Brownian Motion and the Distance to a Submanifold

This is a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. It contains a variety of results, including an inequality for the Laplacian of the distance function derived from a Jacobian comparison theorem, a characterization of local time on a hypersurface which includes a formula for the mean local time, an exit time estimate for tubular neighbourhoods and a concentration inequality. The concentration inequality is derived using moment estimates to obtain an exponential bound, which holds under fairly general assumptions and which is sufficiently sharp to imply a comparison theorem. We provide numerous examples throughout. Further applications will feature in a subsequent article, where we see how the main results and methods presented here can be applied to certain study objects which appear naturally in the theory of submanifold bridge processes.


Introduction
not something which has previously been considered in the literature, the closest reference being the study of mean exit times given by Gray, Karp and Pinksy in [10]. Our main results are Theorems 1, 8 and 9. We will denote by r N the distance function and assume that there exist constants ν ≥ 1 and λ ∈ R such that the Lyapunov-like condition holds off the cut locus of N . Geometric conditions under which such an inequality arises, which allow for unbounded curvature, are given by Theorem 1 (see Corollary 2), which is derived from the classical Heintze-Karcher comparison theorem [12]. Under such assumptions we deduce a variety of probabilistic estimates, which are presented in Section 4. We do so first using a logarithmic Sobolev inequality. Such inequalities were originally studied by Gross in [11] and we refer to the article [7] for the special case of the heat kernel measure. We then prove more general estimates using the Itô-Tanaka formula of Section 3, which is derived from the formula of Barden and Le [17] and which reduces to the formula of Cranston, Kendall and March [6] in the one-point case. The basic method is similar to that of Hu in [14], who studied (uniform) exponential integrability for diffusions in R m for C 2 functions satisfying another Lyapunov-like condition. Indeed, several of our results could just as well be obtained for such functions, but we choose to focus on the distance function since in this case we have a geometric interpretation for the Laplacian inequality (1). Section 3 also includes a characterization of the local time of Brownian motion on a hypersurface and a couple of examples. The probabilistic estimates of Section 4 include Theorem 4 and its generalization Theorem 6, which provide upper bounds on the even moments of the distance r N (X t (x)) for each t ≥ 0. Using properties of Laguerre polynomials we use these estimates to deduce Theorems 7 and 8, which provide upper bounds on the moment generating functions of the distance r N (X t (x)) and squared distance r 2 N (X t (x)) for each t ≥ 0, respectively. The latter estimate improves and generalizes a theorem of Stroock. Using Theorem 8 and Markov's inequality we then deduce Theorem 9, which provides an concentration inequality for tubular neighbourhoods. Note that in this paper we do not assume the existence of Gaussian upper bounds for the heat kernel; the constants appearing in our estimates are all explicit. This paper has been presented in such a way that it should be possible for the reader to read Section 4 either before or after Sections 2 and 3.

Geometric Inequalities
We begin by deriving an inequality for the Laplacian of the distance function. This object can be written in terms of the Jacobian determinant of the normal exponential map, so we begin with a comparison theorem and inequalities for this object. The main references here are [12] and [23].
As regards the extrinsic geometry of the submanifold, denote by A ξ the shape operator associated to ξ and by λ 1 (ξ ), . . . , λ n (ξ ) the eigenvalues of A ξ . These eigenvalues are called the principle curvatures of N with respect to ξ and their arithmetic mean, denoted by H ξ , is called the mean curvature of N with respect to ξ . Let θ N : T N ⊥ → R denote the Jacobian determinant of the normal exponential map exp N : T N ⊥ → T M (which is simply the exponential map of M restricted to T N ⊥ ) and for κ, λ ∈ R define, for comparison, functions S κ , C κ , G κ and F λ κ by If κ ξ (t 1 ) is any constant such that κ ξ (t 1 ) ≤ κ ξ (t 1 ) then, as was proved by Heintze and Karcher in [12] using a comparison theorem for Jacobi fields, and which was generalized somewhat by Kasue in [16], there is the inequality for all 0 ≤ t ≤ t 1 and for n = m − 1 there is the inequality for all 0 ≤ t ≤ t 1 . Heintze and Karcher's method implies that the right-hand sides of inequalities (2), (3) and (4) are finite for all 0 ≤ t ≤ t 1 . Note that the 'empty sum is zero' convention is used to cover the case n = 0 in inequality (2).

