Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak H\"ormander type

We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak H\"ormander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expression for the bridge from 0 to 0 in time 1 of an iterated Kolmogorov diffusion.


Introduction
The small-time asymptotics for hypoelliptic diffusion processes can depend crucially on the drift term. For instance, Ben Arous and Léandre [5,6] showed that an interaction of the flow of the drift vector field with the heat diffusion can lead to an exponential decay of the heat kernel on the diagonal. The current paper discusses and illustrates the effects the drift term can have on the small-time fluctuations for hypoelliptic diffusion bridges. Bailleul, Mesnager and Norris [1] studied the small-time asymptotics of sub-Riemannian diffusion bridges outside the cut locus. Their analysis was extended by us to the diagonal, cf. [8], to describe the asymptotics of sub-Riemannian diffusion loops. Both works are concerned with hypoelliptic diffusion processes whose associated generators satisfy the so-called strong Hörmander condition and where the drift vector fields are nice enough to not affect the small-time asymptotics. In continuation of this work, we would like to analyse the small-time asymptotics for hypoelliptic diffusion bridges, where one assumes a weak Hörmander condition only. As a first step towards this goal, we determine the small-time bridge fluctuations for a model class of hypoelliptic diffusions satisfying a weak Hörmander condition, and we contrast our results with [1] and [8]. We consider the same model class for which Barilari and Paoli [3] describe the small-time heat kernel expansion on the diagonal and give a geometric characterisation of the coefficients in terms of curvature-like invariants. The corresponding model class of hypoelliptic operators already features in the pioneering work of Hörmander [9], and Lanconelli and Polidoro [13] study a notion of principal part as well as the invariance with respect to suitable groups of translations and dilations for this class of operators. Let n denote the minimal N satisfying (1.1). We study the diffusion process whose generator L is Research supported by the German Research Foundation DFG through the Hausdorff Center for Mathematics. the second order differential operator on R d given by For the linear vector field X 0 and the constant vector fields X 1 , . . . , X m on R d defined by the operator L rewrites as We further note that, for i ∈ {1, . . . , m} and k ∈ N, In the analysis of the small-time fluctuations for the corresponding hypoelliptic diffusion bridges, it is of advantage that a diffusion process with generator of the form (1.2) is always Gaussian and in particular, that its bridge processes can be written down explicitly. Additionally, unlike [1], we do not come across any cut locus phenomena for this class of diffusions. Fix x ∈ R d and let ε > 0. There exists a diffusion process (x ε t ) t∈[0,1] starting from x and having generator εL. For y ∈ R d , let (z ε t (y)) t∈[0,1] be the process obtained by conditioning (x ε t ) t∈[0,1] on x ε 1 = y. An explicit expression for the bridge process (z ε t (y)) t∈[0,1] is given in Lemma 2.2. We consider these diffusion bridges in the limit ε → 0. Using the notion of the matrix exponential of a square matrix, we set, for t ∈ [0, 1], According to [13,Proposition A.1], the Kalman rank condition (1.1) implies that the square matrix We see that this path describes the leading order behaviour of the diffusion bridge (z ε t (y)) t∈[0,1] as 1] converge weakly as ε → 0 to the zero process on the set of continuous loops Ω 0,0 .
