The oscillation of the Poisson semigroup associated to parabolic Hermite operator

Let O(P_τ~L) be the oscillation of the Possion semigroup associated with the parabolic Hermite operator L = ?_t-?+|x|~2. We show that O(P_τ~L) is bounded from L~p(R~(n+1))into itself for 1 p ∞, bounded from L~1(R~(n+1)) into weak-L~1(R~(n+1)) and bounded from L_c~∞(R~(n+1)) into BMO(R~(n+1)). In the case p = ∞ we show that the range of the image of the operator O(P_τ~L) is strictly smaller than the range of a general singular operator.     

Quasi-linear PDEs and forward–backward stochastic differential equations: Weak solutions

In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space $H_{\rho }^1({\mathbb{R}}^d)$. For this, we study first the solutions of forward–backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space $L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^d)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^k)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^{k\times d})$. This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.     

Large deviations of kernel density estimator in L1(Rd) for uniformly ergodic Markov processes

In this paper, we consider a uniformly ergodic Markov process $(X_n{)}_{n⩾0}$ valued in a measurable subset E of ${\mathbb{R}}^d$ with the unique invariant measure $\mu (\mathrm{d}x)=f(x)\mathrm{d}x$, where the density f is unknown. We establish the large deviation estimations for the nonparametric kernel density estimator $f_n^{*}$ in $L^1({\mathbb{R}}^d,\mathrm{d}x)$ and for $6f_n^{*}-f{6}_{L^1({\mathbb{R}}^d,\mathrm{d}x)}$, and the asymptotic optimality $f_n^{*}$ in the Bahadur sense. These generalize the known results in the i.i.d. case.     

A cohomological study of local rings of embedding codepth 3

The generating series of the Bass numbers ${\mu }_R^i={\mathrm{rank}}_k{\mathrm{Ext}}_R^i(k,R)$ of local rings $R$ with residue field $k$ are computed in closed rational form, in case the embedding dimension $e$ of $R$ and its depth $d$ satisfy $e?d\le 3$. For each such $R$ it is proved that there is a real number $\gamma >1$, such that ${\mu }_R^{d+i}\ge \gamma {\mu }_R^{d+i?1}$ holds for all $i\ge 0$, except for $i=2$ in two explicitly described cases, where ${\mu }_R^{d+2}={\mu }_R^{d+1}=2$. New restrictions are obtained on the multiplicative structures of minimal free resolutions of length 3 over regular local rings.     

On the gentle properties of anisotropic Besov spaces

In this paper, we prove that anisotropic homogeneous Besov spaces ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ are gentle spaces, for all parameters $s,p,q$ and all anisotropies $u$. Using the Littlewood–Paley decomposition, we study their completeness, separability, duality and homogeneity. We then define the notion of anisotropic orthonormal wavelet basis of $L^2\left({\mathbb{R}}^d\right)$, and we show that the homogeneous version of Triebel families of anisotropic orthonormal wavelet bases associated to the tensor product of Lemarié–Meyer (resp. Daubechies) wavelets are particular examples. We characterize the ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ spaces using Lemarié–Meyer wavelets. In fact, we show that these bases will be either unconditional bases or unconditional $1$-weak bases of ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$, depending on whether ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ is separable or not. By introducing an anisotropic version of the class of almost diagonal matrices related to anisotropic orthonormal wavelet bases, we prove that these spaces are stable under changes of anisotropic orthonormal wavelet bases. As a consequence, we extend the characterization of ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ using Daubechies wavelets.