Jacobian Inequalities
We use the Heintze-Karcher inequalities to deduce secondary estimates.
By Gronwall's inequality and the fact that θ N | N ≡ 1, this differential inequality implies an upper bound on θ N . By a change of variables, upper bounds on θ N imply upper bounds on the volumes of tubular neighbourhoods and the areas of the boundaries. In this way we can obtain estimates on these objects which are more explicit than those found in [9]. These calculations will not be presented here; they can be found in the author's doctoral thesis. To prove the proposition we will use two preliminary lemmas.
We can now prove Propostion 1.
Proof of Proposition 1 By Lemmas 1 and 2 it follows that Note that the factor (m − 1) is reasonable since an orthonormal basis of a tangent space T γ ξ M gives rise to precisely (m − 1) orthogonal planes containing the radial directionγ ξ .

Laplacian Inequalities
Denote by M(N) the largest domain in T N ⊥ whose fibres are star-like and such that the restriction of the exponential map exp N | M(N ) is a diffeomorphism onto its image. Then that image is M \ Cut(N ), where Cut(N ) denotes the cut locus of N . Recall that Cut(N ) is a closed subset of M with vol M -measure zero. With r N : M → R defined by r N (·) := d M (·, N) the vector field ∂ ∂r N will denote differentiation in the radial direction, which is defined off the union of N and Cut(N ) to be the gradient of r N and which is set equal to zero on that union. If as in [9, p.146] we define a function N : M \ Cut(N ) → R by we then have the following corollary of Proposition 1, in which c N (ξ ) denotes the distance to the cut locus along γ ξ .

Corollary 1
Suppose that there is a function κ : [0, ∞) → R such that for each ξ ∈ UT N ⊥ and t 1 ∈ (0, c N (ξ )) we have κ(t 1 ) ≤ κ ξ (t 1 ) and that the principal curvatures of N are bounded in modulus by a constant ≥ 0. Then there is the estimate Proof For each ξ ∈ UT N ⊥ and t 1 ∈ (0, c N (ξ )) we see by Proposition 1 that Since for each p ∈ M \ (N ∪ Cut(N )) there exists a unique ξ p ∈ UT N ⊥ such that γ ξ p (r N (p)) = p, the result follows for such p by setting t 1 = r N (p). For p ∈ N the radial derivative is set equal to zero in which case the result is trivial.
Furthermore, following from remarks made at the end of Section 2.1, if there is a function ρ : [0, ∞) → R such that for each ξ ∈ UT N ⊥ and t 1 ∈ (0, c N (ξ )) we have ρ(t 1 ) ≤ ρ ξ (t 1 ) then for n = 0 there is the estimate on M \ Cut(N ) and for n = m − 1 with |H ξ | ≤ for each ξ ∈ UT N ⊥ there is the estimate on M \ Cut(N ). Thus we arrive at the main results of this section, which are the following theorem and its corollary. Then for N , defined by Eq. 5, we have the inequality where ∂ ∂r N denotes differentiation in the radial direction.
Note that if n = 0 then the mean curvature is not relevant and if m = 1 then the sectional curvatures are not relevant but that the above estimates still make sense in either of these cases. Recall that we are primarily interested in the Laplacian of the distance function. If denotes the Laplace-Beltrami operator on M then, as shown in [9], there is the formula . This yields the following corollary.

Corollary 2 Under the conditions of Theorem 1 we have
For the particular case in which N is a point p, it was proved by Yau in [26] that if the Ricci curvature is bounded below by a constant R then the Laplacian of the distance function r p is bounded above by (m − 1)/r p plus a constant depending on R. Yau then used analytic techniques in [27] to prove that this bound implies the stochastic completeness of M. A relaxation of Yau's condition which allows the curvature to grow like a negative quadratic in the distance function is essentially optimal from the point of view of curvature and nonexplosion; this is why we did not feel it necessary to present Theorem 1 in terms of a general growth function, although one certainly could. We will return to this matter in Section 4.3. If in Yau's example we set (m − 1) = R, then inequality (3) and Taylor . This has the advantage of taking into account the effect of positive curvature and in turn yields the Laplacian estimate on M \Cut(p), which is different to Yau's bound. Note that inequalities (9) and (10) actually hold on the whole of M in the sense of distributions.

