Abstract
We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter $H ~(1/3 < H \le 1/2)$ under the ellipticity assumption at the starting point. In such a case, the law of the solution at a fixed time has a kernel, i.e., a density function with respect to Lebesgue measure. In this paper we prove a short time off-diagonal asymptotic expansion of the kernel under mild additional assumptions. Our main tool is Watanabe’s distributional Malliavin calculus.
Highlights
For the usual d-dimensional Brownian motion and sufficiently regular vector fields Vi (0 ≤ i ≤ d) on Rn, consider the following stochastic differential equation (SDE) of Stratonovich type:d dyt = Vi(yt) ◦ dwti + V0(yt)dt i=1 with y0 = a ∈ Rn.If the vector fields satisfy the hypoellipticity condition at the starting point a, the law of yt has a heat kernel i.e., a density function pt(a, a′) with respect to Lebesgue measure da′ for any t > 0.In probability theory, the short time asymptotic problem of pt(a, a′) has extensively been studied and is a classical topic
In this subsection we introduce several index sets for the exponent of the small parameter ε > 0, which will be used in the asymptotic expansion
We carefully look at the proof in [31, 36] once again and make sure that every operation is ”of at most polynomial order.”
Summary
For the usual d-dimensional Brownian motion (wt) and sufficiently regular vector fields Vi (0 ≤ i ≤ d) on Rn, consider the following stochastic differential equation (SDE) of Stratonovich type:. Bismut [14] was first to prove short time kernel asymptotics via Malliavin calculus. The expansion in the deterministic sense is already known, but we need ”Lp-version” (or ”D∞-version”) of the expansion in this paper These estimates play a crucial role in the proof of the main theorem. Thanks to these propositions, we can use Watanabe’s asymptotic theory in the proof of the main theorem, following the argument in [56, 33]. It can be found on arXiv preprint server (arXiv:1403.3181)
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