We give necessary and sufficent conditions for a smooth, generic, differential one-form w on Rⁿ to decompose into a sum w = a¹du₁ +... +a^kdu_k, where the function a ^l are positive and the u_l convex (or quasi-convex) near the origin.

In this work we prove the equivalence between static and dynamic points of views for certain ballistic random walks in random environment on Z^d, when d greater than or equal to 4 and the disorder is low. Our techniques also enable us to derive in the same setting a functional central limit theorem for almost every realization of the environment. We also provide an example where the equivalence between static and dynamic points of views breaks down.

Suppose that g(t) and W_t are independent Brownian motions starting from g(0) = W₀ = 0. Consider the Brownian motion Y_t reflected on g(t), obtained from W_t by the means of the Skorohod lemma. The upper and lower limiting behaviors of Y_t are presented. The upper tail estimate on exit time is computed via principal eigenvalue.

We consider additive functionals INTt-0 V(n_s )ds of symmetric zero-range processes, where V is a mean zero local function. In dimensions 1 and 2 we obtain a central limit theorem for a^-1(t) INTt-0 V(n_s)ds with a(t) = SQRROOT (tlogt) in d =2 and a(t) = t ^3/4 in d = 1 and an explicit form for the asymptotic variance SIGMA². The transient case d greater than or equal to 3 can be handled by standard arguments [KV, SX,S]. We also obtain corresponding invariance principles. This generalizes results obtained by Port (see [CG]) for noninteracting random walks and Kipnis [K] for the symmetric simple exclusion process. Our main tools are the martingale method together with L² decay estimates [JLQY] for the process semigroup.

Here we establish a Liouville type theorem for minimizing maps from R² (or in general, from R^m) into a compact Riemannian manifold N. As a consequence of this, we prove a local gradient estimate for minimal solutions to a variational problem arise from planar ferromagnetism and anti-ferromagnetism. The latter can be applied to study the asymptotic behavior of entire solutions.

Kleeman has recently demonstrated that the relative entropy provides a significant measure of the information content of a prediction ensemble compared with the climate record in several simplified climate models. Here several additional aspects of utilizing the relative entropy for predictability theory are developed with full mathematical rigor in a systematic fashion which the authors believe will be very useful in practical problems with many degrees of freedom in atmosphere/ocean and biological science. The results developed here include a generalized signal-dispersion decomposition, rigorous explicit lower bound estimators for information content, and rigorous lower bound estimates on relative entropy for many variables, N, through N, one-dimensional relative entropies and N, two-dimensional mutual information functions. These last results provide a practical context for rapid evaluation of the predictive information content in a large number of variables.

In this work we prove the equivalence between static and dynamic points of views for certain ballistic random walks in random environment on Z^d, when d greater than or equal to 4 and the disorder is low. Our techniques also enable us to derive in the same setting a functional central limit theorem for almost every realization of the environment. We also provide an example where the equivalence between static and dynamic points of views breaks down.