Quelques espaces fonctionnels associés à des processus gaussiens

Ciesielski, Z.; Kerkyacharian, Gérard; Roynette, Bernard

doi:10.4064/sm-107-2-171-204

Cited by 155 publications

(149 citation statements)

References 12 publications

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“…Then, almost surely, w α−hld < ∞. (By the way, w 1/H−var = ∞, a.s. See [7,19]. Hence, w H−hld = ∞, a.s.)…”

Section: Outline Of Proof Of Off-diagonal Asymptoticsmentioning

confidence: 99%

Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

Inahama

2016

J. Math. Soc. Japan

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In this paper we study short time asymptotics of a density function of the solution of a stochastic differential equation driven by fractional Brownian motion with Hurst parameter H (1/2 < H < 1) when the coefficient vector fields satisfy an ellipticity condition at the starting point. We prove both on-diagonal and off-diagonal asymptotics under mild additional assumptions. Our main tool is Malliavin calculus, in particular, Watanabe's theory of generalized Wiener functionals. IntroductionLet (w t ) t≥0 be the standard d-dimensional Brownian motion and let V i (0 ≤ i ≤ d) be smooth vector fields on R n with sufficient regularity. Consider the following stochastic differential equation (SDE) of Stratonovich-type;If the set of vector fields satisfies a hypoellipticity condition, the solution y t = y t (a) has a smooth density p t (a, a ′ ) with respect to Lebesgue measure on R n . From an analytic point of view, p t (a, a ′ ) is a fundamental solution of the parabolic equation ∂u/∂t = Lu, where, and is also called a heat kernel of L. In many fields of mathematics such as probability, analysis, mathematical physics, and differential geometry, short time asymptotic of p t (a, a ′ ) is a very important problem and has been studied extensively. Although analytic methods are also well-known, we * Revised on 7 Nov. 2013. Mathematics Subject Classification: 60H07, 60F99, 60H10, 60G15.

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“…Then, almost surely, w α−hld < ∞. (By the way, w 1/H−var = ∞, a.s. See [7,19]. Hence, w H−hld = ∞, a.s.)…”

Section: Outline Of Proof Of Off-diagonal Asymptoticsmentioning

confidence: 99%

Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

Inahama

2016

J. Math. Soc. Japan

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“…It is known from Ciesielski et al (1993) that the subspace We end this section by the following regularity of local time.…”

Section: The Functional Frameworkmentioning

confidence: 99%

Limit Theorems for a Class of Additive Functionals of Symmetric Stable Process and Fractional Brownian Motion in Besov-Orlicz Spaces

Ouahra¹

2012

Journal of Mathematics and Statistics

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“…(i) A path-wise approach that uses the Hölder continuity properties of the sample paths, developed from the works by Ciesielski, Kerkyacharian and Roynette [7] and Zähle [37].…”

Section: Fractional Brownian Motionmentioning

confidence: 99%

Stochastic calculus with respect to fractional Brownian motion

Nualart¹

2009

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Quelques espaces fonctionnels associés à des processus gaussiens

Cited by 155 publications

References 12 publications

Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

Limit Theorems for a Class of Additive Functionals of Symmetric Stable Process and Fractional Brownian Motion in Besov-Orlicz Spaces

Stochastic calculus with respect to fractional Brownian motion

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