On the exit time from a cone for Brownian motion with drift

We investigate the tail distribution of the first exit time of Brownian motion with drift from a cone and find its exact asymptotics for a large class of cones. Our results show in particular that its exponential decreasing rate is a function of the distance between the drift and the cone, whereas the polynomial part in the asymptotics depends on the position of the drift with respect to the cone and its polar cone, and reflects the local geometry of the cone at the point where the drift is orthogonally projected.


Introduction
Let B t be a d-dimensional Brownian motion with drift a ∈ R d . For any cone C ⊂ R d , define the first exit time τ C = inf{t > 0 : B t / ∈ C}. In this article we study the probability for the Brownian motion started at x not to exit C before time t, namely, and its asymptotics (2) κh(x)t −α e −γt (1 + o(1)), t → ∞.
In the literature, these problems have first been considered for Brownian motion with no drift (a = 0). In [24], Spitzer considered the case d = 2 and obtained an explicit expression for the probability (1) for any two-dimensional cone. He also introduced the winding number process θ t = arg B t (in dimension d = 2, the Brownian motion does not exit a given cone before time t if and only if θ t stays in some interval). He proved a weak limit theorem for θ t as t → ∞. Later on, this result has been extended by many authors in several directions (e.g., strong limit theorems, winding numbers not only around points but also around certain curves, winding numbers for other processes), see for instance [21].
In [11], motivated by studying the eigenvalues of matrices from the Gaussian Unitary Ensemble, Dyson analyzed the Brownian motion in the cone formed by the Weyl chamber of type A, namely, {x = (x 1 , . . . , x d ) ∈ R d : x 1 < · · · < x d }. He also defined the Brownian motion conditioned never to exit the chamber. These results have been extended by Biane [3] and Grabiner [15]. In [2], Biane studied some further properties of the Brownian motion conditioned to stay in cones, and in particular generalized the famous Pitman's theorem to that context. In [20] König and Schmid analyzed the non-exit probability (1) of Brownian motion from a growing truncated Weyl chamber.
In [5], Burkholder considered open right circular cones in any dimension and computed the values of p > 0 such that In [8,9], for a fairly general class of cones, DeBlassie obtained an explicit expression for the probability (1) in terms of the eigenfunctions of the Dirichlet problem for the Laplace-Beltrami operator on Θ = S d−1 ∩ C, see [8,Theorem 1.2]. DeBlassie also derived the asymptotics (2), see [8,Corollary 1.3]: he found γ = 0 (indeed, the drift is zero), while α is related to the first eigenvalue and h(x) to the first eigenfunction. The basic strategy in [8,9] was to show that the probability (1) is solution to the heat equation and to solve the latter. In [1], Bañuelos and Smits refined the results of DeBlassie [8,9]: they considered more general cones, and obtained a quite tractable expression for the heat kernel (the transition densities for the Brownian motion in C killed on the boundary), and thus for (1).
We conclude this part by mentioning the work [10], in which Doumerc and O'Connell found a formula for the distribution of the first exit time of Brownian motion from a fundamental region associated with a finite reflection group.
For Brownian motion with non-zero drift, much less is known. Only the case of Weyl chambers (of type A) has been investigated. In [4], Biane, Bougerol and O'Connell obtained an expression for the probability P x [τ C = ∞] = lim t→∞ P x [τ C > t] in the case where the drift is inside of the Weyl chamber (and hence the latter probability is positive). In [23], Pucha la and Rolski gave, for any drift a, the exact asymptotics (2) of the tail distribution of the exit time, in the context of Weyl chambers too. The different quantities in (2) were determined explicitly in terms of the drift a and of a vector obtained by a procedure involving the construction of a stable partition of the drift vector.
In this article, we compute the asymptotics (2) for a very general class of cones C, and we identify κ, h(x), α and γ in terms of the cone C and the drift a. We find that there are six different regimes depending on the position of the drift with respect to (w.r.t.) the cone.
To be more specific, we will consider general cones as defined by Bañuelos and Smits in [1]. Namely, given a proper, open and connected subset Θ of the unit sphere S d−1 ⊂ R d , we consider the cone C generated by Θ, that is, the set of all rays emanating from the origin and passing through Θ: We associate with the cone the polar cone (which is a closed set) C ♯ = {x ∈ R d : x, y 0, ∀y ∈ C}.
See Figure 1 for an example. Below and throughout, we shall denote by D o (resp. D) the interior (resp. the closure) of a set D ⊂ R d . The six cases leading to different regimes are then: A. polar interior drift: a ∈ (C ♯ ) o ; B. zero drift: a = 0; C. interior drift: a ∈ C; D. boundary drift: a ∈ ∂C \ {0}; E. non-polar exterior drift: a ∈ R d \ (C ∪ C ♯ ); F. polar boundary drift: a ∈ ∂C ♯ \ {0}.
These cases will be analyzed in Theorems A, B, C, D, E and F, respectively. Our results show in particular that the exponential decreasing rate e −γ in (2) is related to the distance between the drift and the cone by the formula As for the polynomial part t −α in (2), it depends on the case under consideration and reflects the local geometry of the cone at the point(s) that minimize the distance to the drift, plus the local geometry at the contact points between ∂Θ and the hyperplane orthogonal to the drift in case F. We would like to point out that the formula for γ obtained in [23] in the case of the Weyl chamber of type A is the same as ours. Indeed, though it is not mentioned there, the vector f obtained in [23] via the construction of a stable partition of the drift is the projection of the drift on the Weyl chamber, and their formula (4.10) reads γ = |a − f | 2 /2, as the reader can check.

