Moderate deviations for a random walk in random scenery

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér’s condition. We prove moderate deviation principles in dimensions d≥2$d\ge 2$, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d≥4$d\ge 4$ we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. For d≥3$d\ge 3$, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst for d=2$d=2$ we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.     

Local probabilities for random walks conditioned to stay positive

Let S ₀ = 0, {S _n , n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X ₁, X ₂, . . . and let $\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}$ and $\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} $ . Assuming that the distribution of X ₁ belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as ${n\rightarrow \infty }$ , of the local probabilities ${\bf P}{(\tau ^{\pm }=n)}$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities ${\bf P}{(S_{n} \in [x,x+\Delta )|\tau^{-} > n)}$ with fixed Δ and ${x=x(n)\in (0,\infty )}$ .     

Restrictions on collapsing with a lower sectional curvature bound

We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.     

A Critical Phenomenon for Sublinear Elliptic Equations in Cone-Like Domains

The authors of this paper study positive supersolutions to theelliptic equation -u = c|x|^–su^p in Cone-like domains ofR^N (N 2), where p, s R and c > 0. They prove that in thesublinear case p < 1 there exists a critical exponent p_*> 1 such that the equation has a positive supersolution ifand only if – < p < p_*. The value of p_* is determinedexplicitly by s and the geometry of the cone. 2000 MathematicsSubject Classification 35J60 (primary), 35B05, 35R45 (secondary).     

L^p -uniqueness for Dirichlet operators with singular potentials

We study the problem of strong uniqueness in L^p for the Dirichlet operator perturbed by a singular complex-valued potential. First we construct the generator -H_p of a C₀-semigroup in L^p, with H_p extending the restriction of the perturbed Dirichlet operator to the set of smooth functions. The corresponding sesquilinear form in L² is not assumed to be sectorial. Then we reveal sufficient conditions on the logarithmic derivative # of the measure rdx \rho dx and the potential q which ensure that -H_p is the only extension of D+b·?-q \upharpoonright_C0^￥ \Delta +\beta \cdot \nabla -q \upharpoonright_{C_0^{\infty}} which generates a C₀-semigroup on L^p. The method of a priori estimates of solutions to corresponding differential equations is employed.     

K-shell Auger lifetime variation in doubly ionized Ne and first row hydrides

We consider 1s Auger decay in doubly (core-core and core-valence) ionized Ne and in the isoelectronic first row element hydrides. We show theoretically that the presence of the spectator inner valence vacancy leads to Auger lifetime variation of up to about a factor of 2, relative to the Auger lifetimes in the singly ionized species. The origin of this effect is traced to spin selection rules. Implications on the modelling of the radiation damage in strong x-ray fields are discussed.     

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