Local Time
In this section we show how the distance function relates to the local time of Brownian motion on a hypersurface. Since the boundaries of regular domains are included as examples, this could yield applications related to the study of reflected Brownian motion. The main references here are [4,17] and [6]. The articles [17] and [4] approach geometric local time in the general context of continuous semimartingales from the point of view of Tanaka's formula while [6] approaches the topic for the special case of Brownian motion using Markov process theory.

Itô-Tanaka Formula
Suppose that X is a Brownian motion on M defined up to an explosion time ζ and that U is a horizontal lift of X to the orthonormal frame bundle with antidevelopment B on R m . In [21] it was proved, using the theory of viscosity solutions, that the cut locus of N is given by the disjoint union of two subsetsC(N) andČ(N) where the connected components ofC(N), of which there are at most countably many, are smooth two-sided (m − 1)-dimensional submanifolds of M and whereČ(N) is a closed subset of M of Hausdorff dimension at most m − 2 (and therefore polar for X by [25]). AlsoC(N) ∪ N has vol M -measure zero so it follows that where β is a standard one-dimensional Brownian motion, by Lévy's characterization and the fact that U consists of isometries. Furthermore, points belonging toC(N) can be connected to N by precisely two length-minimizing geodesic segments, both of which are non-focal. Using these observations, it follows from [17, Theorem 1] that r N (X) is a continuous semimartingale. In particular, if τ is a stopping time with 0 ≤ τ < ζ then there exist continuous adapted nondecreasing and nonnegative processes L N (X) and LC (N) (X), whose associated random measures are singular with respect to Lebesgue measure and supported when X takes values in N andC(N), respectively, such that for all t ≥ 0, almost surely, where Here n is any unit normal vector field onC(N) and the Gâteaux derivatives D ± r N are defined for z ∈C(N) and v ∈ T z M by . A detailed explanation of precisely how formula (11) is derived from [17, Theorem 1] can be found in the author's doctoral thesis. Note that the integral appearing in formula (11) is well-defined since the set of times when X ∈ N ∪ Cut(N ) has Lebesgue measure zero. The process LC (N) (X) is given by the local time of d(X,C(N)) at zero so long as the latter makes sense, while for the one-point case the process L Cut(N) (X) coincides with the geometric local time introduced in [6]. The process L N (X), which we will refer to as the local time of X on N , satisfies where L 0 (r N (X)) denotes the (symmetric) local time of the continuous semimartingale for all t ≥ 0, almost surely. Note that the Brownian motion considered here can be replaced with a Brownian motion with locally bounded and measurable drift.

Revuz Measure
In this subsection we consider an application of a different approach, based on the theory of Markov processes. Although it is not always necessary to do so, we will assume that M is compact. We will also assume that N is a closed embedded hypersurface and that X(x) is a Brownian motion on M starting at x. The convexity based argument of [6], which was applied to the one point case, can be adapted to our situation and implies that with respect to the invariant measure vol M the local time L N (X(x)) corresponds in the sense of [22] to the induced measure vol N . By [8] this implies the following theorem, in which p M denotes the transition density function for Brownian motion.
for all t ≥ 0.
In a subsequent article we will calculate, estimate and provide an asymptotic relation for the rate of change d dt E L N t (X(x)) . By a standard change of variables, it follows that the expected value of the occupation times appearing inside the limit on the right-hand side of Eq. 12 converge to the right-hand side of Eq. 13 as ↓ 0. If one could justify exchanging this limit with the expectation, one would obtain a different proof of Theorem 2. The following corollary follows directly from Theorem 2 and basic ergodicity properties of Brownian motion.

