# Small-time fluctuations for sub-Riemannian diffusion loops

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## Abstract

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and the generator of the diffusion process is non-elliptic at that point, the deterministic Malliavin covariance matrix is always degenerate. We identify, after a suitable rescaling, another limiting Malliavin covariance matrix which is non-degenerate, and we show that, with the same scaling, the diffusion Malliavin covariance matrices are uniformly non-degenerate. We further show that the suitably rescaled fluctuations of the diffusion loop converge to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.

## Mathematics Subject Classification

58J65 60H07 35H10## 1 Introduction

The small-time asymptotics of heat kernels have been extensively studied over the years, from an analytic, a geometric as well as a probabilistic point of view. Bismut [9] used Malliavin calculus to perform the analysis of the heat kernel in the elliptic case and he developed a deterministic Malliavin calculus to study Hörmander-type hypoelliptic heat kernels. Following this approach, Ben Arous [4] found the corresponding small-time asymptotics outside the sub-Riemannian cut locus and Ben Arous [5] and Léandre [11] studied the behaviour on the diagonal. In joint work [6, 7], they also discussed the exponential decay of hypoelliptic heat kernels on the diagonal.

In recent years, there has been further progress in the study of heat kernels on sub-Riemannian manifolds. Barilari et al. [3] found estimates of the heat kernel on the cut locus by using an analytic approach, and Inahama and Taniguchi [10] combined Malliavin calculus and rough path theory to determine small-time full asymptotic expansions on the off-diagonal cut locus. Moreover, Bailleul et al. [2] studied the asymptotics of sub-Riemannian diffusion bridges outside the cut locus. We extend their analysis to the diagonal and describe the asymptotics of sub-Riemannian diffusion loops. In a suitable chart, and after a suitable rescaling, we show that the small-time diffusion loop measures have a non-degenerate limit, which we identify explicitly in terms of a certain local limiting operator. Our analysis also allows us to determine the loop asymptotics under the scaling used to obtain a small-time Gaussian limit of the sub-Riemannian diffusion bridge measures in [2]. In general, these asymptotics are now degenerate and need no longer be Gaussian.

*M*be a connected smooth manifold of dimension

*d*and let

*a*be a smooth non-negative quadratic form on the cotangent bundle \(T^*M\). Let \(\mathcal {L}\) be a second order differential operator on

*M*with smooth coefficients, such that \(\mathcal {L}1=0\) and such that \(\mathcal {L}\) has principal symbol

*a*/ 2. One refers to

*a*as the diffusivity of the operator \(\mathcal {L}\). We say that

*a*has a sub-Riemannian structure if there exist \(m\in \mathbb {N}\) and smooth vector fields \(X_1,\dots ,X_m\) on

*M*satisfying the strong Hörmander condition, i.e. the vector fields together with their commutator brackets of all orders span \(T_yM\) for all \(y\in M\), such that

*M*, which we also assume to be smooth. Note that the vector fields \(X_0,X_1,\dots ,X_m\) are allowed to vanish and hence, the sub-Riemannian structure \((X_1,\dots ,X_m)\) need not be of constant rank. To begin with, we impose the global condition

*M*with \(\Vert a(\beta ,\beta )\Vert _\infty <\infty \), and a locally invariant positive smooth measure \(\tilde{\nu }\) on

*M*such that, for all \(f\in C^\infty (M)\),

*x*and having generator \(\varepsilon \mathcal {L}\) may explode with positive probability before time 1. Though, on the event \(\{\zeta >1\}\), the process \((x_t^\varepsilon )_{t\in [0,1]}\) has a unique sub-probability law \(\mu _\varepsilon ^x\) on the set of continuous paths \(\Omega =C([0,1],M)\). Choose a positive smooth measure \(\nu \) on

*M*, which can differ from the locally invariant positive measure \(\tilde{\nu }\) on

*M*, and let

*p*denote the Dirichlet heat kernel for \(\mathcal {L}\) with respect to \(\nu \). We can disintegrate \(\mu _\varepsilon ^x\) to give a unique family of probability measures \((\mu _\varepsilon ^{x,y}{:}\,y\in M)\) on \(\Omega \) such that

*x*,

*y*) lies outside the sub-Riemannian cut locus. Due to the latter condition, their results do not cover the diagonal case unless \(\mathcal {L}\) is elliptic at

*x*. We show how to extend their analysis in order to understand the small-time fluctuations of the diffusion loop measures \(\mu _\varepsilon ^{x,x}\).

As a by-product, we recover the small-time heat kernel asymptotics on the diagonal shown by Ben Arous [5] and Léandre [11]. Even though our approach for obtaining the small-time asymptotics on the diagonal is similar to [5], it does not rely on the Rothschild and Stein lifting theorem, cf. [15]. Instead, we use the notion of an adapted chart at *x*, introduced by Bianchini and Stefani [8], which provides suitable coordinates around *x*. We discuss adapted charts in detail in Sect. 2. The chart Ben Arous [5] performed his analysis in is in fact one specific example of an adapted chart, whereas we allow for any adapted chart. In the case where the diffusivity *a* has a sub-Riemannian structure which is one-step bracket-generating at *x*, any chart around *x* is adapted. However, in general these charts are more complex and for instance, even if \(M=\mathbb {R}^d\) there is no reason to assume that the identity map is adapted. Paoli [14] successfully used adapted charts to describe the small-time asymptotics of Hörmander-type hypoelliptic operators with non-vanishing drift at a stationary point of the drift field.

*M*, we associate a linear scaling map \({\delta }_\varepsilon {:}\,\mathbb {R}^d\rightarrow \mathbb {R}^d\) in a suitable set of coordinates, which depends on the number of brackets needed to achieve each direction, and the so-called nilpotent approximations \({\tilde{X}}_1,\dots ,{\tilde{X}}_m\), which are homogeneous vector fields on \(\mathbb {R}^d\). For the details see Sect. 2. The map \({\delta }_\varepsilon \) allows us to rescale the fluctuations of the diffusion loop to high enough orders so as to obtain a non-degenerate limiting measure, and the nilpotent approximations are used to describe this limiting measure. Let \((U,\theta )\) be an adapted chart around \(x\in M\). Smoothly extending this chart to all of

*M*yields a smooth map \(\theta {:}\,M\rightarrow \mathbb {R}^d\) whose derivative \({\mathrm d}\theta _x{:}\,T_xM\rightarrow \mathbb {R}^d\) at

*x*is invertible. Write \(T\Omega ^{0,0}\) for the set of continuous paths \(v=(v_t)_{t\in [0,1]}\) in \(T_xM\) with \(v_0=v_1=0\). Define a rescaling map \(\sigma _\varepsilon {:}\,\Omega ^{x,x}\rightarrow T\Omega ^{0,0}\) by

### Theorem 1.1

*M*be a connected smooth manifold and fix \(x\in M\). Let \(\mathcal {L}\) be a second order partial differential operator on

*M*such that, for all \(f\in C^\infty (M)\),

*a*on \(T^*M\) has a sub-Riemannian structure and the smooth one-form \(\beta \) on

*M*satisfies \(\Vert a(\beta ,\beta )\Vert _\infty <\infty \). Then the rescaled diffusion loop measures \({\tilde{\mu }}_\varepsilon ^{x,x}\) converge weakly to the probability measure \({\tilde{\mu }}^{x,x}\) on \(T\Omega ^{0,0}\) as \(\varepsilon \rightarrow 0\).

We prove this result by localising Theorem 1.2. As a consequence of the localisation argument, Theorem 1.1 remains true under the weaker assumption that the smooth vector fields giving the sub-Riemannian structure are only locally defined. The theorem below imposes an additional constraint on the map \(\theta \) which ensures that we can rely on the tools of Malliavin calculus to prove it. As we see later, the existence of such a diffeomorphism \(\theta \) is always guaranteed.

### Theorem 1.2

*x*. Then the rescaled diffusion loop measures \({\tilde{\mu }}_\varepsilon ^{x,x}\) converge weakly to the probability measure \({\tilde{\mu }}^{x,x}\) on \(T\Omega ^{0,0}\) as \(\varepsilon \rightarrow 0\).

Note that the limiting measures with respect to two different choices of admissible diffeomorphisms \(\theta _1\) and \(\theta _2\) are related by the Jacobian matrix of the transition map \(\theta _2\circ \theta _1^{-1}\).

*x*and having generator \(\varepsilon \mathcal {L}\). Choose \(\theta {:}\,\mathbb {R}^d\rightarrow \mathbb {R}^d\) as in Theorem 1.2 and define \(({\tilde{x}}_t^\varepsilon )_{t\in [0,1]}\) to be the rescaled diffusion process given by

### Theorem 1.3

We see that the uniform non-degeneracy of the rescaled Malliavin covariance matrices \({\tilde{c}}_1^\varepsilon \) is a consequence of the non-degeneracy of the limiting diffusion process \(({\tilde{x}}_t)_{t\in [0,1]}\) with generator \(\tilde{\mathcal {L}}\). The latter is implied by the nilpotent approximations \({\tilde{X}}_1,\dots ,{\tilde{X}}_m\) satisfying the strong Hörmander condition everywhere on \(\mathbb {R}^d\), as proven in Sect. 2.

*Organisation of the paper* The paper is organised as follows. In Sect. 2, we define the scaling operator \({\delta }_\varepsilon \) with which we rescale the fluctuations of the diffusion loop to obtain a non-degenerate limit. It also sets up notations for subsequent sections and proves preliminary results from which we deduce properties of the limiting measure. In Sect. 3, we characterise the leading-order terms of the rescaled Malliavin covariance matrices \({\tilde{c}}_1^\varepsilon \) as \(\varepsilon \rightarrow 0\) and use this to prove Theorem 1.3. Equipped with the uniform non-degeneracy result, in Sect. 4, we adapt the analysis from [2] to prove Theorem 1.2. The approach presented is based on ideas from Azencott, Bismut and Ben Arous and relies on tools from Malliavin calculus. Finally, in Sect. 5, we employ a localisation argument to prove Theorem 1.1 and give an example to illustrate the result. Moreover, we discuss the occurrence of non-Gaussian behaviour in the \(\sqrt{\varepsilon }\)-rescaled fluctuations of diffusion loops.

## 2 Graded structure and nilpotent approximation

We introduce the notion of an adapted chart and of an associated dilation \(\delta _\varepsilon {:}\,\mathbb {R}^d\rightarrow \mathbb {R}^d\) which allows us to rescale the fluctuations of a diffusion loop in a way which gives rise to a non-degenerate limit as \(\varepsilon \rightarrow 0\). To be able to characterise this limiting measure later, we define the nilpotent approximation of a vector field on *M* and show that the nilpotent approximations of a sub-Riemannian structure form a sub-Riemannian structure themselves. This section is based on Bianchini and Stefani [8] and Paoli [14], but we made some adjustments because the drift term \(X_0\) plays a different role in our setting. At the end, we present an example to illustrate the various constructions.

### 2.1 Graded structure induced by a sub-Riemannian structure

*M*and fix \(x\in M\). For \(k\ge 1\), set

*M*by

*C*of the Lie algebra of smooth vector fields on

*M*. Consider the subspace \(C_n(x)\) of the tangent space \(T_xM\) given by

*N*the step of the filtration \(\mathcal {C}\) at

*x*.

### Definition 2.1

*x*if \(\theta (x)=0\) and, for all \(n\in \{1,\dots ,N\}\),

- (i)
\(\displaystyle C_n(x)= \hbox {span}\left\{ \frac{\partial }{\partial \theta ^1}(x), \dots ,\frac{\partial }{\partial \theta ^{d_n}}(x)\right\} ,\) and

- (ii)\(\left( \hbox {D }\theta ^k\right) (x)=0\) for every differential operator \(\hbox {D}\) of the formand all \(k>d_n.\)$$\begin{aligned} \hbox {D}=Y_1\dots Y_n\quad \text{ with }\quad Y_1,\dots ,Y_n\in \{X_1,\dots ,X_m\} \end{aligned}$$

Note that condition (ii) is equivalent to requiring that \((\hbox {D }\theta ^k)(x)=0\) for every differential operator \(\hbox {D }\in \hbox {span} \{Y_1\cdots Y_j{:}\,Y_l\in C_{i_l} \text{ and } i_1+\dots +i_j\le n\}\) and all \(k>d_n\). The existence of an adapted chart to the filtration \(\mathcal {C}\) at *x* is ensured by [8, Corollary 3.1], which explicitly constructs such a chart by considering the integral curves of the vector fields \(X_1,\dots ,X_m\). However, we should keep in mind that even though being adapted at *x* is a local property, the germs of adapted charts at *x* need not coincide.

*U*of an adapted chart, as this works better with our analysis. Define weights \(w_1,\dots ,w_d\) by setting \(w_k=\min \{l\ge 1{:}\, d_l\ge k\}\). This definition immediately implies \(1\le w_1 \le \dots \le w_d=N\). Let \(\delta _\varepsilon {:}\,\mathbb {R}^d\rightarrow \mathbb {R}^d\) be the anisotropic dilation given by

*w*, a polynomial

*g*on \(\mathbb {R}^d\) is called homogeneous of weight

*w*if it satisfies \(g\circ \delta _\varepsilon =\varepsilon ^{w/2}g\). For instance, the monomial \(y_1^{\alpha _1}\dots y_d^{\alpha _d}\) is homogeneous of weight \(\sum _{k=1}^d\alpha _k w_k\). We denote the set of polynomials which are homogeneous of weight

*w*by \(\mathcal {P}(w)\). Note that the zero polynomial is contained in \(\mathcal {P}(w)\) for all non-negative integers

*w*. Following [8], the graded order \(\mathcal {O}(g)\) of a polynomial

*g*is defined by the property

*g*is the maximal non-negative integer

*i*such that \(g\in \oplus _{w\ge i}\mathcal {P}(w)\) whereas the graded order of the zero polynomial is set to be \(\infty \). Similarly, for a smooth function \(f\in C^\infty (V)\), where \(V\subset \mathbb {R}^d\) is an open neighbourhood of 0, we define its graded order \(\mathcal {O}(f)\) by requiring that \(\mathcal {O}(f)\ge i\) if and only if each Taylor approximation of

*f*at 0 has graded order at least

*i*. We see that the graded order of a smooth function is either a non-negative integer or \(\infty \). Furthermore, for an integer

*a*, a polynomial vector field

*Y*on \(\mathbb {R}^d\) is called homogeneous of weight

*a*if, for all \(g\in \mathcal {P}(w)\), we have \(Y g\in \mathcal {P}(w-a)\). Here we set \(\mathcal {P}(b)=\{0\}\) for negative integers

*b*. The weight of a general polynomial vector field is defined to be the smallest weight of its homogeneous components. Moreover, the graded order \(\mathcal {O}(\hbox {D})\) of a differential operator \(\hbox {D}\) on

*V*is given by saying that

*V*, it holds true that

*X*on

*V*and every integer

*n*, there exists a unique polynomial vector field \(X^{(n)}\) of weight at least

*n*such that \(\mathcal {O}(X-X^{(n)})\le n-1\), namely the sum of the homogeneous vector fields of weight greater than or equal to

*n*in the formal Taylor series of

*X*at 0.

### Definition 2.2

Let *X* be a smooth vector field on *V*. We call \(X^{(n)}\) the graded approximation of weight *n* of *X*.

Note that \(X^{(n)}\) is a polynomial vector field and hence, it can be considered as a vector field defined on all of \(\mathbb {R}^d\).

### 2.2 Nilpotent approximation

Let \((U,\theta )\) be an adapted chart to the filtration induced by a sub-Riemannian structure \((X_1,\dots ,X_m)\) on *M* at *x* and set \(V=\theta (U)\). Note that, for \(i\in \{1,\dots ,m\}\), the pushforward vector field \(\theta _*X_i\) is a vector field on *V* and write \({\tilde{X}}_i\) for the graded approximation \((\theta _*X_i)^{(1)}\) of weight 1 of \(\theta _*X_i\).

### Definition 2.3

The polynomial vector fields \({\tilde{X}}_1,\dots ,{\tilde{X}}_m\) on \(\mathbb {R}^d\) are called the nilpotent approximations of the vector fields \(X_1,\dots ,X_m\) on *M*.

### Remark 2.4

The vector fields \({\tilde{X}}_1,\dots ,{\tilde{X}}_m\) on \(\mathbb {R}^d\) have a nice cascade structure. Since \({\tilde{X}}_i\), for \(i\in \{1,\dots ,m\}\), contains the terms of weight 1 the component \({\tilde{X}}_i^k\), for \(k\in \{1,\dots ,d\}\), does not depend on the coordinates with weight greater than or equal to \(w_k\) and depends only linearly on the coordinates with weight \(w_k-1\). \(\square \)

We show that the nilpotent approximations \({\tilde{X}}_1,\dots ,{\tilde{X}}_m\) inherit the strong Hörmander property from the sub-Riemannian structure \((X_1,\dots ,X_m)\). This result plays a crucial role in the subsequent sections as it allows us to describe the limiting measure of the rescaled fluctuations by a stochastic process whose associated Malliavin covariance matrix is non-degenerate.

### Lemma 2.5

### Proof

The lemma allows us to prove the following proposition.

### Proposition 2.6

The nilpotent approximations \({\tilde{X}}_1,\dots ,{\tilde{X}}_m\) satisfy the strong Hörmander condition everywhere on \(\mathbb {R}^d\).

### Proof

We conclude with an example.

### Example 2.7

## 3 Uniform non-degeneracy of the rescaled Malliavin covariance matrices

*x*in

*U*. We note that \(\theta _*X_0,\theta _*X_1,\dots ,\theta _*X_m\) are also smooth bounded vector fields on \(\mathbb {R}^d\) with bounded derivatives of all orders. In particular, to simplify the notation in the subsequent analysis, we may assume \(x=0\) and that \(\theta \) is the identity map. By Sect. 2, this means that, for Cartesian coordinates \((y_1,\dots ,y_d)\) on \(\mathbb {R}^d\) and for all \(n\in \{1,\dots ,N\}\), we have

- (i)
\(\displaystyle C_n(0)= \hbox {span}\left\{ \frac{\partial }{\partial y^1}(0), \dots ,\frac{\partial }{\partial y^{d_n}}(0)\right\} ,\) and

- (ii)\(\left( \hbox {D } y^k\right) (x)=0\) for every differential operatorand all \(k>d_n.\)$$\begin{aligned} \hbox {D }\in \left\{ Y_1\cdots Y_j{:}\, Y_l\in C_{i_l} \text{ and } i_1+\dots +i_j\le n\right\} \end{aligned}$$

### Corollary 3.1

Hence, the rescaled diffusion processes \(({\tilde{x}}_t^\varepsilon )_{t\in [0,1]}\) have a non-degenerate limiting diffusion process as \(\varepsilon \rightarrow 0\). This observation is important in establishing the uniform non-degeneracy of the rescaled Malliavin covariance matrices \({\tilde{c}}_1^\varepsilon \). In the following, we first gain control over the leading-order terms of \({\tilde{c}}_1^\varepsilon \) as \(\varepsilon \rightarrow 0\), which then allows us to show that the minimal eigenvalue of \({\tilde{c}}_1^\varepsilon \) can be uniformly bounded below on a set of high probability. Using this property, we prove Theorem 1.3 at the end of the section.

### 3.1 Properties of the rescaled Malliavin covariance matrix

### Lemma 3.2

*Y*be a smooth vector field on \(\mathbb {R}^d\). Then

### Proof

The next lemma, which is enough for our purposes, does not provide an explicit expression for the leading-order terms of \({\tilde{c}}_1^\varepsilon \). However, its proof shows how one could recursively obtain these expressions if one wishes to do so. To simplify notations, we introduce \((B_t^0)_{t\in [0,1]}\) with \(B_t^0=t\).

### Lemma 3.3

### Proof

*n*. For all \(u\in C_1(0)^\perp \), we have \(\langle u, X_i(0)\rangle =0\) because \(C_1(0)=\hbox {span}\{X_1(0),\dots ,X_m(0)\}\). From Lemma 3.2, it then follows that

*n*. \(\square \)

### 3.2 Uniform non-degeneracy of the rescaled Malliavin covariance matrices

### Lemma 3.4

### Proof

*K*,

*L*and

*m*but which are independent of \(\alpha ,\beta ,\gamma ,\delta \) and \(\varepsilon \). If we now choose \(\kappa \) and \(\chi \) in such a way that both \(\kappa \le 1/(4\max \{\kappa _1,\kappa _2\})\) and \(\chi ^3\ge 4\max \{\kappa _3,\kappa _4^{1/2}\}\), and provided that \(\chi \varepsilon ^{1/6}\le \alpha \), \(\beta =\gamma =\alpha \) as well as \(\delta =\kappa \alpha ^2\), then

As a consequence of this lemma, we are able to control \(\det \left( {\tilde{c}}_1^\varepsilon \right) ^{-1}\) on the good set \(\Omega (\alpha ,\beta ,\gamma ,\delta ,\varepsilon )\). This allows us to prove Theorem 1.3.

### Proof of Theorem 1.3

*S*and, for all \(p<\infty \), constants \(C(p)<\infty \) such that, for all \(\varepsilon \in (0,1]\),

*q*and

*r*in such a way that we can control both \(\varepsilon ^{-S/2}\alpha ^{q/2p}\) and \(\varepsilon ^{-S/2}\delta ^{-r/2p}\varepsilon ^{r/4p}\). Since \(\delta = \kappa \alpha ^2\) and \(\alpha =\chi ^{3/4}\varepsilon ^{1/8}\), we have

## 4 Convergence of the diffusion bridge measures

We prove Theorem 1.2 in this section with the extension to Theorem 1.1 left to Sect. 5. For our analysis, we adapt the Fourier transform argument presented in [2] to allow for the higher-order scaling \(\delta _\varepsilon \). As in Sect. 3, we may assume that the sub-Riemannian structure \((X_1,\dots ,X_m)\) has already been pushed forward by the global diffeomorphism \(\theta {:}\,\mathbb {R}^d\rightarrow \mathbb {R}^d\) which is an adapted chart at \(x=0\) and which has bounded derivatives of all positive orders.

*g*on \((\mathbb {R}^d)^k\) of polynomial growth and consider the smooth cylindrical function

*G*on \(T\Omega ^0\) defined by \(G(v)=g(v_{t_1},\dots ,v_{t_k})\). For \(y\in \mathbb {R}^d\) and \(\varepsilon >0\), set

### Lemma 4.1

*G*on \(T\Omega ^0\) there are constants \(C(G)<\infty \) such that, for all \(\varepsilon \in (0,1]\) and all \(\xi \in \mathbb {R}^d\), we have

With this setup, we can prove Theorem 1.2.

### Proof of Theorem 1.2

*p*and

*q*are the Dirichlet heat kernels, with respect to the Lebesgue measure on \(\mathbb {R}^d\), associated to the processes \((x_t^1)_{t\in [0,1]}\) and \(({\tilde{x}}_t^1)_{t\in [0,1]}\), respectively. From (4.5), it follows that

It remains to establish Lemma 4.1. The proof closely follows [2, Proof of Lemma 4.1], where the main adjustments needed arise due to the higher-order scaling map \(\delta _\varepsilon \). In addition to the uniform non-degeneracy of the rescaled Malliavin covariance matrices \({\tilde{c}}_1^\varepsilon \), which is provided by Theorem 1.3, we need the rescaled processes \(({\tilde{x}}_t^\varepsilon )_{t\in [0,1]}\) and \((\tilde{v}_t^\varepsilon )_{t\in [0,1]}\) defined in Sect. 3.1 to have moments of all orders bounded uniformly in \(\varepsilon \in (0,1]\). The latter is ensured by the following lemma.

### Lemma 4.2

### Proof

*t*. Observe that \(x_t^\varepsilon (0)=0\) and \(x_t^\varepsilon (1)=x_t^\varepsilon \) for all \(t\in [0,1]\), almost surely. Moreover, for \(n\ge 1\), the rescaled

*n*th derivative in \(\tau \)

*t*, almost surely. For instance, \((x_t^{\varepsilon ,(1)}(\tau ))_{t\in [0,1]}\) is the unique strong solution of the Itô stochastic differential equation

*t*. We note that \(v_t^\varepsilon (0)=I\) and \(v_t^\varepsilon (1)=v_t^\varepsilon \) for all \(t\in [0,1]\), almost surely. For \(n\ge 1\), set

*t*, almost surely. For \(n_1,n_2\in \{1,\dots ,N\}\) and \(u^1\in C_{n_1}(0)\cap C_{n_1-1}(0)^\perp \) as well as \(u^2\in C_{n_2}(0)\cap C_{n_2-1}(0)^\perp \), we have

We finally present the proof of Lemma 4.1. For some of the technical arguments which carry over unchanged, we simply refer the reader to [2].

### Proof of Lemma 4.1

*k*th component of the vector \(\sqrt{\varepsilon }{\delta }_\varepsilon ^{-1} (v_t^\varepsilon X_i(x_t^\varepsilon ))\) in \(\mathbb {R}^d\). Write \((x_t^{\varepsilon ,\eta })_{t\in [0,1]}\) for the strong solution of the stochastic differential equation (4.10) with the driving Brownian motion \((B_t)_{t\in [0,1]}\) replaced by \((B_t^{\eta })_{t\in [0,1]}\). We choose a version of the family of processes \((x_t^{\varepsilon ,\eta })_{t\in [0,1]}\) which is almost surely smooth in \(\eta \) and set

## 5 Localisation argument

In proving Theorem 1.1 by localising Theorem 1.2, we use the same localisation argument as presented in [2, Section 5]. This is possible due to [2, Theorem 6.1], which provides a control over the amount of heat diffusing between two fixed points without leaving a fixed closed subset, also covering the diagonal case. After the proof, we give an example to illustrate Theorem 1.1 and we remark on deductions made for the \(\sqrt{\varepsilon }\)-rescaled fluctuations of diffusion loops.

*M*satisfying the conditions of Theorem 1.1 and let \((X_1,\dots ,X_m)\) be a sub-Riemannian structure for the diffusivity of \(\mathcal {L}\). Define \(X_0\) to be the smooth vector field on

*M*given by requiring

*U*be a domain in

*M*containing

*x*and compactly contained in \(U_0\). We start by constructing a differential operator \(\bar{\mathcal {L}}\) on \(\mathbb {R}^d\) which satisfies the assumptions of Theorem 1.2 with the identity map being an adapted chart at 0 and such that \(\mathcal {L}(f)=\bar{\mathcal {L}}(f\circ \theta ^{-1})\circ \theta \) for all \(f\in C^\infty (U)\).

*V*,

*V*. Additionally, we see that the nilpotent approximations of \(({\bar{X}}_1,\dots ,{\bar{X}}_m,{\bar{X}}_{m+1},\dots ,{\bar{X}}_{m+d})\) are \(({\tilde{X}}_1,\dots ,{\tilde{X}}_m,0,\dots ,0)\) which shows that the limiting rescaled processes on \(\mathbb {R}^d\) associated to the processes with generator \(\varepsilon \bar{\mathcal {L}}\) and \(\varepsilon \mathcal {L}\), respectively, have the same generator \(\tilde{\mathcal {L}}\). Since \((U_0,\theta )\), and in particular the restriction \((U,\theta )\) is an adapted chart at

*x*, it also follows that the identity map on \(\mathbb {R}^d\) is an adapted chart to the filtration induced by the sub-Riemannian structure \(({\bar{X}}_1,\dots ,{\bar{X}}_m,{\bar{X}}_{m+1},\dots ,{\bar{X}}_{m+d})\) on \(\mathbb {R}^d\) at 0. Thus, Theorem 1.2 holds with the identity map as the global diffeomorphism and we associate the same anisotropic dilation \(\delta _\varepsilon {:}\,\mathbb {R}^d\rightarrow \mathbb {R}^d\) with the adapted charts \((U,\theta )\) at

*x*and (

*V*,

*I*) at 0. We use this to finally prove our main result.

### Proof of Theorem 1.1

*M*which satisfies \(\nu =(\theta ^{-1})_*\lambda \) on

*U*and let

*p*denote the Dirichlet heat kernel for \(\mathcal {L}\) with respect to \(\nu \). Write \(\mu _\varepsilon ^{0,0,\mathbb {R}^d}\) for the diffusion loop measure on \(\Omega ^{0,0}(\mathbb {R}^d)\) associated with the operator \(\varepsilon \bar{\mathcal {L}}\) and write \({\tilde{\mu }}_\varepsilon ^{0,0,\mathbb {R}^d}\) for the rescaled loop measure on \(T\Omega ^{0,0}(\mathbb {R}^d)\), which is the image measure of \(\mu _\varepsilon ^{0,0,\mathbb {R}^d}\) under the scaling map \({\bar{\sigma }}_\varepsilon {:}\,\Omega ^{0,0}(\mathbb {R}^d)\rightarrow T\Omega ^{0,0}(\mathbb {R}^d)\) given by

*U*of the restriction of \(\mathcal {L}\) to

*U*and write \(\mu _\varepsilon ^{x,x,U}\) for the diffusion bridge measure on \(\Omega ^{x,x}(U)\) associated with the restriction of the operator \(\varepsilon \mathcal {L}\) to

*U*. For any measurable set \(A\subset \Omega ^{x,x}(M)\), we have

*U*, we obtain

*B*be a bounded measurable subset of the set \(T\Omega ^{x,x}(M)\) of continuous paths \(v=(v_t)_{t\in [0,1]}\) in \(T_xM\) with \(v_0=0\) and \(v_1=0\). For \(\varepsilon >0\) sufficiently small, we have \(\sigma _\varepsilon ^{-1}(B)\subset \Omega ^{x,x}(U)\) and so (5.2) and (5.3) imply that

*x*to

*x*through \(M\setminus U\). Since \(d(x,M\setminus U,x)\) is strictly positive, it follows that

We close with an example and a remark.

### Example 5.1

### Remark 5.2

*x*, the latter is a degenerate measure which is supported on the set of paths \((v_t)_{t\in [0,1]}\) in \(T\Omega ^{0,0}\) which satisfy \(v_t\in C_1(x)\), for all \(t\in [0,1]\). Hence, the rescaled diffusion process \((\varepsilon ^{-1/2}\theta (x_t^\varepsilon ))_{t\in [0,1]}\) conditioned by \(\theta (x_1^\varepsilon )=0\) localises around the subspace \((\theta _*C_1)(0)\).

## Notes

### Acknowledgements

I would like to thank James Norris for suggesting this problem and for his guidance and many helpful discussions.

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