Integrability and tail estimates for Gaussian rough differential equations

Cass, Tony; Litterer, Christian; Lyons, Terry

doi:10.1214/12-aop821

Cited by 95 publications

(176 citation statements)

References 33 publications

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“…Actually, the periodicity of the solutions is mainly used in Section 5. It should however be possible to simplify those arguments with the help of the results obtained in [7,12] after the redaction of the present work is completed. Remark 1.2.…”

Section: A Motivation: Path Samplingmentioning

confidence: 99%

Rough stochastic PDEs

Hairer

2011

Comm. Pure Appl. Math.

112

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In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too-high spatial roughness for classical analytical methods to apply. In fact, the class of SPDEs that we consider is genuinely ill-posed in the sense that different approximations to the nonlinearity may converge to different limits. Using rough path theory, a pathwise notion of solution to these SPDEs is formulated, and we show that this yields a well-posed problem that is stable under a large class of perturbations, including the approximation of the rough-driving noise by a mollified version and the addition of hyperviscosity.We also show that under certain structural assumptions on the coefficients, the SPDEs under consideration generate a reversible Markov semigroup with respect to a diffusion measure that can be given explicitly.

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Section: A Motivation: Path Samplingmentioning

confidence: 99%

Rough stochastic PDEs

Hairer

2011

Comm. Pure Appl. Math.

112

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“…Cass, Litterer, and Lyons [5] recently proved it in rough path setting for Gaussian rough path including fractional Brownian rough path with 1/4 < H ≤ 1/2. [9].…”

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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“…Rough paths have found numerous successful applications in stochastic analysis, among them the study of the properties of stochastic di¤erential equations driven by Gaussian processes, see e.g. [1], [3], [2] and the analysis of broad classes of stochastic di¤erential equations, see e.g. [9], [8].…”

Section: Introductionmentioning

confidence: 99%