Assumptions on the cone and statements of results
Though our results are stated precisely in Theorems A, B, C, D, E and F, we would like to give now a brief overview as well as precise statements.
2.1. Assumptions on the cone. Our main assumption on the cones studied here is the following: With this assumption (see [6, page 169]), there exists a complete set of eigenfunctions (m j ) j 1 orthonormal w.r.t. the surface measure on Θ with corresponding eigenvalues 0 < λ 1 < λ 2 λ 3 · · · , satisfying for any j 1 where L S d−1 denotes the Laplace-Beltrami operator on S d−1 . We shall say that the cone is normal if Θ is normal. For any j 1, we set Figure 1. Cones C with opening angle β and polar cones C ♯ in dimension 2. The set Θ (the arc of circle) and its boundary are particularly important in our analysis. and (6)

Example 1. In dimension 2, any (connected and proper) open cone is a rotation of
{ρe iθ : ρ > 0, 0 < θ < β} for some β ∈ (0, 2π], see Figure 1. A direct computation starting from Equation (4) yields λ j = (jπ/β) 2 , and thus p j = α j = jπ/β, for any j 1. Further, the eigenfunctions (4) are given in polar coordinates by where the term 2/β comes from the normalization β 0 m j (θ) 2 dθ = 1. The functions m j and constants α j are particularly important in this study because they allow to write a series expansion for the heat kernel of the cone (Lemma 2) to which the non-exit probability is explicitly related (Lemma 1).
Cases A, B and C are treated with full generality under the sole assumption (C1). Thus we extend the corresponding results of Pucha la and Rolski in [23] about Weyl chambers of type A in these cases. (Note that case A is new since the polar cone of a Weyl chamber of type A has an empty interior, whereas case B has already been settled in [1], but is presented here for the sake of completeness.) Cases D, E and F will be considered under an additional smoothness assumption on the cone that excludes Weyl chambers from our analysis. The reason is that we will need estimates for the heat kernel of the cone at boundary points, and those are only available (to our knowledge) in the case of smooth cones or, on the other hand, in the case of Weyl chambers. More precisely, we shall assume in these cases that: Notice that under this assumption Θ is normal (in other words, (C1) implies (C2)).
We have already mentioned the formula for the exponential decreasing rate: and the reader can already imagine the importance of the set Indeed, the formula for the non-exit probability involves an integral of Laplace's type, and only neighborhoods of the points of Π(a) will contribute to the asymptotics. It follows by elementary topological arguments that Π(a) is a non-empty compact set. This holds if the cone is convex for example. Our final comment concerns the case F. Surprisingly, it is the most difficult: it is a mixture between cases A and B, and its analysis reveals an unexpected (at first sight) contribution of the contact points (see section 5.6 for a precise definition) between ∂Θ and the hyperplane orthogonal to the drift. Here again, we shall add a technical assumption, namely: (C4) The set of contact points Θ c is finite.
Moreover, we will consider case F only in dimension 2 (where (C4) always holds) and 3. The reason is that we are technically not able to handle more general cases.

2.2.
Main results. The following theorem summarizes our results. Some important comments may be found below.
Theorem. Let C be a normal cone in R d (hypothesis (C1)). For Brownian motion with drift a, in each of the six cases A, B, C, D, E and F, the asymptotic behavior of the non-exit probability is given by if a is an interior drift (case C), 1/2 if a is a boundary drift (case D) and Θ is real-analytic (C2), 3/2 if a is a non-polar exterior drift (case E), (C2) and (C3), p 1 /2 + 1 if a is a polar boundary drift (case F) and C is two-dimensional.
The constants κ and the functions h(x) are also explicit, but their expression is rather complicated in some cases. For this reason they are given in the corresponding sections. As a matter of example, let us give them in case A: where m 1 is defined in (4) and α 1 in (5), and where for any y = 0, we denote by y = y/|y| its projection on the unit sphere S d−1 . Above, case F is presented in dimension 2 only, because the value of α in dimension 3 is quite complicated (we refer to Theorem F for the full statement).

