The Lie Bracket of Adapted Vector Fields on Wiener Spaces

Let W(M) be the based (at o∈ M) path space of a compact Riemannian manifold M equipped with Wiener measure ν . This paper is devoted to considering vector fields on W(M) of the form X _s ^h ( σ )=P _s ( σ )h _s ( σ ) where P _s ( σ ) denotes stochastic parallel translation up to time s along a Wiener path σ ∈ W(M) and {h _s } _{s∈ [0,1]} is an adapted T _o M -valued process on W(M). It is shown that there is a large class of processes h (called adapted vector fields) for which we may view X ^h as first-order differential operators acting on functions on W(M) . Moreover, if h and k are two such processes, then the commutator of X ^h with X ^k is again a vector field on W(M) of the same form. Accepted 5 May 1997     

Invariant Subspaces of Collectively Compact Sets of Linear Operators

In this paper, we first give some invariant subspace results for collectively compact sets of operators in connection with the joint spectral radius of these sets. We then prove that any collectively compact set M in algΓ satisfies Berger-Wang formula, where Γ is a complete chain of subspaces of X.      

Differentiable and analytic families of continuous martingales in manifolds with connection

Summary.  We prove that the derivative of a differentiable family X _t (a) of continuous martingales in a manifold M is a martingale in the tangent space for the complete lift of the connection in M, provided that the derivative is bicontinuous in t and a. We consider a filtered probability space (Ω,(ℱ _t )_{0≤ t ≤1}, ℙ) such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a↦L(a) of random variable with values in M and such that L(0)≡x ₀∈M, there exists an analytic family a↦X(a) of continuous martingales such that X ₁(a)=L(a). For this, we investigate the convexity of the tangent spaces T ^{( n )} M, and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time T such that ℙ(T<1) is arbitrarily small. We use this construction of families of martingales in complex analytic manifolds to prove that every ℱ₁-measurable random variable with values in a compact convex set V with convex geometry in a manifold with a C ¹ connection is reachable by a V-valued martingale. Received: 14 March 1996/In revised form: 12 November 1996     

Stochastic Volterra equations driven by cylindrical Wiener process

In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of α-times resolvent families. Both authors are supported partially by project “Proyecto Anillo: Laboratorio de Analisis Estocastico; ANESTOC”.     

Séries de poincaré des groupes géométriquement finis

In this paper, we study the behaviour of the Poincaré series of a geometrically finite group Γ of isometries of a riemannian manifoldX with pinched curvature, in the case when Γ contains parabolic elements. We give a sufficient condition on the parabolic subgroups of Γ in order that Γ be of divergent type. When Γ is of divergent type, we show that the Sullivan measure on the unit tangent bundle ofX/Γ is finite if and only if certain series which involve only parabolic elements of Γ are convergent. We build also examples of manifoldsX on which geometrically finite groups of convergent type act.

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Durant la rédaction de cet article, M. Peigné a bénéficié d'un détachement au Centre National de la Recherche Scientifique, URA 305.     

Invariant measures for the linear stochastic Cauchy problem and R-boundedness of the resolvent

We study the asymptotic behaviour of solutions of the stochastic abstract Cauchy problem $$ \left\{ {\begin{array}{*{20}l} {dU\left( t \right) = AU\left( t \right)dt + BdW_H \left( t \right),\quad t \geqslant 0,} \hfill\ {U\left( 0 \right) = 0,} \hfill\ \end{array}} \right. $$ where A is the generator of a C₀-semigroup on a Banach space E, W_H is a cylindrical Brownian motion over a separable Hilbert space H, and $$ B \in \user1{\mathscr L}\left( {H,E} \right) $$ is a bounded operator. Assuming the existence of a solution U, we prove that a unique invariant measure exists if the resolvent R(λ, A) is R-bounded in the right half-plane {Reλ > 0}, and that conversely the existence of an invariant measure implies the R-boundedness of R(λ, A)B in every half-plane properly contained in {Re λ > 0}. We study various abscissae related to the above problem and show, among other things, that the abscissa of R-boundedness of the resolvent of A coincides with the abscissa corresponding to the existence of invariant measures for all γ -radonifying operators B provided the latter abscissa is finite. For Hilbert spaces E this result reduces to the Gearhart-Herbst-Prüss theorem. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Metrics on products of surfaces with non-positive sectional curvature

Let (S _i, g_i),i=1, 2 be two compact riemannian surfaces isometrically embedded in euclidean spaces. In this paper we show that ifM=S ₁×S₂,then for any functionF: M→R, the graph ofF, i.e. the manifold {(x, F(x)): x∈M}, does not have positive sectional curvature.     

