Invariant measures of stochastic partial differential equations and conditioned diffusions

This work establishes and exploits a connection between the invariant measure of stochastic partial differential equations (SPDEs) and the law of bridge processes. Namely, it is shown that the invariant measure of $u_t=u_{xx}+f(u)+\sqrt{2?}\eta (x\text{,}t)$, where $\eta (x\text{,}t)$ is a space–time white-noise, is identical to the law of the bridge process associated to $\mathrm{d}U=a(U)\mathrm{d}x+\sqrt{?}\mathrm{d}W(x)$, provided that a and f are related by $?a^{″}(u)+2a^{\prime }(u)a(u)=?2f(u)$, $u\in \mathbb{R}$. Some consequences of this connection are investigated, including the existence and properties of the invariant measure for the SPDE on the line, $x\in \mathbb{R}$. To cite this article: M.G. Reznikoff, E. Vanden-Eijnden, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Existence and blow-up rate of large solutions of p(x)-Laplacian equations with gradient terms

In this paper we investigate boundary blow-up solutions of the problem

$\{\begin{array}{c}?{\mathrm{\Delta }}_{p(x)}u+f(x,u)=\pm K(x)|\mathrm{?}u{|}^{m(x)}\text{ in }\mathrm{\Omega },\hfill \\ u(x)\to +\infty \text{as }d(x,?\mathrm{\Omega })\to 0,\hfill \end{array}$

where ${\mathrm{\Delta }}_{p(x)}u=\mathrm{div}(|\mathrm{?}u{|}^{p(x)?2}\mathrm{?}u)$ is called the $p(x)$-Laplacian. Our results extend the previous work [25] of Y. Liang, Q.H. Zhang and C.S. Zhao from the radial case to the non-radial setting, and [46] due to Q.H. Zhang and D. Motreanu from the assumption that $K(x)|\mathrm{?}u(x){|}^{m(x)}$ is a small perturbation, to the case in which $\pm K(x)|\mathrm{?}u{|}^{m(x)}$ is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $d(x,?\mathrm{\Omega })$ and in terms of the growth of the exponents. Furthermore, the comparison principle is no longer applicable in our context, since $f(x,?)$ is not assumed to be monotone in this paper.     

On the semiclassical approximation to the eigenvalue gap of Schrödinger operators

We consider two types of Schrödinger operators $H(t)=?d^2/d{x}^2+q(x)+t\cos x$ and $H(t)=?d^2/d{x}^2+q(x)+A\cos (tx)$ defined on $L^2(\mathbb{R})$, where q is an even potential that is bounded from below, A is a constant, and $t>0$ is a parameter. We assume that $H(t)$ has at least two eigenvalues below its essential spectrum; and we denote by ${\lambda }_1(t)$ and ${\lambda }_2(t)$ the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap $\Gamma (t)={\lambda }_2(t)?{\lambda }_1(t)$ in the limit as $t\to \infty$.     

Generalized Ricci bounds and convergence of metric measure spaces

We introduce and analyze curvature bounds $\underset{?}{\mathbb{C}\mathrm{urv}}(M\text{,}\mathsf{d}\text{,}m)?K$ for metric measure spaces $(M\text{,}\mathsf{d}\text{,}m)$, based on convexity properties of the relative entropy $\mathrm{Ent}(?|m)$. For Riemannian manifolds, $\underset{?}{\mathbb{C}\mathrm{urv}}(M\text{,}\mathsf{d}\text{,}m)?K$ if and only if ${\mathrm{Ric}}_M(\xi \text{,}\xi )?K?{\left|\xi \right|}^2$ for all $\xi \in TM$. We define a complete separable metric $\mathbb{D}$ on the family of all isomorphism classes of normalized metric measure spaces. It has a natural interpretation in terms of mass transportation. Our lower curvature bounds are stable under $\mathbb{D}$-convergence. We also prove that the family of normalized metric measure spaces with doubling constant $?C$ is closed under $\mathbb{D}$-convergence. Moreover, the subfamily of spaces with diameter $?R$ is compact. To cite this article: K.-T. Sturm, C. R. Acad. Sci. Paris, Ser. I 340 (2005).