The example of two-dimensional Brownian motion in cones
For the one-dimensional Brownian motion and the cone C = (0, ∞), there are three regimes for the asymptotics of the non-exit probability, according to the sign of the drift a ∈ R. Precisely, for any x > 0, as t → ∞ one has, with obvious notations (see [17, section 2.8]), For some specific two-dimensional cones, the asymptotics of the non-exit probability is easy to determine. This is for example the case of the upper half-plane since this is essentially a one-dimensional case. It is also an easy task to deal with the quarter plane Q. Indeed, by independence of the coordinates (B (1) t , B (2) t ) of the Brownian motion B t , the non-exit probability can be written as the product where x = (x 1 , x 2 ). Denoting by a = (a 1 , a 2 ) the coordinates of the drift and making use of (8), one readily deduces the asymptotics P x [τ Q > t] = κh(x)t −α e −γt (1 + o(1)), as summarized in Figure 2, where the value of α is given, according to the position of the drift (a 1 , a 2 ) in the quarter plane. We focus on α and not on γ, since the value of γ is always obtained in the same way. More generally, our results show that the value of α for any two-dimensional cone is given as in Figure 3. This can be understood as follows: when the drift is negative (i.e., Figure 3. Value of α in terms of the position of the drift (a 1 , a 2 ) in the plane (case of a general cone of opening angle β, for which α 1 = π/β, see Figure 1) when it belongs to the polar cone C ♯ ), one sees the influence of the vertex of the cone (α is expressed with the opening angle β) since the trajectories that do not leave the cone will typically stay close to the origin. In all other cases, the Brownian motion will move away from the vertex, and will see the cone as a half-space (boundary drift and non-polar exterior drift) or as a whole-space (interior drift).

Preliminary results
In this section we introduce all necessary tools for our study. We first give the expression of the non-exit probability (1) in terms of the heat kernel of the cone C (see Lemmas 1 and 3). Then we guess the value of the exponential decreasing rate of this probability, by simple considerations on its integral expression. Finally we present our general strategy to compute the asymptotics of the non-exit probability.

