Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels

Watanabe, Satoshi

doi:10.1214/aop/1176992255

Cited by 239 publications

(223 citation statements)

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“…In this subsection we state two main results of ours, which are basically analogous to the corresponding ones in Watanabe [21]. However, there are some differences.…”

Section: Statement Of the Main Resultssupporting

confidence: 66%

“…First, the exponents of t are not (a constant multiple of) natural numbers. Second, cancellation of "odd terms" as in p. 20 and p. 34, [21] does not happen in general in our case. (If the drift term in Young ODE (2.1) is zero, then this kind of cancellation takes place as in [1,2]).…”

Section: Statement Of the Main Resultsmentioning

confidence: 64%

“…We basically follow Watanabe [21]. In this paper, however, we can localize around the energy minimizing path in the abstract Wiener space since Itô map is continuous in our setting.…”

Section: Off-diagonal Short Time Asymptoticsmentioning

confidence: 99%

“…If the case of the usual stochastic analysis (i.e., H = 1/2), (A2) and (A3) have a geometric meaning. (See Remark 3.2, [21], which was originally in [15,4].) First, (A2) means that there is a unique shortest geodesics between a and a ′ .…”

Section: Proof Of Off-diagonal Short Time Asymptoticsmentioning

confidence: 99%

“…(See [21] or Sections 5.8-5.10, [9].) His theory of distributional Malliavin calculus is not only very powerful, but also user-friendly.…”

Section: Introductionmentioning

confidence: 99%

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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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“…In this subsection we state two main results of ours, which are basically analogous to the corresponding ones in Watanabe [21]. However, there are some differences.…”

Section: Statement Of the Main Resultssupporting

confidence: 66%

Section: Statement Of the Main Resultsmentioning

confidence: 64%

“…We basically follow Watanabe [21]. In this paper, however, we can localize around the energy minimizing path in the abstract Wiener space since Itô map is continuous in our setting.…”

Section: Off-diagonal Short Time Asymptoticsmentioning

confidence: 99%

Section: Proof Of Off-diagonal Short Time Asymptoticsmentioning

confidence: 99%

“…(See [21] or Sections 5.8-5.10, [9].) His theory of distributional Malliavin calculus is not only very powerful, but also user-friendly.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations