We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B_{\pi ,q}^s$ on a regular domain of ${ℝ}^d.$ The result is: if then the Kolmogorov metric entropy satisfies . This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper...

Let ( be a $\mathbb{Z}$-random walk and ${\left({\xi }_x\right)}_{x\in \mathbb{Z}}$ be a sequence of independent and identically distributed $ℝ$-valued random variables, independent of the random walk. Let be a measurable, symmetric function defined on ${ℝ}^2$ with values in $ℝ$. We study the weak convergence of the sequence ${𝒰}_n,n\in \mathbb{N}$, with values in the set of right continuous real-valued functions with left limits, defined by $$\sum _{i,j=0}^{\left[nt\right]}h({\xi }_{S_i},{\xi }_{S_j}),t\in [0,1].$$ Statistical applications are presented, in particular we prove a strong law of large numbers for -statistics indexed by a...

We give necessary and sufficient conditions which characterize the Young measures associated to two oscillating sequences of functions, on ${\omega }_1\times {\omega }_2$ and on ${\omega }_2$ satisfying the constraint $v_n\left(y\right)=\frac{1}{|{\omega }_1|}{\int }_{{\omega }_1}{u}_n(x,y)dx$. Our study is motivated by nonlinear effects induced by homogenization. Techniques based on equimeasurability and rearrangements are employed.

In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation $u_t(x,t)+\text{div}F(x,t,u(x,t))=0$ with the initial condition $u_0\in L^{\infty }\left({ℝ}^N\right)$. Passing to the limit in these schemes, we prove the existence of an entropy solution $u\in L^{i}nfty({ℝ}^N\times {ℝ}_{+})$. Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in $L_{loc}^p({ℝ}^N\times {ℝ}_{+})$, 1 ≤ ≤ +∞. Furthermore, if $u_0\in L^{\infty }\cap {\text{BV}}_{loc}\left({ℝ}^N\right)$, we show that $u\in {\text{BV}}_{loc}({ℝ}^N\times {ℝ}_{+})$, which leads to an “$h^{\frac{1}{4}}$” error estimate between the approximate and the entropy solutions (where defines the...