Weak existence and uniqueness for forward–backward SDEs

We aim to establish the existence and uniqueness of weak solutions to a suitable class of non-degenerate deterministic FBSDEs with a one-dimensional backward component. The classical Lipschitz framework is partially weakened: the diffusion matrix and the final condition are assumed to be space Hölder continuous whereas the drift and the backward driver may be discontinuous in x$x$. The growth of the backward driver is allowed to be at most quadratic with respect to the gradient term.     

Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle

Summary Given two pointsx, yS ¹ randomly chosen independently by a mixing absolutely continuous invariant measure of a piecewise expanding and smooth mapf of the circle, we consider for each >0 the point process obtained by recording the timesn>0 such that |f ⁿ (x)–f ⁿ (y)|. With the further assumption that the density of is bounded away from zero, we show that when tends to zero the above point process scaled by –1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicity by an average on the rate of expansion off.Partially supported by FAPESP grant number 90/3918-5     

Naturally reductive homogeneous Randers spaces

In this paper, we study naturally reductive Randers metrics on homogeneous manifolds. We first prove that naturally reductive Randers metrics are of Berwald type. We then give an explicit formula for the flag curvature of naturally reductive Randers metrics. Finally a necessary and sufficient condition for invariant Randers metrics on homogeneous manifolds being naturally reductive is given.     

Singularities of the quad curl problem

We consider the quad curl problem in smooth and non smooth domains of the space. We first give an augmented variational formulation equivalent to the one from [25] if the datum is divergence free. We describe the singularities of the variational space which correspond to the ones of the Maxwell system with perfectly conducting boundary conditions. The edge and corner singularities of the solution of the corresponding boundary value problem with smooth data are also characterized. We finally obtain some regularity results of the variational solution.     

Criteria for a ring to have a left Noetherian left quotient ring

Two criteria are given for a ring to have a left Noetherian left quotient ring (to find a criterion was an open problem since 70's). It is proved that each such ring has only finitely many maximal left denominator sets.     

On the discretisation in time of the stochastic Allen–Cahn equation

We consider the stochastic Allen–Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension , and study the semidiscretisation in time of the equation by an Euler type split‐step method with step size . We show that the method converges strongly with a rate . By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.     

Gradient estimates of Dirichlet heat kernels for unimodal Lévy processes

Under some mild assumptions on the Lévy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure‐jump isotropic unimodal Lévy processes in .     

Densely defined non‐closable curl on carpet‐like metric measure spaces

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one.     

Determinacy of refinements to the difference hierarchy of co-analytic sets

In this paper we develop a technique for proving determinacy of classes of the form ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+\mathrm{\Gamma }$ (a refinement of the difference hierarchy on ${\mathrm{\Pi }}_1^1$ lying between ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1$ and $({\omega }^2+1)\text{-}{\mathrm{\Pi }}_1^1$) from weak principles, establishing upper bounds for the determinacy-strength of the classes ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+{\mathrm{\Sigma }}_{\alpha }^0$ for all computable α and of ${\omega }^2\text{-}{\mathrm{\Pi }}_1^1+{{\mathrm{\Delta }}^1}_1$. This bridges the gap between previously known hypotheses implying determinacy in this region.