# Prépublications du Laboratoire de Mathématiques

• 01--2006  Olivier Rahavandrainy

Résolution de fibrés généraux stables de rang 2
sur P^3 de classes de Chern c1 = -1, c2 = 2p >= 6 : I
(version pdf)

Résumé :
On considère l'espace de modules M(c1,c2) des fibrés stables de rang 2 sur P^3, de classes de Chern c1, c2. Si (c1 = 0, c2 > 0) ou (c1 = -1, c2 = 2p >= 6), on sait que M(c1,c2) a une composante irréductibledont le point générique F(c1,c2) a la cohomologie naturelle. Nous avons calculé la résolution minimale de F(0,c2). Dans cet article, nous voulons déterminer celle de F(-1,c2) si c2 (v+2)(2v^2+3v-1)/(6v+7), où v est le plus petit entier tel que h^0(F(v)) > 0.

• 02--2006  Johannes Huisman

Modules over twisted group rings and vector bundles
over the anisotropic real conic

Résumé :
We present another proof of a recent result of Biswas and Nagaraj that states that a locally free sheaf of finite rank over the anisotropic real conic is the direct sum of indecomposable locally free sheaves of rank 1 or 2. Our proof is purely algebraic, and is based on a classification of graded C[X,Y]-modules endowed with a certain action of the cyclic group Z/4Z.

• 03--2006  Francoise Pene

A limit theorem for a random walk in a stationary scenery
coming from a hyperbolic dynamical system
(version pdf)

Résumé :
In this paper, we extend a result of Kesten and Spitzer (1979). Let us consider a stationary sequence $(\xi_k:=f(T^k(.)))_k$ given by an invertible probability dynamical system and some centered function $f$. Let $(S_n)_n$ be a simple symmetric random walk on $Z$ independent of $(\xi_k)_k$. We give examples of partially hyperbolic dynamical systems and of functions $f$ such that $n^{-3/4}(\xi(S_1)+...+\xi(S_k))$ converges in distribution as $n$ goes to infinity.

• 04--2006  Aurélien Monteillet

Integral formulations of the geometric eikonal equation
(version pdf)

Abstract:
We prove integral formulations of the eikonal equation, equivalent to the notion of viscosity solution in the framework of the set-theoretic approach to front propagation problems. We apply these integral formulations to investigate the regularity of the front: we prove that under regularity assumptions on the velocity c, the front has locally finite perimeter in the region where c does not vanish, and we give a time-integral estimate of its perimeter.

The moduli space of anisotropic Gaussian curves

Abstract :
Let X be a real hyperelliptic curve. Its opposite curve X- is the curve obtained from X by twisting the real structure by the hyperelliptic involution. The curve X is said to be Gaussian if X- is isomorphic to X. In an earlier paper, we have studied Gaussian curves having real points. In the present paper we study Gaussian curves without real points, i.e. anisotropic Gaussian curves. We prove that the moduli space of such curves is a reducible connected real analytic subset of the moduli space of all anisotropic hyperelliptic curves, and determine its irreducible components.

• 06--2006  Bertrand Banos

Monge-Ampère equations and generalized complex geometry

Abstract :
We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Ampère equation of divergent type and, using the Gualtieri \overline{\partial} operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects.

• 07--2006  M. Benyounes E. Loubeau et C.M. Wood

Harmonic sections of Riemannian bundles and metrics of Cheeger-Gromoll type

Abstract :

We study harmonic sections of Riemannian vector bundles equipped with a metric generalising the Sasaki and Cheeger-Gromoll metrics.

• 08--2006  E. Loubeau, S. Montaldo et C. Oniciuc

The stress-energy tensor for biharmonic maps

Abstract :

We study the stress-energy tensor associated to the bienergy.

• 09--2006  Dan Goreac

Approximate controllability for linear stochastic
differential equations with control acting on the noise

Abstract :
In this paper we study approximate controllability for a linear stochastic differential equation

dy(t)=(Ay(t)+Bu(t))dt+(Cy(t)+Du(t))dW(t),

for the case when the control acts also on the noise. This may be considered as a generalization of the work of Buckdahn, Quincampoix and Tessitore where the problem is solved for D=0 and of Peng for D of full rank. We prove, using the dual BSDE and Riccati methods that approximate controllability is equivalent to the local in time viability for a suitable set. Finally, an invariance criterion is given..

