Tube estimates for diffusion processes under a weak Hörmander condition

Pigato, Paolo

doi:10.1214/16-aihp805

Cited by 7 publications

(4 citation statements)

References 31 publications

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“…There is also a relation with density estimates of the associated diffusion over small time intervals, see e.g. Pigato [19]. In the proof we shall use tools developed in the recent paper by Delarue and Menozzi [6] where the same "cascade"-structure of the drift as in our case is present.…”

Section: Notice That Writingmentioning

confidence: 79%

Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes

Löcherbach

2017

J Theor Probab

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We consider oscillatory systems of interacting Hawkes processes introduced in [9] to model multi-class systems of interacting neurons together with the diffusion approximations of their intensity processes. This diffusion, which incorporates the memory terms defining the dynamics of the Hawkes process, is hypo-elliptic. It is given by a high dimensional chain of differential equations driven by 2−dimensional Brownian motion. We study the large-population-, i.e., small noise-limit of its invariant measure for which we establish a large deviation result in the spirit of Freidlin and Wentzell. 

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Section: Notice That Writingmentioning

confidence: 79%

Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes

Löcherbach

2017

J Theor Probab

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“…Again, we stress the fact that C N,α B (]0, T [×D) not only does contain C ∞ (]0, T [×D), but also includes the standard Hölder space C N,α (]0, T [×D). Recent results on the local density assuming standard regularity of the coefficients were proved in [3], under local strong Hörmander-type conditions, and in [15], under local weak Hörmander-type conditions for two-dimensional diffusions.…”

Section: Main Results and Comparison With The Literaturementioning

confidence: 99%

Local densities for a class of degenerate diffusions

Lanconelli¹,

Pagliarani²,

Pascucci³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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We study a class of R d -valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is a local Itô formula for functions that are not twice-differentiable in the classical sense, but only intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the Hörmander condition. The second main contribution, which builds upon the first one, is an existence and regularity result for the local transition density.

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“…The case of a diffusion process whose infinitesimal generator is hypoelliptic has been studied recently in [15], which we briefly describe below. Suppose X t is a two-dimensional diffusion solution to…”

Section: Introductionmentioning

confidence: 99%

“…where φ ∈ L 2 ([0, T ]) is considered as a control. One of the results in [15] gives upper and lower bounds of the probability that the paths of X t are contained in a tube around x t (φ). While these bounds are sharper than the ones we use to find the asymptotics in Theorem 3.13, the radius of the tube considered by P. Pigato cannot be arbitrarily small.…”

Section: Introductionmentioning

confidence: 99%

On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group

Marco¹,

Gordina²

2019

Preprint

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Given a stochastic process, the problem of finding the corresponding Onsager-Machlup functional is well known in probability theory, and it has been intensively studied over the last fifty years. For a stochastic process taking values in a Riemannian manifold and whose infinitesimal generator is an elliptic operator, the Onsager-Machlup functional is already known. In this paper, we present a way to find the Onsager-Machlup functional associated to the Brownian motion taking values in the Heisenberg group. Its generator is a hypoelliptic operator instead of an elliptic one, and the state space changes from a Riemannian manifold to the Heisenberg group. This space is the simplest non-trivial example of a sub-Riemannian manifold. As the standard geometric and analytic tools from Riemannian geometry are not available, we need to use new techniques. Contents 1. Introduction 2 2. The Onsager-Machlup functional for an elliptic diffusion 4 3. The setting and the main result 6 3.1. Heisenberg group as Lie group 6 3.2. Heisenberg group as a sub-Riemannian manifold 9 3.3. Main result 13 4. Proof of Theorem 3.13 13 4.1. Reduction of the asymptotics 13 4.2. Girsanov's transformation 16 Appendix A. Girsanov's theorem 21 References 23

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Tube estimates for diffusion processes under a weak Hörmander condition

Cited by 7 publications

References 31 publications

Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes

Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes

Local densities for a class of degenerate diffusions

On the Onsager-Machlup functional for the Brownian motion on the Heisenberg group

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