Abstract
We obtain bounds for the density of \(\left (x_{1,n}+W_{t}, x_{n+1}+{{\int }_{0}^{t}} |x_{1,n}+\right .\\\left .W_{s}|^{k}ds\right )\) where (Wt)t≥0 is a standard Brownian motion of ℝn, k∈ℕ∗ is even and x=(x1,n,xn+1)∈ℝn+1. This process satisfies a weak Hormander condition but the support of its density is not the whole space. Also, the Density has various asymptotic regimes depending on the starting/final points considered (which are as well related to the number of brackets needed to span the space in Hormander’s theorem). The proofs of lower and upper bounds are based on Harnack inequalities and Malliavin calculus respectively. The case of the joint law of Brownian motion and the integral of odd powers of its coordinates is also considered.
Highlights
We present a methodology to derive two-sided bounds for the density of some RN -valued degenerate processes of the form
Where the (Yi)i∈[[0,n]] are smooth vector fields defined on RN, ((Wti)t≥0)i∈[[1,n]] stand for nstandard monodimensional independent Brownian motions defined on a filtered probability space (Ω, F, (Ft)t≥0, P) satisfying the usual conditions
We investigate from a quantitative viewpoint what can be said for a drastic reduction of this simplified model, that is when considering 2 equations only, when the noise only acts on one component and is transmitted through the system thanks to the Hormander condition
Summary
Introduced at the end of the 70s by Malliavin, [30], [29], the stochastic calculus of variations, known as Malliavin calculus, turned out to be a very fruitful tool It allows to give probabilistic proofs of the celebrated Hormander theorem, see e.g. Stroock [37] or Norris [31]. The most striking achievement in this direction is the series of papers by Kusuoka and Stroock, [26], [27], [28] Anyhow, in those works the authors always considered “strong” Hormander conditions, that is the underlying space is assumed to be spanned by brackets involving only the vector fields of the diffusive part. We refer to the monograph of Nualart [32], from which we borrow the notations, or Chapter 5 in Ikeda and Watanabe [21], for further details
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