Expression of the non-exit probability.
In what follows we consider (B t ) t 0 a d-dimensional Brownian motion with drift a and identity covariance matrix. Under P x , the Brownian motion starts at x ∈ R d .
The lemma hereafter gives an expression of the non-exit probability for Brownian motion with drift a in terms of an integral involving the transition probabilities of the Brownian motion with zero drift killed at the boundary of the cone. This is a quite standard result (see [23,Proposition 2.2] for example) and an easy consequence of Girsanov theorem. Notice that this result is not at all specific to cones and is valid for any domain in R d .
x, y) denote the transition probabilities of the Brownian motion with zero drift killed at the boundary of the cone C. We have We shall now write a series expansion for the transition probabilities of the Brownian motion killed at the boundary of C (or equivalently, see [16, section 4], for the heat kernel p C (t, x, y) of the cone C), as given in [1]. We denote by I ν the modified Bessel function of order ν: .
It satisfies the second order differential equation Its leading asymptotic behavior near 0 is given by: We refer to [25] for proofs of the facts above and for any further result.
). Under (C1), the heat kernel of the cone C has the series expansion where the convergence is uniform for (t, x, y) ∈ [T, ∞) × {x ∈ C : |x| R} × C, for any positive constants T and R.
Making the change of variables y → ty in (9) and using (12), we easily obtain the following lemma, where the expression of the non-exit probability now involves an integral of Laplace's type.
Lemma 3. Let C be a normal cone. For Brownian motion with drift a, the non-exit probability is given by The aim now is to understand the asymptotic behavior as t → ∞ of the integral in the right-hand side of (13). First, we notice that it suffices to analyze the asymptotic behavior of (14) To do this, we shall use Laplace's method [7,Chapter 5]. The basic question when applying this method is to locate the points y ∈ C where the function in the exponential reaches its minimum value, for it is expected that only a neighborhood of these points will contribute to the asymptotics. And indeed, we shall prove that the exponential decreasing rate e −γ of the non-exit probability in (2) is given, for the six cases A-F, by (3), namely Specifically, let Π(a) be the set of minimum points, that is, It follows by elementary topological arguments that Π(a) is a non-empty compact set. The lemma below shows that if the domain of integration is restricted to the complement of any neighborhood of Π(a), then the integral in (14) becomes negligible w.r.t. the expected exponential rate e −tγ . To be precise, consider the open δ-neighborhood of Π(a): where γ is the quantity defined in (3).
Proof. Let δ > 0 and define From the inequality |y| 2 (|y − a| + |a|) 2 2|y − a| 2 + 2|a| 2 , we obtain the upper bound e |y| 2 /2 ce 2|y−a| 2 , from which we deduce that Since y → |a − y| 2 /2 is coercive and continuous, its infimum on the closed set C \ Π δ (a) is a minimum. Thus, by definition of Π(a), we have In other words, there exists η > 0 such that |a − y| 2 /2 γ + η on C \ Π δ (a). Hence, for all s 0, we have This concludes the proof of the lemma.
It is now clear that the strategy to analyze the non-exit probability is to determine the asymptotic behavior of the integral I δ (t), which is defined by and to check that it has the right exponential decreasing rate e −γ , as expected. Indeed, in this case, the asymptotic behavior of I(t), and consequently that of the non-exit probability, can be derived from the asymptotics of I δ (t), as explained in the next lemma, which will constitute our general proof strategy.
Lemma 5. Suppose that g(t) is a function satisfying conditions (i) and (ii) below: (i) g(t) = κt −α e −tγ for some κ > 0 and α ∈ R; (ii) For all ǫ > 0, there exists δ > 0 such that Then Proof. It follows from Lemma 4 as an easy exercise.
In our study of I δ (t), it will be important that the elements of Π(a) be isolated from each other. By compactness, this condition is equivalent to the fact that Π(a) be finite. In that case, for δ > 0 small enough, I δ (t) decomposes into the finite sum where B(p, δ) does not contain any other minimum point than p. The contribution of each minimum point p can then be analyzed separately. The reason to do that is that we simply don't know how to handle the general case.
In most cases, it is not much of a restriction. Indeed, for a convex cone (or any convex set), the set Π(a) reduces to a single point, namely the projection p C (a) of a on C. Though the projection may not be unique in general (that is, when the cone is not convex), it is still true in cases A, B, C, D and F that Π(a) has only one element, namely p = 0 (cases A, B, F) or p = a (cases C and D), and that this point satisfies the usual property a − p, y − p 0 for all y ∈ C. Therefore, we call this point the projection and write it p C (a). The condition that Π(a) be finite is a restriction only in case E: according to the cone, the minimum could be reached at infinitely many different points, but we leave this general setting as an open problem.

