A Class of Vector Fields on Path Spaces

Lyons, Terry; Qian, Zhongmin

doi:10.1006/jfan.1996.3013

Cited by 12 publications

(6 citation statements)

References 27 publications

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“…Even for bounded variation paths it is not so easy to establish the existence of the flow (which corresponds to a nonlinear hyperbolic PDE). Spaces of rough paths seem to provide the correct domain for these vector fields as I f (x) exists; it is proved in [20] that an evolution or flow does exist even for rough initial conditions. The solution may explode, but exists at least for a finite period of time; the existence of a flow makes it clear that the functional F (I f (x)) really is a vector field, and is not just a formal object; we can differentiate functions on rough path space in these directions.…”

Section: Introductionmentioning

confidence: 99%

Smoothness of Itô maps and diffusion processes on path spaces (I)

Li¹,

Lyons

2006

Annales Scientifiques de l’École Normale Supérieure

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Let p ∈ [1, 2) and α, ε > 0 be such that α ∈ (p − 1, 1 − ε). Let V , W be two Euclidean spaces. Let Ωp(V ) be the space of continuous paths taking values in V and with finite p-variation. Let k ∈ N and f : W → Hom(V, W ) be a Lip(k + α + ε) map in the sense of E.M. Stein [Stein E.M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970]. In this paper we prove that the Itô map, defined by I(x) = y, is a local C k, ε 1+ε map (in the sense of Fréchet) between Ωp(V ) and Ωp(W ), where y is the solution to the differential equationThis result strengthens the continuity results and Lipschitz continuity results in [Lyons T., Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young, Math. Res. Lett. 1 (4) (1994) 451-464; Lyons T., Qian Z., System Control and Rough Paths, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2002] particularly to the non-integer case. It allows us to construct the fractional like Brownian motion and infinite dimensional Brownian motions on the space of paths with finite p-variation. As a corollary in the particular case where p = 1, we obtain that the development from the space of finite 1-variation paths on R d to the space of finite 1-variation paths on a d-dimensional compact Riemannian manifold is a smooth bijection. © 2006 Elsevier Masson SASDans cet article, nous montrons que l'application de Itô I : Ωp(V ) → Ωp(W ), définie par I(x) = y, où y est la solution de l'équation différentielle suivante :est localement de classe C k, ε 1+ε au sens de Fréchet. Cela nous permet de construire des processus de type mouvement brownien fractionnaire ainsi que des mouvements browniens de dimension infinie sur l'espace des chemins de p-variation finie. Comme corollaire, nous obtenons, dans le cas particulier où p = 1, que l'application de développement de l'espace des chemins de 1-variation finie sur R d dans l'espace des chemins de 1-variation finie sur une variété riemannienne compacte d-dimensionnelle est une bijection régulière.

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Section: Introductionmentioning

confidence: 99%

Smoothness of Itô maps and diffusion processes on path spaces (I)

Li¹,

Lyons

2006

Annales Scientifiques de l’École Normale Supérieure

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“…Vector fields on the path space of M-valued Brownian motion or continuous semimartingales have attracted the attention of many authors, e.g. [7], [ 141, [ 171, [ 131, [.5], [20]. Recently, T. Lyons and Z.-M. Qian 1211 studied vector fields along semimartingales obtained by varying a metric connection.…”

Section: Introductionmentioning

confidence: 99%

Complete lifts of connections and stochastic Jacobi fields

Arnaudon

Thalmaier

1998

Journal de Mathématiques Pures et Appliquées

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Differentiable families of V-martingales on manifolds are investigated: their infinitesimal variation provides a notion of stochastic Jacobi fields. Such objects are known [2] to be martingales taking values in the tangent bundle when the latter is equipped with the complete lift of the connection V. We discuss various characterizations of TM-valued martingales. When applied to specific families of V-martingales which appear in connection with the heat flow for maps between Riemannian manifolds, our results allow to establish formulas giving a stochastic representation for the differential of solutions to the nonlinear heat equation. As an application, we prove local and global gradient estimates for harmonic maps of bounded dilatation. 0 Elsevier, Paris

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“…Both authors used the Picard's iteration method or the Euler approximation method to prove the existence of a flow associated to the vector field. Enchev and Stroock [11] and Lyons and Qian [18] also constructed the flow on path spaces by different methods. Driver [8] and Enchev and Stroock [12] extended their results in [7,11] to loop spaces, respectively, and X.D.…”

Section: Introduction and The Main Resultsmentioning

confidence: 98%

Flows associated to adapted vector fields on the Wiener space

Gong

Zhang

2007

Journal of Functional Analysis

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The existence and uniqueness of a flow associated to an adapted vector field ξ on the Wiener space with d τ ξ α = a β α dω β (τ ) + b α dτ are proved by a modified Picard's iteration method, mainly under the conditions of exponential integrability concerning b α as well as the first-order Malliavin gradients of a β α and b α . A Newton-Leibnitz type inequality for a kind of Malliavin differentiable functionals is also proved, which is the key point to prove the above result, and has an independent interest.

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A Class of Vector Fields on Path Spaces

Cited by 12 publications

References 27 publications

Smoothness of Itô maps and diffusion processes on path spaces (I)

Smoothness of Itô maps and diffusion processes on path spaces (I)

Complete lifts of connections and stochastic Jacobi fields

Flows associated to adapted vector fields on the Wiener space

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