In our discussion of examples in Section 4, we observe that the path (φ ε t (y)) t∈[0,1] can exhibit a blow-up behaviour in the limit ε → 0. Hence, this path compensates for any blow-up occurring in the process (z ε t (y)) t∈ [0,1] . In comparison to the law of large number type theorem [1, Theorem 1.1] for sub-Riemannian diffusion bridges, we note that in the weak Hörmander setting the minimal-like path (φ ε t (y)) t∈[0,1] depends on ε > 0. However, as in [1, Section 2], the path (φ ε t (y)) t∈[0,1] can still be obtained as projection of a solution to an appropriate Hamiltonian system. Let us consider the Hamiltonian H ε : T * R d → R given by The description in [3, Section 2] implies that (φ ε t (y)) t∈[0,1] is the projection onto R d of the unique solution in T * R d to the Hamiltonian equations associated with H ε subject to starting in T * x R d at time 0 and ending in T * y R d at time 1. Theorem 1.1 is a consequence of our study of the small-time fluctuations for the bridge (z ε t (y)) t∈[0,1] . To state our fluctuation result, cf. Theorem 1.2, we first introduce a basis for R d which simplifies the analysis, also see [3] and [13]. For k ∈ {1, . . . , n}, set As detailed in Lemma 3.1, in the limit ε → 0 and in our chosen basis, U ε (r) takes the form where u k is a (d k − d k−1 )× m matrix with constant entries. Here we use the convention that d 0 = 0. Let D ε and J t be the d × d diagonal matrices whose j th diagonal element, for d k−1 < j ≤ d k , equals ε k−1 and t k−1/2 , respectively. The natural rescaled fluctuation process to study is (F ε t ) t∈[0,1] given by , where we show that the fluctuations indeed neither depend on x ∈ R d nor on y ∈ R d . As in [8] and due to (1.3), the orders of ε which we rescale the fluctuations by are determined in terms of a filtration induced by the commutator brackets of the vector fields X 0 , X 1 , . . . , X m . To describe the limit fluctuation process, we set, for r ∈ R, and further introduce the d×d matrix V which is an n×n block matrix whose (k, l) th block element As established in Lemma 3.3, the matrix V is invertible. This allows us to describe the small-time fluctuations for the bridge process (z ε t (y)) t∈[0,1] as follows.
Then, for all x, y ∈ R d , the rescaled fluctuations (F ε t ) t∈[0,1] converge weakly to (F t ) t∈[0,1] as ε → 0. It is of interest by itself that after compensating for a blow-up in the process (z ε t (y)) t∈[0,1] through the path (φ ε t (y)) t∈[0,1] , the small-time fluctuations do not exhibit any further blow-ups as ε → 0. Moreover, the example discussed in Section 4.2 demonstrates that, while the bridge processes and the rescaled fluctuations can always be computed explicitly due to the Gaussian nature of the considered diffusion, Theorem 1.2 indeed simplifies the determination of the small-time fluctuations for the bridge. We observe that since D ε , J t ,Û (r) and V are uniquely determined in terms of n ∈ N and u 1 , . . . , u n , processes which give rise to the same n ∈ N and u 1 , . . . , u n for the same orthonormal basis of R d exhibit the same small-time fluctuations for the bridge, according to Theorem 1.2. A formulation of this property in terms of the generator L is given in Remark 3.4. It is similar to [8] where, in a suitable chart, the small-time fluctuations for sub-Riemannian diffusion loops only depend on the nilpotent approximations of the vector fields X 1 , . . . , X m . The paper is organised as follows. In Section 2, we discuss in more detail the hypoelliptic diffusions in our model class, and we derive an expression for the associated bridge processes. The small-time analysis, which leads to the proofs of Theorem 1.1 and Theorem 1.2, is then performed in Section 3. We close by presenting a collection of examples in Section 4 to illustrate our results. As part of the discussions in Section 4.4, we find an explicit expression for the bridge from 0 to 0 in time 1 of an iterated Kolmogorov diffusion.

Diffusion bridge in the model class
We analyse the diffusion processes whose generators are of the form (1.2) for matrices A and B satisfying condition (1.1). We further derive explicit expressions for the associated bridge processes. Let (W t ) t∈[0,1] be a standard Brownian motion in R m , which we assume is realised as the coordinate process on the path space 1] be the unique strong solution to the Itô stochastic differential equation in R d 1] has generator εL, where L is given by (1.2). From the discussions in the Introduction, we know that operators of this form satisfy a weak Hörmander condition and hence, that (x ε t ) t∈[0,1] is a hypoelliptic diffusion. It has the explicit expression as can be checked by direct computation. We see that (x ε t ) t∈[0,1] is a Gaussian process with e εtA x and whose covariance structure is given as follows in terms of Γ ε t defined by (1.4).
Proof. Using the expression (2.1), the property (2.2) and the Itô isometry, we obtain as claimed.
With the covariance structure for the Gaussian process (x ε t ) t∈[0,1] at hand, we can find an explicit expression for the corresponding bridge processes. The derivation relies on the fact that Gaussian random variables are independent if and only if they are uncorrelated.
Then, for y ∈ R d , the stochastic process (z ε t (y)) t∈[0,1] in R d given by has the same law as the process ( Proof. For all t ∈ [0, 1], we can write Applying Lemma 2.1, we compute that The analysis of this expression in the limit ε → 0 is performed in the next section.

