2] studied the small fluctuations of the diffusion bridge measurementsμεx,y in the limitε→0 under the assumption that(x,y) lies outside the sub-Riemannian cutting locus. We show how to extend their analysis to understand the small fluctuations of the diffusion loop measurementsμxε,x. As a result of the localization argument, Theorem 1.1 remains true under the weaker assumption that the smooth vector fields yielding the sub-Riemannian structure are only locally defined.
The additional technical result needed in our analysis is the uniform non-degeneracy of the rescaled Malliavin covariance matrices. In Section 2, we define the scaling operator, with which we rescale the fluctuations of the diffusion loop to obtain a non-degenerate boundary. In Section 3, we characterize the leading-order terms of the rescaled Malliavinc˜1εasε→0 covariance matrices and use this to prove Theorem 1.3.
Therefore, C = {Cn}n≥0 is an increasing filtering of the subalgebraC of the Lie algebra of smooth vector fields onM. The existence of an adapted diagram for filtrationCatxis ensured by [8, Corollary 3.1], which explicitly constructs such a diagram by considering the integral curves of the vector fieldsX1,. 3 Uniform non-degeneracy of the rescaled Malliavin covariance matrices We prove the uniform non-degeneracy of suitably rescaled Malliavin covariance matrices under the global condition.
This observation is important to determine the uniform non-degeneracy of the rescaled Malliavin covariance matrix sc˜1ε.
Properties of the rescaled Malliavin covariance matrix
In the following, we first gain control over the leading-order terms ofc˜ε1asε→0, which then allows us to show that the minimal eigenvalue ofc˜1ε can be bounded uniformly below in a high probability set. It turns out that we get a more consistent forc˜εt expression if we write it in terms of (xtε, vεt)t∈[0,1], which is the unique strong solution of the following system of Itô stochastic differential equations. The next lemma, which is sufficient for our purposes, does not give a clear expression for the leading-order terms of c˜ε1.
However, its proof shows how one could obtain these expressions recursively if one wishes.
This together with the integrability (3.6) ofλ−min1 implies the uniform non-degeneracy of the rescaled Malliavin covariance matrixsec˜1ε. Let μ˜0ε denote the law of the rescaled process(˜xεt)t∈[0,1]onT0 and write q(ε,0,·) for the law of v1 under the measure μ˜0ε. To obtain the rescaled diffusion bridge measures, we uniquely disintegrate μ˜0ε, with respect to the Lebesgue measure onRd, as.
Similarly, writeμ˜0 for the law of the limiting rescaled diffusion process(x˜t)t∈[0,1]opT0, denote the law ofv1underμ˜0byq¯(·) and let(μ˜0,y: y ∈ Rd ) is the unique family of probability measures that we obtain by disintegrating the measure μ˜0as. 4.2) To keep track of the paths of the diffusion bridges, fixt1,. In order to use this convergence result to draw conclusions about the behavior of the functions Gε and G0, we need Gˆε to be uniformly integrable inε∈(0,1). This is provided by the following lemma, which is proven at the end of the section.
The proof closely follows [2, Proof of Lemma 4.1], where the most important adjustments needed arise due to the higher-order scale mapδε. We exploit the graded structure induced by the sub-Riemannian structure (X1, . . . ,Xm) and we make use of the properties of a modified graph. In general, proceeding in the same way and by appealing to the Faà di Bruno formula, we iteratively prove that, for alln∈ {1,.
Since the coefficient bounds of the graded structure are uniform inτ ∈ [0,1]andε∈ (0,1], we obtain, uniform inτ andε, moment bounds of all orders for (xtε(τ),xtε,(1)( τ) ,.For some of the technical arguments that are conveyed unchanged, we simply refer the reader to [2] Proof of Lemma4.1 Let (xtε)t∈[0,1] be the process inRd and(uεt)t ∈[0 ,1] as well as (vtε)t∈[0,1]be the processes inRd ⊗(Rd)∗ which are defined as the unique strong solutions of the following system of Itô stochastic differential equations.
Moreover, we see that the nilpotent approximations (X¯1, . . ,X¯m,X¯m+1, . . ,X¯m+d) are (X˜1, . . ,X˜m, which shows that limit repeated processes on Rd associated with processes with generators εL¯ and εL have the same generator L. Since ˜ (U0, θ) and in particular the constraint (U, θ) is a fitted chart atx, it also follows that the identity map onRd is fitted plot for the filtration induced by the sub-Riemann structure(X¯1, . . ,X¯m,X¯m+1, . . ,X¯m+d)onRdat 0. Proof of Theorem 1.1 Let p¯ be Dirichlet the thermal kernel for L¯ with respect to the Lebesgue measure λonRd.
Furthermore, let μ˜0,0,Rd be the loop size onT0,0(Rd) associated with the stochastic process(˜xt)t∈[0,1]onRd, starting from 0 and with generatorL˜and letq¯ be the probability density function denote vanx ˜1. 5.1) Let pU denote the Dirichlet heat kernel inU of the constraint of LtoU and write μεx,x,U for the diffusion bridge measure onx,x(U)associated with the constraint of the operatorεLtoU. Recalling that these coordinates are not adapted to the filtration caused by (X1,X2) at 0, we begin by illustrating why this graph is not suitable for our analysis. Since dθ0: R2 → R2 is the identity, Theorem 1.1 says that the appropriately rescaled fluctuations of the diffusion loop at 0 associated with the stochastic process with generator L = 12(X21 + X22) converge weakly to the loop obtained by conditioning ( Bt1 ,−-t.
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