Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part

Let be a compact set with interior G. Let L ¹ (G,dx), >0 dx-a.e. on G, and m:=dx. Let A=(a _ij ) be symmetric, and globally uniformly strictly elliptic on G. Let be such that ; f, , is closable in L ² (G,m) with closure ( ^r ,D( ^r )). The latter is fulfilled if satisfies the Hamza type condition, or _i L ¹ _loc (G,dx), 1id. Conservative, non-symmetric diffusion processes X _t related to the extension of a generalized Dirichlet form where satisfies are constructed and analyzed. If G is a bounded Lipschitz domain, H ^1,1 (G), and a _ij D( ^r ), a Skorokhod decomposition for X _t is given. This happens through a local time that is uniquely associated to the smooth measure 1_{{ Tr ()>0}} d, where Tr denotes the trace and the surface measure on G.This research has been financially supported by TMR grant HPMF-CT-2000-00942 of the European Union. Mathematics Subject Classification (2000): 60J60, 60J55, 31C15, 31C25, 35J25     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

Intersection Local Times of Independent Brownian Motions as Generalized White Noise Functionals

A 'chaos expansion' of the intersection local time functional of two independent Brownian motions in R ^d is given. The expansion is in terms of normal products of white noise (corresponding to multiple Wiener integrals). As a consequence of the local structure of the normal products, the kernel functions in the expansion are explicitly given and exhibit clearly the dimension dependent singularities of the local time functional. Their L ^p -properties are discussed. An important tool for deriving the chaos expansion is a computation of the 'S-transform' of the corresponding regularized intersection local times and a control about their singular limit.     

Variational representations of Varadhan functionals

Motivated by the theory of large deviations, we introduce a class of non-negative non-linear functionals that have a variational ``rate function" representation.

     

Some Generic Results on Nonattaining Functionals

We prove that (a) in a reflexive space, for any linearly bounded but unbounded closed convex subset the nonsupport functionals are a dense G subset of the polar set, and (b) any nonsemicoercive proper convex lsc [weak^*-lsc] function in a [dual] Banach space has a generic [dense G ] set of L -perturbations which do not attain their infimum. We also characterize the proper convex functions that have inf-nonattaining L -perturbations. This results also in a criterion for reflexivity.     

Variations of Constrained Domain Functionals Associated with Boundary-Value Problems

We study variational formulas for maximizers for domain functionalsF(x0, u(x0)), x0, and F(x,u(x))dxover all Lipschitz domains satisfying the constraintg(x) dx=1. Here, u is the solution ofa diffusion equation in . Functional variations arecomputed using domain variations which preserve the constraint exactly. Weshow that any maximizer solves a moving boundary problem for the diffusionequation. Further, we show that, for problems with symmetry, the optimaldomains are balls.     

Open subgroups of

Let be a locally compact group and let denote the -algebra generated by left translation operators on . Let and be the spaces of almost periodic and weakly almost periodic functionals on the Fourier algebra , respectively. It is shown that if contains an open abelian subgroup, then (1) if and only if is norm dense in ; (2) is a -algebra if is norm dense in , where denotes the set of elements in with compact support. In particular, for any amenable locally compact group which contains an open abelian subgroup, has the dual Bohr approximation property and is a -algebra.

     

Dynamics of Cohen-Grossberg neural networks with variable and distributed delays

This paper is concerned with the analysis problem for the exponential stability of a class of Cohen-Grossberg neural networks with variable and distributed delays. Some sufficient conditions ensuring the existence, uniqueness and exponential stability of the equilibrium point are obtained by employing Brouwer’s fixed-point theorem and by applying the inequality technique. In the results, we do not assume that the activation function satisfies the boundedness and the Lipschitz condition. Three numerical examples are given to show the effectiveness of the obtained results.     

An implementation of the exponential time differencing scheme to the magnetohydrodynamic equations in a spherical shell

Over the last decade there has been renewed interest in applying exponential time differencing (ETD) time stepping schemes to the solution of stiff systems. In this paper, we present an implementation of such a scheme to the fully spectral solution of the incompressible magnetohydrodynamic equations in a spherical shell. One problem associated with ETD schemes is the accurate calculation of the necessary matrices; we implement and discuss in detail a variety of different methods including direct computation, contour integration, spectral expansions and recurrence relations. We compare the accuracy of six different second-order methods in determining the evolution of a three-dimensional magnetic field under the action of a prescribed time-dependent flow of electrically conducting fluid, and find that for the timestep restriction imposed by the nonlinear terms, ETD methods are no more accurate than linearly implicit methods which have the significant advantage of being easier to implement. However, ETD methods are more readily extendable than those which are linearly implicit and will become much more advantageous at higher order.     

Asymptotic exponential cones of Metzler matrices and their use in the solution of an algebraic problem

The aim of this paper is that of investigating the asymptotic exponential cone of a single Metzler matrix, introduced in [23], and of defining and analysing the new concept of asymptotic exponential cone of a family of Metzler matrices (along a certain direction). These results will provide necessary and/or sufficient conditions for the solvability of an interesting algebraic problem that arises in the context of continuous-time positive switched systems and, specifically, in the investigation of the reachability property [21,22,25].