1,721,262 research outputs found
Uniqueness results for the generators of the two-dimensional Euler and Navier-Stokes flows. The case of Gaussian invariant measures
No full textUniqueness of solutions of the stochastic Navier-Stokes equation with invariant measure given by the enstrophy
No full textA stochastic Navier–Stokes equation with space-time Gaussian white
noise is considered, having as infinitesimal invariant measure a Gaussian
measure whose covariance is given in terms of the enstrophy. Pathwise
uniqueness for a.e. initial velocity (a.e. with respect to this Gaussian measure) is proven for solutions having this invariant measure
Invariant Gibbs measures for the 2D vortex motion of fluids
No full textInvariant quantities of the classical motion of an ideal incompressible fluid in a two-dimensional bounded domain are used to construct a family {Pi(alpha)}(alpha) of probability measures of the Gibbs form, which are invariant under the flow. The Gibbs exponent H is given by the renormalized energy. These measures are supported by the space of configuration Gamma, i.e. the fluid vorticity is concentrated in a finite number of distinct points. Properties of a deterministic vortex dynamics having Pi(alpha) as invariant measure are investigated; in particular Markov uniqueness is proven. The classical (pre-)Dirichlet form associated to Pi(alpha) is also introduced and analyzed
Erratum to “Uniqueness of the generators of 2D Euler and Stokes flows” [Stochastic Process. Appl. 118 (11) (2008) 2071–2084]
No full textSmall noise asymptotic expansions for stochastic PDE's.The case of a dissipative polynomiallly bounded non linearity
No full textWe study a reaction-diffusion evolution equation perturbed by a Gaussian
noise. Here the leading operator is the infinitesimal generator of a C0-semigroup of strictly
negative type, the nonlinear term has at most polynomial growth and is such that the whole
system is dissipative.
The corresponding Itô stochastic equation describes a process on a Hilbert space with
dissipative nonlinear, non globally Lipschitz drift and a Gaussian noise.
Under smoothness assumptions on the nonlinearity, asymptotics to all orders in a small
parameter in front of the noise are given, with uniform estimates on the remainders. Applications
to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded
domain are included. As a particular example we consider the small noise asymptotic expansions
for the stochastic FitzHugh-Nagumo equations of neurobiology around deterministic
solutions
On the essential self-adjointness of Wick powers of relativistic fields and of fields unitary equivalent to random fields
No full textThe essential self-adjointness on a natural domain of the sharp-time Wick powers of the relativistic free field in two space-time dimension is proven. Other results on Wick powers are reviewed and discussed
Invariant measures for stochastic differential equations on networks
No full textAbstract. We study existence and uniqueness of an invariant measure for infinite dimensional
stochastic differential equations with dissipative polynomially bounded nonlinear terms. We also
exhibit the existence of a density with respect to a Gaussian measure. Moreover, we decompose
the solution process into a stationary component and a component which vanishes asymptotically
in the L 2 -sense. Applications are given to neurobiological networks where the signals propagation
is modelled by a system of coupled stochastic FitzHugh-Nagumo equations
Small noise asymptotic expansions for stochastic PDE's driven by dissipative nonlinearity and L\'evy noise
Full text linkInvariant measures of Lévy-Khinchine type for 2D fluids
No full textA survey of results on invariant measures of the Levy-Khinchine type for 2D Euler and stochastic Navier-Stokes equations is given. Uniqueness results of the corresponding Liouville respectively Kolmogorov flows are discussed. Stochastic dynamics associated with the invariant measures are also discussed (stochastic Stokes equation for the vorticity in the Gaussian case, Doob's independent Brownian motions process in the compound Poisson case)
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