Spectral flexibility of symplectic manifolds <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> × <Emphasis Type="Italic">M</Emphasis>

We consider Riemannian metrics compatible with the natural symplectic structure on T ² × M, where T ² is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ₁. We show that λ₁ can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic question.     

Markov processes associated with L p -resolvents and applications to stochastic differential equations on Hilbert space

We give general conditions on a generator of a C₀-semigroup (resp. of a C₀-resolvent) on L^p(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Existence of natural and projectively equivariant quantizations

We study the existence of natural and projectively equivariant quantizations for differential operators acting between order 1 vector bundles over a smooth manifold M. To that aim, we make use of the Thomas-Whitehead approach of projective structures and construct a Casimir operator depending on a projective Cartan connection. We attach a scalar parameter to every space of differential operators, and prove the existence of a quantization except when this parameter belongs to a discrete set of resonant values.     

The maximum maximum of a martingale constrained by an intermediate law

Let (M _t ) be any martingale with M ₀≡ 0, an intermediate law M ₁∼μ₁, and terminal law M ₂∼μ₂, and let Mˉ ₂≡ sup_{0≤ t ≤2} M _t . In this paper we prove that there exists an upper bound, with respect to stochastic ordering of probability measures, on the law of Mˉ ₂. We construct, using excursion theory, a martingale which attains this maximum. Finally we apply this result to the robust hedging of a lookback option. Received: 26 December 1998 / Revised version: 20 April 2000 /?Published online: 15 February 2001     

Riemannian geometry of finite rank positive operators

A riemannian metric is introduced in the infinite dimensional manifold Σn of positive operators with rank n<∞ on a Hilbert space H. The geometry of this manifold is studied and related to the geometry of the submanifolds Σp of positive operators with range equal to the range of a projection p (rank of p=n), and Pp of selfadjoint projections in the connected component of p. It is shown that these spaces are complete in the geodesic distance.     

Semi-primary hereditary rings

A semi-primary hereditary ring Σ, with radicalM and residue ring Γ=Σ/M, is uniquely determined by Γ and a Γ-bimoduleA=M/M ², whenever Σ admits a splitting Σ=Γ+A+M ².     

Geometry of the Normal Bundle of a Submanifold*

 We study the geometric behavior of the normal bundle T ^⊥ M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T ^⊥ M and look at the relation between the minimality of T ^⊥ M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T ^⊥ M, are null. Received March 14, 2001; in revised form February 11, 2002     

Intégration Symplectique des Variétés de Poisson Régulières

Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ₀ with the induced Poisson structure Λ₀ is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability ([2], [6], [11], [17], [28]). The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s third theorem in the infinite dimensional case.

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Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90)     

Stochastic Jacobi fields and vector fields induced by varying area on path spaces

Summary. We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Lévy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Lévy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space. Received: 8 August 1996 / In revised form: 8 January 1997     

Decomposition of stochastic flows and Lyapunov exponents

Let φ _t be the stochastic flow of a stochastic differential equation on a compact Riemannian manifold M. Fix a point m∈M and an orthonormal frame u at m, we will show that there is a unique decomposition φ _t = ξ _t ψ _t such that ξ _t is isometric, ψ _t fixes m and Dψ _t (u) = us _t , where s _t is an upper triangular matrix. We will also establish some convergence properties in connection with the Lyapunov exponents and the decomposition Dφ _t (u) = u _t s _t with u _t being an orthonormal frame. As an application, we can show that ψ_t preserves the directions in which the tangent vectors at m are dilated at fixed exponential rates. Received: 19 November 1998 / Revised version: 1 October 1999 / Published online: 14 June 2000     

Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Let E be a real Banach space with property (α) and let W _Γ be an E-valued Brownian motion with distribution Γ. We show that a function is stochastically integrable with respect to W _Γ if and only if Γ-almost all orbits Ψx are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of Γ-measurable linear extensions of γ-radonifying operators with values in spaces of γ-radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution equations driven by an E-valued Brownian motion. The first named author gratefully acknowledges the support by a ‘VIDI subsidie’ in the ‘Vernieuwingsimpuls’ programme of The Netherlands Organization for Scientific Research (NWO) and the Research Training Network HPRN-CT-2002–00281. The second named author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1–1).     

On the existence of prolongation of connections

We classify all bundle functors G admitting natural operators transforming connections on a fibered manifold Y → M into connections on GY → M. Then we solve a similar problem for natural operators transforming connections on Y → M into connections on GY → Y. Dedicated to Professor Ivan Kolář on the occasion of his 70th birthday