Densities for rough differential equations under Hörmander’s condition

Cass, Tony; Friz, Peter K.

doi:10.4007/annals.2010.171.2115

Cited by 120 publications

(137 citation statements)

References 34 publications

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“…(Remark that we could have considered perturbation h ∈ H β for some β < 0, which en passant shows that the effective tangent space to gPAM is larger than the Cameron-Martin space. 3 …”

Section: Introductionmentioning

confidence: 99%

Malliavin calculus for regularity structures: The case of gPAM

Cannizzaro

Friz

Gassiat

2017

Journal of Functional Analysis

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Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accomodates a robust, and purely deterministic, translation operator, in L 2 -directions, between "models". In the concrete context of the generalized parabolic Anderson model in 2D -one of the singular SPDEs discussed in the afore-mentioned article -we establish existence of a density at positive times.

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“…(Remark that we could have considered perturbation h ∈ H β for some β < 0, which en passant shows that the effective tangent space to gPAM is larger than the Cameron-Martin space. 3 …”

Section: Introductionmentioning

confidence: 99%

Malliavin calculus for regularity structures: The case of gPAM

Cannizzaro

Friz

Gassiat

2017

Journal of Functional Analysis

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“…(This was a crucial ingredient in the development of Malliavin calculus for rough differential equations driven by fBm, see e.g. [CF10,CHLT15]. )…”

Section: Introductionmentioning

confidence: 99%

On the Regularity of Sle Trace

Friz

Tran

2017

Forum of Mathematics, Sigma

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We revisit regularity of SLE trace, for all κ = 8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia-Rodemich-Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1 + κ/8, 2), improving on previous works of Werness, and also (optimal) Hölder regularityà la Johansson Viklund and Lawler.

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“…This theory allows one to solve (deterministically) differential equations driven by rough signals at the expense of "enhancing" the rough signal with some additional information. Lyons' theory has found numerous applications to stochastic calculus and stochastic differential equations, for example see [4], [5], [6], [8], and the references therein. For some more recent applications, see [1], [19], [18], [9] , and [2].…”

Section: Introductionmentioning

confidence: 99%

Controlled rough paths on manifolds I

Semko¹

2017

Rev. Mat. Iberoam.

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In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of controlled rough one-forms along such a controlled path and their resulting integrals are then defined. This general integration theory does require the introduction of an additional geometric structure on the manifold which we refer to as a "parallelism." The transformation properties of the theory under change of parallelisms is explored. Using these transformation properties, it is shown that the integration of a smooth one-form along a manifold valued controlled rough path is in fact well defined independent of any additional geometric structures. We present a theory of push-forwards and show how it is compatible with our integration theory. Lastly, we give a number of characterizations for solving a rough differential equation when the solution is interpreted as a controlled rough path on a manifold and then show such solutions exist and are unique.

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Densities for rough differential equations under Hörmander’s condition

Cited by 120 publications

References 34 publications

Malliavin calculus for regularity structures: The case of gPAM

Malliavin calculus for regularity structures: The case of gPAM

On the Regularity of Sle Trace

Controlled rough paths on manifolds I

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