Stochastic variational inequalities with jumps

This work is devoted to the study of a stochastic variational inequality with a Wiener–Poisson driving term. Existence and uniqueness are proven for Lipschitz coefficients and under general conditions for the unbounded term. One of the main tools used in order to obtain the existence result is a penalization method involving Moreau–Yosida regularization.     

BSDEs driven by time-changed Lévy noises and optimal control

We study backward stochastic differential equations (BSDEs) for time-changed Lévy noises when the time-change is independent of the Lévy process. We prove existence and uniqueness of the solution and we obtain an explicit formula for linear BSDEs and a comparison principle. BSDEs naturally appear in control problems. Here we prove a sufficient maximum principle for a general optimal control problem of a system driven by a time-changed Lévy noise. As an illustration we solve the mean–variance portfolio selection problem.     

Phase reduction in the noise induced escape problem for systems close to reversibility

We consider n$n$-dimensional deterministic flows obtained by perturbing a gradient flow. We assume that the gradient flow admits a stable curve of stationary points, and thus if the perturbation is not too large the perturbed flow also admits an attracting curve. We show that the noise induced escape problem from a stable fixed point of this curve can be reduced to a one-dimensional problem: we can approximate the associated quasipotential by the one associated to the restricted dynamics on the stable curve. The error of this approximation is given in terms of the size of the perturbation.     

Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs II: Voltage Graph Constructions and Applications

In an earlier article the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two‐part series, we extend those results to orientable surfaces for all . In part II, a voltage graph construction is presented for building embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that p is prime, completing the proof started by part I (which covers the case ) that there exists an orientable hamilton cycle embedding of for all , . These embeddings are then used to determine the genus of several families of graphs, notably for and, in some cases, for .     

Nonlocal Harnack inequalities

We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.     

Unitary representations of the groups of measurable and continuous functions with values in the circle

We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theorems for linear operators are used.     

A generalization of a theorem of Ore

Let (K,v)$(K,v)$ be a discrete rank one valued field with valuation ring _Rv$R_v$. Let L/K$L/K$ be a finite extension such that the integral closure S   of _Rv$R_v$ in L   is a finitely generated _Rv$R_v$-module. Under a certain condition of v  -regularity, we obtain some results regarding the explicit computation of _Rv$R_v$-bases of S, thereby generalizing similar results that had been obtained for algebraic number fields in El Fadil et al. (2012) [7]. The classical Theorem of Index of Ore is also extended to arbitrary discrete valued fields. We give a simple counter example to point out an error in the main result of Montes and Nart (1992) [12] related to the Theorem of Index and give an additional necessary and sufficient condition for this result to be valid.     

Vanishing of self-extensions over symmetric algebras

We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type Z_A∞$\mathbb{Z}{A}_{\infty }$. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander–Reiten Conjecture and the Extension Conjecture.