Small-time analysis for the model diffusion bridge
We study the dependence of e εrA B and α ε t given by (2.3) on ε → 0 and then use the expression (2.5) to give the proofs of Theorem 1.1 and Theorem 1.2. Recall from (1.7) that, for r ∈ R, we define In a suitable basis, U ε (r) takes the following form.
Lemma 3.1. Let {e 1 , . . . , e d } be an orthonormal basis of R d such that {e 1 , . . . , e d k } is a basis of the subspace E k given by (1.6), for k ∈ {1, . . . , n}. In such a basis, U ε (r) has the form, as ε → 0, Proof. Write ·, · for the standard inner product on R d . Since E k is the subspace of R d spanned by the columns of A l B for l ∈ {0, . . . , k − 1}, these columns can be written as a linear combination of the vectors e 1 , . . . , e d k . It follows that, for j ∈ {d k + 1, . . . , d} and for all v ∈ R m , Due to the properties of the matrix exponential, we have, as ε → 0, uniformly in r ∈ R on compact intervals. By using (3.2) we obtain that, for all j ∈ {d k + 1, . . . , d} and all v ∈ R m , uniformly in r on compact intervals. This establishes that U ε (r) is indeed of the form (3.1).
We work in such an orthonormal basis of R d which respects the filtration of subspaces {E k } 1≤k≤n for the remainder of the section. According to Lemma 3.1, for the rescaling matrix D ε and forÛ (r) defined by (1.9), we have uniformly in r ∈ R on compact intervals. We deduce that, uniformly in t ∈ [0, 1], We use the following lemma to obtain a concise expression of Proof. We prove this identity by induction over k ∈ N with l ∈ N fixed. For k = 1, we compute which settles the base case for all t ∈ [0, 1]. To establish the induction step, consider the functions We The induction hypothesis implies that for all t ∈ [0, 1]. Due to f k (0) = 0 = g k (0), the result follows upon integrating (3.6).
For t ∈ [0, 1], the matrix t 0Û (t − s)Û (−s) * ds is an n × n block matrix whose (k, l) th block element is the (d Using Lemma 3.2 we deduce that, with the n×n block matrix V defined by (1.10) and the rescaling matrix J t , Following on from (3.5), we end up with the expression uniformly in t ∈ [0, 1]. To use (3.8) to obtain an alternative expression for α ε t , we first show that the square matrix V is invertible. Lemma 3.3. The n × n block matrix V whose (k, l) th block element is given by (1.10) is invertible.
Proof. As shown in [13, Proposition 2.1], in our chosen basis of R d , the matrix A takes the form of an n×n block matrix whose (k, l) th block element, for k, l ∈ {1, . . . , n}, is a (d k −d k−1 )×(d l −d l−1 ) matrix, where all the blocks with k ≥ l + 2 vanish. LetÂ be an n × n block matrix of the same block structure. We set its block elements to zero unless k = l + 1, in which case we set that block element to equal the (k, l) th block element of A. By definition of the subspace E 1 of R d , we further observe that in our chosen basis, for all j ∈ {d 1 + 1, . . . , d}, the j th row of B vanishes.
For l ∈ {1, . . . , n − 1}, let A l denote the (l + 1, l) th block element of the matrix A and let B 1 be the d 1 × m matrix obtained by considering the first d 1 rows of B only. For k ∈ {1, . . . , n}, we set By construction ofÂ, the d×m matrixÂ l B is an n×1 block matrix, whose (k, 1) th block element is a (d k − d k−1 ) × m matrix, which vanishes unless k = l + 1, in which case it equals A l · · · A 1 B 1 . From this form it follows that, in the chosen basis {e 1 , . . . , e d } of R d , we have, for all l ∈ {0, . . . , n − 1} and all v ∈ R m , (3.9) e j ,Â l Bv = 0 unless j ∈ {d l + 1, . . . , d l+1 } .
Moreover, for l ≥ n, we obtainÂ l B = 0, which implies that, for r ∈ R, Combining (3.9) and (3.10) yields, for all v ∈ R m , (3.11) e j , e εrÂ Bv = ε l r l l! e j ,Â l Bv for j with d l < j ≤ d l+1 .