Corollary 3 Suppose that M is compact, that N is a closed embedded hypersurface and that X is a Brownian motion on M. Then
Example 1 Suppose M = S 1 (i.e. the unit circle with the standard metric) and let X(x) be a Brownian motion starting at x ∈ S 1 . By formula (11) it follows that for t ≥ 0, where β is a standard one-dimensional Brownian motion. But r x (x) = 0 and Cut(x) is antipodal to x, which is a distance π away from x, so as dL for Thus by Eqs. 14 and 15 it follows that which implies for large times t the approximation Corollary 3 and the previous example concern the behaviour of E L N (X) for large times, in the compact case. In the next example we fix the time, instead considering the effect of expanding size of the submanifold, using balls in Euclidean space as the example.
Example 2 For r > 0 denote by S m−1 (r) the boundary of the open ball in R m of radius r centred at the origin. If X is a Brownian motion on R m starting at the origin then r S m−1 (r) (X) is a Markov process. It follows from general theory, as in [8], that formula (13) holds with N = S m−1 (r) and M = R m and therefore where (a, b) = ∞ b s a−1 e −s ds is the upper incomplete Gamma function. In this setting the process L S m−1 (r) (X) corresponds to the local time of an Bessel process of dimension m, started at the origin, at the value r. In particular, for the case m = 2 we obtain By differentiating the exponential of the right-hand side of Eq. 16 we can then deduce the curious relation lim t↑∞ log 2t where γ denotes the Euler-Mascheroni constant.

Probabilistic Estimates
In this section we combine the geometric inequalities of the first section with the Itô-Tanaka formula of the second section and deduce probabilistic estimates for the radial moments of Brownian motion with respect to a submanifold.

A Log-Sobolev Inequality Approach
It is fairly standard practice to deduce exponential integrability from a logarithmic-Sobolev inequality. In this subsection, we show how this can be done in a restricted version of the situation in which we are otherwise interested. For this, denote by {P t : t ≥ 0} the heat semigroup on M (acting on some suitable space of functions).
for all θ, t ≥ 0 and be an exhaustion of M by regular domains and denote by τ D i the first exit time of X(x) from D i . Using Itô's formula and formula (11), Corollary 2 and Jensen's inequality, we see that Bihari's inequality [5], which is a nonlinear integral form of Gronwall's inequality, implies for all t ≥ 0, by Fatou's lemma. Now, Bakry and Ledoux discovered (see [3] or [7]) that condition (17) implies the heat kernel logarithmic Sobolev inequality for all f ∈ C ∞ (M) and t > 0. By a slight generalization of the classical argument of Herbst (see [19]) it follows that for Lipschitz F with F Lip ≤ 1 and θ ∈ R we have for all t ≥ 0 while it was proved in [1] (see also [2]) that the log-Sobolev inequality (21) implies for all 0 ≤ θ < C −1 (t). Since r N is Lipschitz with r N Lip = 1, inequality (19) follows from (20) by the estimate (23) while inequality (18) is proved similarly, by applying Jensen's inequality to (20) and using the estimate (22).
To obtain exponential integrability for the heat kernel under relaxed curvature assumption we will use a different approach, which is developed in the next subsection. While the estimates (18) and (19) are, roughly speaking, the best we have under the conditions of Theorem 3, the estimate (19) does not reduce to the correct expression in R m , as we will see, and our later estimates will take into account positive curvature whereas the estimates of this subsection do not. Thus the later estimates are preferable from the point of view of geometric comparison. On the other hand, the estimates given by [24,Theorem 8.62], which concern the case N = {x}, suggest that the 'double exponential' feature of the estimates (18) and (19) (which is the inevitable result of using Herbst's argument and Bakry and Ledoux's log-Sobolev constant, as opposed to being a consequence of our moment estimates) is not actually necessary.

First and Second Radial Moments
Suppose now that X(x) is a Brownian motion on M with locally bounded and measurable drift b starting from x ∈ M, defined upto an explosion time ζ(x), and that N is a closed embedded submanifold of M of dimension n ∈ {0, . . . , m − 1}. We will assume for the majority of this section that there exist constants ν ≥ 1 and λ ∈ R such that the inequality holds off the cut locus. Unless otherwise stated, future references to the validity of this inequality will refer to it on the domain M\Cut(N ). If b satisfies a linear growth condition in r N then geometric conditions under which such an inequality arises are given by Theorem 1 (see Corollary 2), the content of which the reader might like to briefly review. In particular, there are various situations in which one can choose λ = 0. If N is a point and the Ricci curvature is bounded below by a constant R then inequality (24) holds with ν = m and λ = −R/3, as stated by inequality (10). Of course, if N is an affine linear subspace of R m then inequality (24) holds as an equality with ν = m − n and λ = 0. Note that inequality (24) does not imply nonexplosion of X(x). This is clear by considering the products of stochastically incomplete manifolds with ones which are not. We therefore use localization arguments to deal with the possibility of explosion.
be an exhaustion of M by regular domains and denote by τ D i the first exit time of X(x) from D i . Note that τ D i < τ D i+1 and that this sequence of stopping times announces the explosion time ζ(x). Then, by formula (11), it follows that for all t ≥ 0, almost surely. Since the domains D i are of compact closure the Itô integral in (26) is a true martingale and it follows that for all t ≥ 0, where exchanging the order of integrals in the last term is easily justified by the use of the stopping time and the assumptions of the theorem. Before applying Gronwall's inequality we should be careful, since we are allowing the coefficient λ to be negative. For this, note that and that the two functions are nondecreasing. If we define a function f x,i,2 by then f x,i,2 is differentiable, since the boundaries of the D i are smooth, and it follows from (27) and (28) that we have the differential inequality for all t ≥ 0. Now applying Gronwall's inequality to (29) yields for all t ≥ 0, from which the result follows by the monotone convergence theorem.

