on a UMD Banach space with type 2,
{equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t)))
dt + B(t,U(t)) dW_H(t), \quad t\in [0,T],
U(0) & = u_0. {aligned}. {equation}
Here are unbounded operators with domains
which may be time dependent. We assume that
satisfies the conditions of Acquistapace and Terreni. The
functions and are nonlinear functions defined on certain interpolation
spaces and is the initial value. is a cylindrical Brownian
motion on a separable Hilbert space .
Under Lipschitz and linear growth conditions we show that there exists a
unique mild solution of \eqref{eq:SEab}. Under assumptions on the interpolation
spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk,
to obtain space-time regularity results for the solution of
\eqref{eq:SEab}. For Hilbert spaces we obtain a maximal regularity result.
The results improve several previous results from the literature.
The theory is applied to a second order stochastic partial differential
equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to
several improvements of their result.Comment: Accepted for publication in Journal of Evolution Equation
Coercivity condition for higher moment a priori estimates for nonlinear SPDEs and existence of a solution under local monotonicity
Higher order moment estimates for solutions to nonlinear SPDEs governed by
locally-monotone operators are obtained under appropriate coercivity condition.
These are then used to extend known existence and uniqueness results for
nonlinear SPDEs under local monotonicity conditions to allow derivatives in the
operator acting on the solution under the stochastic integral.Comment: 32 page
Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition
Using a weak convergence approach, we prove a LPD for the solution of 2D
stochastic Navier Stokes equations when the viscosity converges to 0 and the
noise intensity is multiplied by the square root of the viscosity. Unlike
previous results on LDP for hydrodynamical models, the weak convergence is
proven by tightness properties of the distribution of the solution in
appropriate functional spaces
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