Precise statements and proofs of the theorems
In this section, we study the case where the drift a belongs to the interior of the polar cone C ♯ . It might be thought of as the natural generalization of the one-dimensional negative drift case. Define (with p 1 as in (6)) The function u is the unique (up to multiplicative constants) positive harmonic function of Brownian motion killed at the boundary of C. We also define (with α 1 as in (5)) Notice that κ A is finite because a ∈ (C ♯ ) o (see Lemma 8). Our main result in this section is the following: Theorem A. Let C be a normal cone. If the drift a belongs to the interior of the polar cone C ♯ , then Proof. Since a ∈ (C ♯ ) o , the projection p C (a) is 0 and γ = |a| 2 /2. According to our general strategy, we focus our attention on {y∈C:|y| δ} e |y| 2 /2 p C (1, x, y)e −t|a−y| 2 /2 dy.
By making the change of variables v = ty and using the homogeneity of u, this expression becomes Now, since a ∈ (C ♯ ) o implies that a, v −c|v| for all v ∈ C, for some c > 0 (see Lemma 8 below), the function u(v)e a,v is integrable on C. Therefore, we can apply the dominated convergence theorem to obtain Hence, the bound for I δ (t) can finally be written as and a direct application of Lemma 5 gives The theorem then follows thanks to the expression (13) of the non-exit probability.
We now state and prove a lemma that was used in the proof of Theorem A. Similar estimates can be found in [13, section 5].
uniformly on {x ∈ C : |x| R}, for any positive constant R.
Proof. For brevity, let us write x = ρθ and y = rη, with ρ, r > 0 and θ, η ∈ Θ, and set M = ρr. It follows from the expression of the heat kernel (12) that Using then equation (20) from Lemma 7 below, we find the upper bound (below and throughout, c will denote a positive constant, possibly depending on the dimension d, which can take different values from line to line) Now, using the integral expression (10) for I α j , we obtain We conclude that Using the latter estimation in (18), we deduce that It is easily seen from equation (19) in Lemma 7 below that ∞ j=1 M α j −α 1 cosh(M ) is a uniformly convergent series for M ∈ [0, 1 − ǫ], for any ǫ ∈ (0, 1]. This implies that the series is uniformly convergent for (M, θ, η) ∈ [0, 1 − ǫ] × Θ × Θ, for any ǫ ∈ (0, 1]. Therefore we can take the limit term by term. Since uniformly in (θ, η) ∈ Θ × Θ (see (11) and Lemma 7 below), we reach the conclusion that where the convergence is uniform for (θ, η) ∈ Θ×Θ. The proof of Lemma 6 is complete.
The following facts in the lemma below, concerning the eigenfunctions (4), are proved in [1].
). If C is normal, then there exist two constants 0 < c 1 < c 2 such that In addition, there exists a constant c such that , ∀j 1, ∀η ∈ Θ.
We conclude this section with a useful characterization of the interior of the polar cone, which was used in the proof of Theorem A: Proof. Assume first that a satisfies the above condition. For all x such that |a − x| < δ and all y ∈ C, we have by Cauchy-Schwarz inequality x, y = a, y + x − a, y < −δ|y| + δ|y| = 0, hence C ♯ contains the open ball B(a, δ), and a is an interior point. Conversely, suppose that there exists r > 0 such that the closed ball B(a, r) is included in C ♯ . It is easily seen that Hence it remains to prove that γ < 0. To that aim, we select a family {x 1 , . . . , x d } of vectors of ∂B(a, r) which forms a basis of R d . One of them, say x 1 , must satisfy x 1 , u 0 < 0, since else we would have x i , u 0 = 0 for all i ∈ {1, . . . , d}, and therefore u 0 = 0. Let x 1 = 2a − x 1 be the opposite of x 1 on ∂B(a, r). Since x 1 , u 0 < 0 and x 1 , u 0 0, it follows that γ = a, u 0 = ( x 1 , u 0 + x 1 , u 0 )/2 < 0. B (zero drift). The case of a driftless Brownian motion, that we consider in the present section, has already been investigated by many authors, see [24,8,9,1]. Define (with α 1 as in (5) and u(y) as in (16))

Case
Theorem B. Let C be a normal cone. If the drift a is zero, then Although a proof of Theorem B can be found in [8,1], for the sake of completeness we wish to write down some of the details below. As we shall see, the arguments are very similar to those used for proving Theorem A.
Proof of Theorem B. We have a = 0 and γ = 0. Thus, the lower and upper bounds (17) for I δ (t) write Since the function u(v)e −|v| 2 /2 is integrable on C, it comes from the dominated convergence theorem that Hence, the bounds for I δ (t) can finally be written as The theorem then follows by an application of Lemma 5 and formula (13).

5.3.
Case C (interior drift). Now we turn to the case when the drift a is inside the cone C.
Theorem C. Let C be a normal cone. If a belongs to C, then x, a).
By the change of variables v = √ t(y − a), this expression becomes Hence, the theorem follows from Lemma 5 and formula (13).

Example 2.
In the case where C is the Weyl chamber of type A, the heat kernel is given by the Karlin-McGregor formula (see [18,Theorem 1]): . An easy computation then shows that p C (1, x, a) is equal to Hence This result was derived earlier by Biane, Bougerol and O'Connell in [4, section 5]. Indeed, in [4] the authors first find the probability P x [τ C = ∞] in the case of a drift a ∈ C via the reflection principle and a change of measure. As an application of this, they show that the Doob h-transform of the Brownian motion with the harmonic function given by the non-exit probability P x [τ C = ∞] has the same law that a certain path transformation of the Brownian motion (defined thanks to the Pitman operator, which is one of the main topics studied in [4]).