Remark 1
If one wishes to include on the right-hand of inequality (24) a term that is linear in r N , as in the estimate (9), or simply a continuous function of r N , as in the estimates (6), (7) and (8), with suitable integrability properties, then one can do so and use a nonlinear version of Gronwall's inequality, such as Bihari's inequality, to obtain an estimate on the left-hand side of Eq. 25.
We will refer the object on the left-hand side of inequality (25) as the second radial moment of X(x) with respect to N . To find an inequality for the first radial moment of X(x) with respect to N one can simply use Jensen's inequality. Note that lim λ→0 R(t)e λt = t and this provides the sense in which Theorem 4 and similar statements should be interpreted if λ = 0.

Nonexplosion
That a Ricci curvature lower bound implies stochastic completeness was originally proved by Yau in [27], as mentioned earlier. This was extended by Ichihara in [15] and Hsu in [13] to allow the Ricci curvature to grow in the negative direction in a certain way (like, for example, a negative quadratic in the distance function). Thus the following theorem is well-known in the one point case (in terms of which it can be proved). Using Theorem 4 our proof is short, so we may as well include it so as to keep the presentation of this article reasonably self-contained. So suppose that r > 0, let B r (N ) := {y ∈ M : r N (y) < r} and denote by τ B r (N) the first exit time of X(x) from the tubular neighbourhood B r (N ).

Theorem 5
Suppose that N is compact and that there exist constants ν ≥ 1 and λ ∈ R such that inequality (24) holds. Then X(x) is nonexplosive.
Proof By following the proof of Theorem 4, with the stopping times τ D i replaced by τ B i (N) , one deduces P{τ B i (N) ≤ t} ≤ r 2 N (x) + νR(t) e λt i 2 for all t ≥ 0. This crude exit time estimate implies that X(x) is nonexplosive since the compactness of N implies that the stopping times τ B i (N) announce the explosion time ζ(x).

Higher Radial Moments
Recall that if X is a real-valued Gaussian random variable with mean μ and variance σ 2 then for p ∈ N one has the formula where L α p (z) are the Laguerre polynomials, defined by the formula L α p (z) = e z z −α p! ∂ p ∂z p e −z z p+α for p = 0, 1, 2, . . . and α > −1 (for the properties of Laguerre polynomials used in this article, see [18]). In particular, if X(x) is a standard Brownian motion on R starting from x ∈ R then for all t ≥ 0. With this in mind we prove the following theorem, a special case of which is Theorem 4, which will be used in the next section to obtain exponential estimates. Theorem 4 was stated separately because it constitutes the base case in an induction argument.

Theorem 6
Suppose that there exist constants ν ≥ 1 and λ ∈ R such that inequality (24) holds and let p ∈ N. Then Proof By the assumption of the theorem we see that on M \ Cut(N ) and for p ∈ N we have and by formula (11) we have N (X s (x))ds for all t ≥ 0, almost surely, where the stopping times τ D i are defined as in the proof of Theorem 4. It follows that if we define functions f x,i,2p by then, arguing as we did in the proof of Theorem 4, there is the differential inequality for x, y ∈ H 3 κ and t > 0 where By a change of variables it follows that for each p ∈ N we have for all t ≥ 0, where 1 F 1 is the confluent hypergeometric function of the first kind, so in particular E[r 2 x (X t (x))] = 3t − κt 2 for all t ≥ 0. This ties in with the fact, proved by Liao and Zheng in [20], that on general M if X(x) is a Brownian motion starting at x ∈ M and if τ is the first exit time of X(x) from the geodesic ball B (x) then as t ↓ 0, where o(t 2 ) might depend upon and where scal(x) denotes the scalar curvature at x (since on H 3 κ the scalar curvature is constant and equal to 6κ). The asymptotics for the distance function r N have yet to be investigated; this is a direction for future research.

