Abstract:
We compare the rate of convergence to the time average of a function over an integrable Hamiltonian flow with the one obtained by a stochastic perturbation of the same flow. Precisely, we provide detailed estimates in different Fourier norms and we prove the convergence even in a Sobolev norm for a special vanishing limit of the stochastic perturbation.

Abstract:
We give estimates of the distance between the densities of the laws of two functionals $F$ and $G$ on the Wiener space in terms of the Malliavin-Sobolev norm of $F-G.$ We actually consider a more general framework which allows one to treat with similar (Malliavin type) methods functionals of a Poisson point measure (solutions of jump type stochastic equations). We use the above estimates in order to obtain a criterion which ensures that convergence in distribution implies convergence in total variation distance; in particular, if the functionals at hand are absolutely continuous, this implies convergence in $L^{1}$ of the densities.

Abstract:
First of all, we establish compactness of continuous mappings of the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$ in ${\Bbb R}^n\,,$ $n\ge 3\,,$ with one fixed point. Then we give a series of theorems on convergence of the Orlicz--Sobolev homeomorphisms and on semicontinuity in the mean of dilatations of the Sobolev homeomorphisms. These results lead us to closeness of the corresponding classes of homeomorpisms. Finally, we come on this basis to criteria of compactness of classes of Sobolev's homeomorphisms with the corresponding conditions on dilatations and two fixed points.

Abstract:
We introduce a new family of refined Sobolev-Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the It\={o} formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.

Abstract:
We obtain estimates for convergence rates, in a scale of anisotropic Sobolev spaces, for certain singularly perturbed parabolic problems. The equations we consider correspond to the two primary spatially continuous models used in modelling stochastic reaction diffusion. One model, due to Smoluchowskii, uses a Dirichlet boundary condition to capture the reaction mechanism while the other model uses an interaction potential with a coupling constant $\lambda$ and corresponds to a Schr\"odinger type operator. Thus we show that the two models agree in the large coupling limit and estimate the rate of convergence.

Abstract:
This work is concerned with the accuracy of Gaussian beam superpositions, which are asymptotically valid high frequency solutions to linear hyperbolic partial differential equations and the Schr\"odinger equation. We derive Sobolev and max norms estimates for the difference between an exact solution and the corresponding Gaussian beam approximation, in terms of the short wavelength $\varepsilon$. The estimates are performed for the scalar wave equation and the Schr\"odinger equation. Our result demonstrates that a Gaussian beam superposition with $k$-th order beams converges to the exact solution as $O(\varepsilon^{k/2-s})$ in order $s$ Sobolev norms. This result is valid in any number of spatial dimensions and it is unaffected by the presence of caustics in the solution. In max norm, we show that away from caustics the convergence rate is $O(\varepsilon^{\lceil k/2\rceil})$ and away from the essential support of the solution, the convergence is spectral in $\varepsilon$. However, in the neighborhood of a caustic point we are only able to show the slower, and dimensional dependent, rate $O(\varepsilon^{(k-n)/2})$ in $n$ spatial dimensions.

Abstract:
We provide sufficient conditions for norm convergence of various projection and reflection methods, as well as giving limiting examples regarding convergence rates.

Abstract:
Self-interacting diffusions are processes living on a compact Riemannian manifold defined by a stochastic differential equation with a drift term depending on the past empirical measure of the process. The asymptotics of this measure is governed by a deterministic dynamical system and under certain conditions it converges almost surely towards a deterministic measure (see Bena\"im, Ledoux, Raimond (2002) and Bena\"im, Raimond (2005)). We are interested here in the rate of this convergence. A central limit theorem is proved. In particular, this shows that greater is the interaction repelling faster is the convergence.

Abstract:
In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincar\'{e} inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincar\'{e} inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.

Abstract:
We improve the Sobolev-type embeddings due to Gagliardo and Nirenberg in the setting of rearrangement invariant (r.i.) spaces. In particular we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between r.i. spaces and mixed norm spaces. As a consequence, we prove that the classical estimate for the standard Sobolev space by Poornima, O'Neil and Peetre (1 <=p