Case D (boundary drift).
In this section and the following ones, we make the additional hypothesis that the cone is real-analytic, that is, hypothesis (C2). Notice that under this assumption Θ is normal. This assumption ensures that the heat kernel can be locally and analytically continued across the boundary, and thus admits a Taylor expansion at any boundary point different from the origin. To our knowledge, for more general cones like those which are intersections of smooth deformations of half-spaces, the boundary behavior of the heat kernel at a corner point (i.e., a point located at the intersection of two or more half-spaces) is not known, except in the particular case of Weyl chambers [18,4]. This behavior will determine the polynomial part t −α in the asymptotics of the non-exit probability. The case of Weyl chambers is treated in [23]. Here, we deal with the opposite (i.e., smooth) setting.
Define the function h D (x) = e |x−a| 2 /2 ∂ n p C (1, x, a), where n stands for the inner-pointing unit vector normal to C at a, and ∂ n p C (1, x, a) denotes the normal derivative of the function y → p C (1, x, y) at y = a. Function h D (x) is non-zero thanks to Lemma 10 below. Define also the constant Theorem D. Let C be a real-analytic cone. If a = 0 belongs to ∂C, then Proof. As in case C, we have p C (a) = a and γ = 0, and the formula (15) for I δ (t) writes where f (y) = e |y| 2 /2 p C (1, x, y). In the present case, f (y) vanishes at y = a, contrary to case C. Since the function y → p C (1, x, y) is infinitely differentiable in a neighborhood of a (see Lemma 9), it follows from Taylor's formula that, for any (sufficiently small) δ > 0, there exists M > 0 such that Therefore, for any fixed δ > 0, one has Making the change of variables v = √ t(y − a) implies that the above equation is the same as Now, due to the regularity of ∂C at a, the set goes to {v ∈ R d : v, n > 0} as t → ∞. Furthermore, an easy computation shows that Hence, we deduce that Since ∂ n f (a) = e |a| 2 /2 ∂ n p C (1, x, a) = 0 by Lemma 10, Theorem D follows from Lemma 5 and formula (13).
The two following lemmas have been used in the proof of Theorem D. The first of the two lemmas follows from [19,Theorem 1], which proves analyticity of solutions to general parabolic problems, both in the interior and on the boundary. As for the second one, it is a consequence of [12,Theorem 2].  Example 1 (continued). In the particular case of the dimension 2, with a cone of opening angle β (see Figure 1), one has the following expression for the normal derivative at a: which gives a simplified expression for function h D (x). The above identity is elementary: it follows from the expression (7) of the eigenfunctions together with the definition of function f and some uniform estimates (to be able to exchange the summation and the derivation in the series defining the heat kernel).