After understanding A l B as an n × 1 block matrix of the same structure as the matrixÂ l B, we further see that the (l + 1, 1) th block element of A l B also equals A l · · · A 1 B 1 . This is a consequence of the observation that a block element in A with k ≥ l + 2 vanishes. In particular, for v ∈ R m and j with d l < j ≤ d l+1 , we have e j ,Â l Bv = e j , A l Bv , and (3.3) together with (3.11) implies that, for r ∈ R, Using (3.7), we conclude that (1 − s)Û (−s) * ds = eÂ 1 0 e −sÂ BB * e −sÂ * ds .
Our discussion above shows that E k =Ê k , for all k ∈ {1, . . . , n}, and especiallyÊ n = R d . Therefore, the matricesÂ and B satisfy the Kalman rank condition, which ensures that 1 0 e −sÂ BB * e −sÂ * ds is invertible. Since eÂ has the matrix inverse e −Â , the invertibility of V follows.
For completeness, we note that Lemma 3.3 implies that, for all k ∈ {1, . . . , n}, the matrix u k has maximal rank. If it did not then, since d k − d k−1 ≤ m by construction, its rows would be linearly dependent leading to V having a collection of linearly dependent rows, which is not possible.
Remark 3.4. LetÂ be the d × d matrix constructed from the matrix A as in the previous proof, and letL be the operator on R d given bŷ In [13], the operatorL− ∂ ∂t is called the principal part of L− ∂ ∂t , and it is shown that the fundamental solution with pole at zero of L − ∂ ∂t can be controlled in terms of the fundamental solution with pole at zero ofL − ∂ ∂t , cf. [13,Theorem 3.1]. Similarly, let us callL the principal part of L. In our model class of hypoelliptic diffusions the small-time fluctuations for the bridge are given by Theorem 1.2 in terms of D ε , J t ,Û (r) and V , which due to the proof of Lemma 3.3 can be uniquely determined fromÂ and B. Therefore, the small-time fluctuations for the bridge are fully governed by the principal partL of the generator L. A similar property was observed in [8].
We now proceed with our analysis to find an alternative expression for α ε t . Since the set of invertible matrices is open, Lemma 3.3 shows that, for ε > 0 sufficiently small, the inverse of V + O (ε) exists. It satisfies From (3.8) and as J 1 equals the identity matrix, it follows that and therefore, Hence, the covariances of the mean-zero Gaussian processes (F ε t ) t∈[0,1] converge uniformly as ε → 0 to the covariance of the mean-zero Gaussian process (F t ) t∈[0,1] given by From [12,Section 3], it follows that the rescaled fluctuations (F ε t ) t∈[0,1] indeed converge weakly to (F t ) t∈[0,1] as ε → 0.
Before we move on to a discussion of four examples in the following section, we make an observation regarding the process By integration by parts, we have that, for k ∈ N, Thus, the process (3.14) can be expressed solely in terms of the matrices u 1 , . . . , u n and an iterated Kolmogorov diffusion, that is, a standard Brownian motion together with a finite number of its iterated time integrals. Since the iterated Kolmogorov diffusion arises as a canonical example, we determine its small-time fluctuations for the bridge in Section 4.4. The Kolmogorov diffusion is discussed separately as a first example in Section 4.1 as it already exhibits interesting features.

Illustrating examples
We discuss four examples which illustrate different aspects of Theorem 1.1 and Theorem 1.2.
4.1. Kolmogorov diffusion. The Kolmogorov diffusion, named after Kolmogorov [11], is the simplest example of a stochastic process which satisfies a weak Hörmander condition but not the strong Hörmander condition. It is the diffusion (x t ) t∈[0,1] in R 2 which pairs a standard Brownian motion (W t ) t∈[0,1] in R with its time integral, that is, It is the unique strong solution to the stochastic differential equation subject to x 0 = 0. This process falls into our model class of hypoelliptic diffusions by taking A = 0 0 1 0 and B = 1 0 , which corresponds to the operator on R 2 . The Kalman rank condition (1.1) is satisfied because We first use Lemma 2.2 to determine the expressions for the associated diffusion bridges in small time to then explicitly see that Theorem 1.1 and Theorem 1.2 hold. For ε > 0, the rescaled Kolmogorov diffusion (x ε t ) t∈[0,1] with generator εL is given by Since A 2 = 0, we obtain, for r ∈ R, e εrA = I + εrA = 1 0 εr 1 .