Exponential Estimates
Before using the estimates of the previous subsection to obtain exponential inequalities, we need the following lemma.
Proof With the stopping times τ D i as in the proof of Theorem 4, for p ∈ N with p even we see by inequality (36) that and so, by Jensen's inequality, if p is odd then It follows from this and Lemma 3, since ν ≥ 2, that which can be seen directly from the definition of 1 F 1 as a generalized hypergeometric series, we deduce that and the theorem follows by monotone convergence.

Remark 2
The right-hand side of Eq. 40 is a continuous function of t, θ and x and since the function 1 F 1 satisfies 1 F 1 it follows that if x ∈ N then the right-hand side of inequality (40) converges to the limit of the left-hand side as t ↓ 0. Furthermore, for the values of ν considered in the theorem the right-hand side of Eq. 40 grows exponentially with R(t, θ, x) (in particular .The theorem shows that under the assumptions of the theorem there is no positive time at which the left-hand side of Eq. 40 is infinite.
For |γ | < 1 the Laguerre polynomials also satisfy the identity which is proved in [18]. It follows from this identity and Eq. 31 that for a real-valued Gaussian random variable X with mean μ and variance σ 2 we have for θ ≥ 0 that so long as θσ 2 < 1 (and there is a well-known generalization of this formula for Gaussian measures on Hilbert spaces). In particular, if X(x) is a standard Brownian motion on R starting from x ∈ R then for t ≥ 0 it follows that so long as θt < 1. With this in mind we state the following theorem.
Theorem 8 Suppose there exists constants ν ≥ 1 and λ ∈ R such that inequality (24) holds. Then for all t, θ ≥ 0 such that θ R(t)e λt < 1, where R(t) := (1 − e −λt )/λ. Above it is a dotted curve, which is the analogous object for R = −1, and below it is a dashed curve, which is the analogous object for R = 1. The solid curve on the right represents the graph of the right-hand side of inequality (44) with R = 0. Above it is a dotted curve, which is the analagous object for R = −1, and below it is a dashed curve, which is the analagous object for R = 1. We have set θ = 1 6 and m = 3 in all cases and the horizontal axes represent the time t. Although not obvious from the two plots, the dotted and solid curves plotted on the left do not explode in finite time while the dotted and solid curves plotted on the right explode at times t = 3 log 3 3.3 and t = 6 respectively Proof Using inequality (36) and Eq. 43 we see that where we safely switched the order of integration using the stopping time. The result follows by the monotone convergence theorem.
Theorem 8 improves upon the estimate given by the second part of a similar theorem due to Stroock (see [24,Theorem 5.40]), since the latter concerns only the one point case, does not take into account positive curvature or the possibility of drift and does not reduce to the correct expression in flat Euclidean space.
Example 4 For a qualitative picture of the behaviour of the above estimates, fix x ∈ M and consider the case in which X(x) is a Brownian motion starting at x. Denote by R the infimum of the Ricci curvature of M and assume that R > −∞. Then inequality (10) implies that the assumptions of Theorems 7 and 8hold when N = {x} with ν = m and λ = −R/3. For these parameters we plot in Fig. 1 the right-hand sides of the inequalities (40) and (44) as functions of time for the three cases R ∈ {−1, 0, 1} with θ = 1 6 and m = 3. Note that if R > 0 then the left-hand sides of these inequalities are bounded, by Myer's theorem.
The sharpness of our estimates allows for the following comparison inequality, for which we note that if n = 0 then N is vacuously both totally geodesic and minimal, since in this case it would be an at most countable collection of isolated points.