5.5.
Case E (non-polar exterior drift). In addition to the real-analyticity (C2) of the cone, we shall assume in this section that (C3) holds, i.e., that the set Π(a) = {y ∈ C : |y − a| = d(a, C)} of minimum points is finite. Define the function where κ E (p) denotes the positive constant defined in equation (24). We shall prove the following: Proof. Since a belongs to R d \ (C ∪ C ♯ ), every p ∈ Π(a) belongs to ∂C and is different from 0 and a. Here γ = |p − a| 2 /2. Because Π(a) is finite, we can choose δ > 0 so that the balls B(p, δ) for p ∈ Π(a) are pairwise disjoint. Then I δ (t) can be written as e |y| 2 /2 p C (1, x, y)e −t|a−y| 2 /2 dy, and B(p, δ) does not contain any other element of Π(a) than p.
The beginning of the analysis of I δ,p (t) is similar to the proof of Theorem D, except that we have to make a Taylor expansion with three (and not two) terms, for reasons that will be clear later. For the same reasons as in case D, for any δ > 0 small enough, we have × e −t|y−a| 2 /2 dy, where f (y) = e |y| 2 /2 p C (1, x, y), (y −p) ⊤ is the transpose of the vector y −p, ∇ 2 f (p) denotes the Hessian matrix of f at p, and the domain of integration is {y ∈ C : |y − p| < δ}. To compute the asymptotics of the integral I δ,p (t) as t → ∞, we shall make a series of two changes of variables. First, the change of variables u = y − p and the use of the identity e −t|y−a| 2 /2 = e −tγ e −t|y−p| 2 /2−t y−p,p−a give the following alternative expression In what follows, we will assume (without loss of generality) that the inner-pointing unit normal to ∂C at p is equal to e 1 , the first vector of the standard basis. With this convention p − a = |p − a|e 1 , and the only non-zero component of ∇f (p) is in the e 1direction. Indeed, since f (y) = 0 for y ∈ ∂C, the boundary of the cone is a level set for the function f , and it is well known that the gradient is orthogonal to the level curves. Therefore, the quantity u, ∇f (p) is equal to u 1 ∂ 1 f (p).
Our last change is v = φ t (u); it sends (u 1 , u 2 , . . . , u d ) onto (tu 1 , √ tu 2 , . . . , √ tu d ). Note that the scalings in the normal and tangential directions are not the same; this entails that in (21) the second term in the integrand is not negligible w.r.t. the first one, and this is the reason why we have to make a Taylor expansion with three terms and not two. Note also that the Jacobian of this transformation is t (d+1)/2 . From this and (22) we deduce that, as t → ∞, The aim is now to understand the behavior of the domain φ t (D) as t → ∞. Since the cone C is tangent to the hyperplane {u ∈ R d : u 1 = 0} at p and its boundary is real-analytic, there exists a real-analytic function g with g(0) = 0 and ∇g(0) = 0, such that, for δ small enough, the domain D coincides with An application of Taylor formula then gives that (up to a set of Lebesgue measure zero) Let us compare the limit domain φ ∞ (D) and the integrand in equation (23). Since f vanishes on the boundary of the cone, we have f (p 1 + g(u 2 , . . . , u d ), p 2 + u 2 , . . . , p d + u d ) = 0, for any u in some neighborhood of 0. Differentiating twice this identity, we obtain Therefore, equation (23) can be rewritten as as t → ∞. Notice that the limit domain φ ∞ (D) is exactly the subset of R d where the integrand is positive. Thus, the constant x, p) = 0 by Lemma 10, we obtain that To conclude the proof of Theorem E, it suffices to sum the estimates for I δ,p (t) over p ∈ Π(a), and then to apply Lemma 5 and to use equation (13).
Example 1 (continued). In the particular case of two-dimensional cones, ∇ 2 g(0) = 0 and the limit domain of integration φ ∞ (D) is the half-space {v ∈ R 2 : v 1 0}. The constant κ E (p) can then be computed: 5.6. Case F (polar boundary drift). We finally consider the case where the drift a = 0 belongs to ∂C ♯ . Let us first notice that the existence of such a vector a implies that the cone C is included in some half-space. More precisely, by definition of the polar cone, the inner product of a with any y ∈ C is non-positive, so that C is included in the halfspace {y ∈ R d : a, y 0}. Moreover, there must exist some θ c ∈ ∂Θ = ∂(C ∩ S d−1 ) such that a, θ c = 0, for else a would belong to the interior of C ♯ , as seen in Lemma 8. We call Θ c the set of all these contact points θ c between ∂Θ and the hyperplane a ⊥ = {y ∈ R d : a, y = 0}. As we shall see, the asymptotics of P x [τ C > t] is determined by the local geometry of the cone C near these points.
We first present some general aspects of our approach, and then we will treat the case d = 2 for cones with opening angle β ∈ (0, π), and the case d = 3 for cones with a realanalytic boundary and a finite number of contact points. Other cases are left as open problems. In the sequel, we will assume (without loss of generality) that a = −|a|e d , where e d stands for the last vector of the standard basis.
As in case A, we have p C (a) = 0 and γ = |a| 2 /2, so that the formula (15) for I δ (t) can be written as {y∈C:|y| δ} Let ǫ > 0 be given. Arguing as in case A, we can pick δ > 0 small enough so that I δ (t) be bounded from above and below by (25) (1 ± ǫ)bu(x)e −|x| 2 /2 t d/2 {y∈C:|y| δ} u(y)e −t|a−y| 2 /2 dy, where b = (2 α 1 Γ(α 1 + 1)) −1 . Thus, we are led to study the asymptotic behavior of Making the change of variables z = √ ty and using the homogeneity property of u (see (16)), we obtain Now, Laplace's method suggests that only some neighborhood of the hyperplane {z ∈ R d : z d = 0} will contribute to the asymptotics. More precisely, we have the following result: Lemma 11. For any η > 0, we have Proof. Since |u(z)| M |z| p 1 , the integral above is bounded from above by which is equal to O(t −(p 1 +d)/2 ). Lemma 11 follows since p 1 > 0.
From now on, we shall assume that (C4) holds, i.e., that the set of contact points Θ c is finite.
Let η > 0 be so small that the d-dimensional balls B(θ c , η) for θ c ∈ Θ c are disjoints. Since the set of all θ ∈ Θ that do not belong to any of these open balls is compact and does not contain any contact point, there exists some η ′ > 0 such that θ d > η ′ for all such θ. For θ c ∈ Θ c , we define the cone Then C can be written as the disjoint union of these (thin) cones and of a (big) remaining cone whose points z all satisfy the inequality z d /|z| > η ′ . Thus, according to formula (26) and Lemma 11, we have Two-dimensional cones. Here the cone is C = {ρe iθ : ρ > 0, θ ∈ (0, β)} with β ∈ (0, π). Define h F (x) = e −a,x u(x) and the constant Theorem F (Case of the dimension 2). Let C be any two-dimensional cone with β ∈ (0, π). If a = 0 belongs to ∂C ♯ , then Proof. Since β < π, there is only one contact point, namely θ c = (1, 0). Let us analyze its contribution. According to (29), we have as soon as η is small enough. (In fact, the condition is arcsin η < β, and η in the integral should be tan(arcsin η).) We now proceed to the change of variables v = φ t (z) = (z 1 , √ tz 2 ), which leads to It follows from the Taylor-Lagrange inequality that (if η is small enough) there exists M such that u(1, h) = ∂ 2 u(1, 0)h + h 2 R(h), with |R(h)| M for all |h| η. Therefore, using the homogeneity of u, we obtain and |hR(h)| ηM for all (v 1 , v 2 ) ∈ D t . As t → ∞, the domain D t converges to the quarter plane R 2 + , and it follows from the dominated convergence theorem that, as t → ∞, where we have used the fact that ∂ 2 u(1, 0) = 2π/β 2 (see (7) for j = 1). For β < π, there is no other contribution and, therefore, combining equations (30), (28) and (25) shows that upper and lower bounds for I δ (t) are given by Hence, as in the other cases, the result follows from Lemma 5 and formula (13).
Remark 12. When β = π, the point (−1, 0) is a second contact point. By symmetry, its contribution is exactly the same as that of (1, 0). Hence the result of Theorem F is still valid if κ F is replaced by 2κ F .
Three-dimensional cones with real-analytic boundary. Recall that (by convention) a = −|a|e 3 and the cone C is contained in the half space {z 3 > 0}, see Figure 4. Thanks to (28), the asymptotic behavior of P x [τ C > t] will follow from the study of the contributions of the contact points θ c ∈ Θ c between ∂Θ and the hyperplane a ⊥ = {z ∈ R 3 : z 3 = 0}. As we shall see, the behavior of the integral above will depend on the geometry of Θ at the point θ c .  Contribution of one fixed contact point. Without loss of generality, let us assume that θ c = e 1 . Since the cone is tangent to the plane {z ∈ R 3 : z 3 = 0} at the point θ c and since its boundary is assumed to be real-analytic, there exists a real-analytic function g(z 2 ) with g(0) = 0 and g ′ (0) = 0, such that the intersection of C with {z ∈ R 3 : z 1 = 1} coincides (in a neighborhood of θ c ) with the set Define q = q(θ c ) = inf{n 2 : g (n) (0) = 0}, and c = c(θ c ) = g (q) (0) q! .
Since θ c is isolated from the other contact points (recall that Θ c is assumed to be finite), the function g(z 2 ) must be positive for all z 2 = 0 in a neighborhood of 0. Thus, by real-analyticity, q must be finite, even, and such that g (q) (0) > 0. Set Then we have: Lemma 13. For any δ > 0 and η > 0 small enough, the contribution of each contact point θ c to the asymptotics of the non-exit probability is given by where ∂ n u(θ c ) stands for the (inner-pointing) normal derivative of the function u at θ c .
We postpone the proof of Lemma 13 after the statement and the proof of Theorem F.
Statement of Theorem F. Let q 1 be the maximum value of q(θ c ) for θ c ∈ Θ c . We define as well as