Since we consider the rescaled fluctuations (F ε t ) t∈[0,1] defined by (1.8) this corresponds to rescaling the first component by ε 1/2 and the second component by ε 3/2 , as above. We further obtain that By integration by parts, we have This together with the computation shows that Theorem 1.2 indeed yields the same small-time fluctuations for a Kolmogorov bridge as derived above. Irrespective of the initial and final positions, the small-time fluctuations are equal in law to a Kolmogorov bridge from 0 to 0 in time 1.

4.2.
Ornstein-Uhlenbeck process paired with its area. Performing the small-time analysis for the bridge of an Ornstein-Uhlenbeck process paired with its area demonstrates that Theorem 1.2 can greatly simplify the determination of the small-time fluctuations for the bridge. Let (W t ) t∈[0,1] be a standard Brownian motion in R and fix x ∈ R 2 . We consider the diffusion (x t ) t∈[0,1] in R 2 which is the unique strong solution to the stochastic differential equation In the following, we first use Lemma 2.2 to find explicit expressions for the corresponding bridge processes in small time to then determine the small-time fluctuations for the bridge by hand, before we show that Theorem 1.2 greatly simplifies the analysis. Using A k = (−1) k−1 A for k ∈ N, we compute, for ε > 0 and r ∈ R, e εrA = e −εr 0 1 − e −εr 1 .

4.3.
Compensating for blow-ups in the bridge process. While the small-time fluctuations for the bridge are uniquely determined in terms of the matrices u 1 , . . . , u n , we present an example which shows that knowledge of u 1 , . . . , u n is not sufficient to construct a path which approximates the minimal-like path well enough to recover the limit fluctuations as in the previous section. We consider the hypoelliptic diffusion corresponding to the matrices uniformly in r on compact intervals. Thus, as for the Ornstein-Uhlenbeck process paired with its area and the Kolmogorov diffusion, U ε (r) is of the form (3.1) with u 1 = u 2 = 1. By Theorem 1.2, these three processes exhibit the same small-time fluctuations for the bridge. We further compute that, for t ∈ [0, 1], which has the expansion , uniformly in t ∈ [0, 1]. Setting 1]. Let I denote the 2 × 2 identity matrix. Since V is invertible, we deduce that, for ε > 0 sufficiently small, and therefore, due to J 1 = I, We compute as well as In particular, for y = (a, b), we obtain It follows that in our current example for an approximate minimal-like path to lead to well-defined small-time fluctuations for the bridge from x = 0 to y = (a, b) with respect to the rescaling D ε , we have to at least subtract the path This differs from the minimal-like path (φ ε t (y)) t∈[0,1] considered for the Kolmogorov diffusion, and the approximate minimal-like path (ψ ε t (y)) t∈[0,1] found for the Ornstein-Uhlenbeck process paired with its area starting from 0. 4.4. Iterated Kolmogorov diffusion. The diffusions studied in Section 4.2 and Section 4.3 both exhibit the same small-time fluctuations for the bridge as the Kolmogorov diffusion. Similarly, there is a family of diffusions which all have the same small-time fluctuations for the bridge as the iterated Kolmogorov diffusion, that is, a standard Brownian motion together with a finite number of its iterated time integrals. Banerjee and Kendall [2] study maximal and efficient couplings for iterated Kolmogorov diffusions, and Baudoin, Gordina and Mariano [4] obtain gradient bounds for this hypoelliptic diffusion. We close by explicitly determining the small-time fluctuations for the bridge of an iterated Kolmogorov diffusion. By the independence of the components of a Brownian motion in R m , it is sufficient to focus on a standard Brownian motion in R and its iterated time integrals. In our model class, this diffusion corresponds to the choice of the d × d matrix A and the d × 1 matrix B, understood as a column vector, whose entries are, for i, j ∈ {1, . . . , d}, is again the iterated Kolmogorov diffusion. Using Theorem 1.2, this observation and the formula for J t V J t V −1 together give an explicit expression of the small-time fluctuations for the bridge of an iterated Kolmogorov diffusion. Moreover, since U ε (r) = D εÛ (r) for r ∈ R, these small-time fluctuations are equal in law to the bridge from 0 to 0 in time 1 of an iterated Kolmogorov diffusion with the same number of iterated time integrals.