Corollary 4
Suppose that X(x) is a Brownian motion on M starting at x and that one of the following conditions is satisfied: (I) n ∈ {0, . . . , m − 1}, the sectional curvature of planes containing the radial direction is non-negative and N is totally geodesic; (II) n ∈ {0, m − 1}, the Ricci curvature in the radial direction is non-negative and N is minimal.
If Y (y) denotes a Brownian motion on R m−n starting at y ∈ R m−n with r 2 Proof This follows directly from Theorem 8 and Corollary 2.
To find a comparison theorem which takes into account negative curvature seems harder. We can, however, perform an explicit calculation for the following special case, which compares favourably with our best estimate.

Example 5 Suppose that X(x) is a Brownian motion on H 3
κ starting at x. Then, using formulae (37) and (38), one can show that for all t > 0 and θ ≥ 0 such that θt < 1. Note that the explosion time of the right-hand side of formula (45) is independent of κ.

Concentration Inequalities
If X(x) is a Brownian motion on R m starting at x then it is easy to see that for all t > 0. Note that the right-hand side of the asymptotic relation (46) For the general setting considered in this article, we have the following theorem.

Theorem 9
Suppose there exists constants ν ≥ 1 and λ ∈ R such that inequality (24) holds and suppose that X(x) is nonexplosive. Then Proof The proof follows a standard argument. In particular, for θ ≥ 0 and r > 0 it follows from Markov's inequality and Theorem 8 that so long as θ R(t)e λt < 1. If t > 0 then choosing θ = δ(R(t)e λt ) −1 shows that for any δ ∈ [0, 1) and r > 0 we have the estimate from which the theorem follows since δ can be chosen arbitrarily close to 1 after taking the limit.
While Theorem 9 is trivial if M is compact, the concentration inequality (47) is still valid in that setting. In fact, for r > 0 suppose that ν ≥ 1 and λ ≥ 0 are constants such that the inequality 1 2 + b ≤ ν + λr 2 N holds on the tubular neighbourhood B r (N ) (such constants always exist if N is compact, by Corollary 1). Assuming that X(x) is non-explosive (which would be the case if N is compact, by Theorem 5) then the methods of this chapter can also be used to estimate quantities involving the process X(x) stopped on the boundary of the tubular neighbourhood. We will not include such calculations here, to avoid extensive repetition, but doing so actually yields the exit time estimate P sup s∈ [0,t] r N (X s (x)) ≥ r ≤ (1 − δ) − ν 2 exp r 2 N (x)δ 2R(t)(1 − δ) − δr 2 2R(t)e λt for all t > 0 and δ ∈ (0, 1), which improves inequality (47) for the λ ≥ 0 case.

Feynman-Kac Estimates
The following two propositions and their corollaries constitute simple applications of Theorems 7 and 8 and provide bounds on the operator norm of certain Feynman-Kac semigroups, when acting on suitable Banach spaces of functions. Proof Using the stopping times τ D i to safely exchange the order of integrals, we see by Jensen's inequality that The result follows from this by the monotone convergence theorem and Theorem 7, since the right-hand side of inequality (40) is nondecreasing in t (which is evident by the right-hand side of inequality (42) and the fact that R(t, θ, x) is nondecreasing in t).

Corollary 5
Suppose there exists constants ν ≥ 2 and λ ≥ 0 such that inequality (24) holds and that V is a measurable function on M such that V ≤ C(1 + r N ) for some constant C ≥ 0. Then for all t ≥ 0, where R is defined by Eq. 41.
Using Theorem 8 the following proposition and its corollary can be proved in much the same way.

Corollary 6
Suppose there exists constants ν ≥ 1 and λ ∈ R such that inequality (24) holds and that V is a measurable function on M such that V ≤ C(1 + 1 2 r 2 N ) for some constant C ≥ 0. Then for all t ≥ 0 such that CtR(t)e λt < 1.

Further Applications
Important applications of the results and methods presented in this article have been explored in the author's doctoral thesis and will feature in a subsequent article. In particular, we will show how one can use Theorem 1 and moment estimates for a certain elementary bridge process, related to Brownian motion, to deduce lower bounds and an asymptotic relation for the integral of the heat kernel over a submanifold. This object appears naturally in the study of submanifold bridge processes, for which the lower bounds imply a gradient estimate sufficient to prove a semimartingale property. This is a new area of study which could lead to future developments related to the geometries of path or loop spaces.