• 10--2006  Dan Goreac

Non compact-valued stochastic control under state constraints

Abstract :

In the present paper, we study a necessary condition under which the solutions of a stochastic differential equation governed by unbounded control processes, remain in an arbitrarily small neighborhood of a given set of constraints. We prove that, in comparison to the classical constrained control problem with bounded control processes, a further assumption on the growth of control processes is needed in order to obtain a necessary and sufficient condition in terms of viscosity solution of the associated Hamilton-Jacobi-Bellman equation. A rather general example illustrates our main result.

• 11--2006  Françoise Pène

Transient random walk in Z^2 with stationary orientations
(version pdf)

Abstract :
In this paper, we extend a result of Campanino and Petritis. We study a random walk in $Z^2$ with a random environment. We suppose that the orientations of the horizontal floors are given by a stationary sequence of random variables (xi_k)_k. Once the environment fixed, the random walk can go either up or down or with respect to the orientation of the present floor (with the same probability). Campanino and Petritis have studied the case when the $xi_k$ are i.i.d..

Guillotin and Le Ny have extended this result is to some cases of independent orientations choosen with stationary probabilities (not equal to 0 and to 1). In the present paper, we generalize the result of Campanino and Petritis to some cases when $(xi_k)_k$ is stationary. Moreover we give a slight extension of a result of one result of Guillotin and Le Ny.

• 12--2006  Rainer Buckdahn, Marc Quincampoix and Gianmario Tessitore

Controlled Stochastic differential equations under constraints in infinite dimensional spaces

Abstract :
In this paper we study the compatibility (or viability) of a given state constraint $K$ with respect to a controlled stochastic evolution equation in a real Hilbert space $H$. We allow the noise to be a cylindrical Wiener process and admit an unbounded linear operator in the state equation. Our assumptions cover, for instance, controlled heat equations with space-time white noise. Our main result is to prove that if $K$ is $\varepsilon$-viable then the square of the distance from $K$: $d_K^2(x):= \inf_{y\in K}|x-y|^2$ is a viscosity supersolution of a suitable class of fully nonlinear Hamilton-Jacobi-Bellman equations in $H$. This extend already obtained results in the finite dimensional case. We use the definition of viscosity supersolutions for `unbounded' elliptic equations in infinite variables that has been recently introduced in by Swiech and Kelome. We discuss several cases where the above necessary condition is also sufficient.

• 13--2006  Y. Derriennic, K. Fraczek, M. Lemanczyk, F. Parreau

Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings

Abstract :   fichier pdf

• 14--2006       Yves Derriennic

The central limit theorem for random walks on orbits of probability preserving transformations

Abstract :
Let $\nu =\{p_k: k \in \Bbb Z\}$ be an ergodic probability distribution on $\Bbb Z$. Given an ergodic probability preserving transformation $\tau$ of $(S,\Sigma,m)$, let $\{X_n\}$ be the Markov chain with state space $S$ and space of trajectories $\Omega$ obtained from the random walk of law $\nu$ on the orbits of $\tau$, with corresponding Markov operator $Pf = \sum_{k\in \Bbb Z} p_k f\circ \tau^k$. It is known, by normality of $P$ in $L_2(m)$, that when $f \in \sqrt{I-P}L_2(m)$ the CLT holds for $\{f(X_n)\}$ in $(\Omega,\bold P_m)$. For $\nu$ centered with finite variance, we strengthen this result: we show that when $f \in \sqrt{I-P}L_2(m)$, the quenched CLT holds for $\{f(X_n)\}$, i.e., for a.e. $x$ the sequence $\frac1{\sqrt n} \sum_{k=1}^n f(X_k)$ converges in distribution, in $(\Omega,\bold P_x)$, to a normal distribution (with variance independent of $x$). In the non-centered case the quenched CLT is obtained when $f \in \sqrt{I-P}L_r(m)$ for some $r>2$.

• 15--2006       Rainer Buckdahn, Jin Ma, Catherine Rainer

Stochastic Control Problems for Systems Driven by Normal Martingales

Abstract :

In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. Assuming that the driving normal martingale is one-dimensional, we prove the Bellman Principle for such a control problem, and derive the corresponding Hamilton-Jacobi-Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Further we prove an uniqueness result for the viscosity solution of such an HJB equation.