Then we have:
Theorem F (Case of the dimension 3). Let C be a real-analytic three-dimensional cone. If a = 0 belongs to ∂C ♯ and the set of contact points Θ c between ∂Θ and the hyperplane a ⊥ is finite, then Proof. Since K δ,η,θc (t) is of order t −(1+1/(2q)) by Lemma 13, only those θ c with q(θ c ) = q 1 will contribute in (28) to the asymptotics of J δ (t). Thus, we obtain that J δ (t) = e −tγ t −(p 1 /2+1+1/(2q 1 )) κ(q 1 ) Now, equation (25) shows that bounds for I δ (t) are given by Hence, the result follows from Lemma 5 and formula (13).
We call D the limit domain in (32). It remains to analyze the behavior of the integrand in (31), i.e., to find the asymptotics of √ t for v 1 > 0, as t → ∞. To this end, we shall use a Taylor expansion of u(1, x, y) in a neighborhood of (0, 0). This can be done since it is known that the real-analyticity of Θ ensures that u can be extended to a strictly bigger cone, inside of which u is (still) harmonic, see [22,Theorem A]. Since u is equal to zero on the boundary of C, the relation u(1, z 2 , g(z 2 )) = 0 holds for all z 2 in a neighborhood of 0, and a direct application of Lemma 16 below for n = 1 and k ∈ {0, . . . , q − 1} shows that Hence, the Taylor expansion of u(1, z 2 , z 3 ) leads to and we finally obtain the upper bound (35) |u(1, z 2 , z 3 ) − (a 0,1 z 3 + a q,0 z q 2 )| C 1 |z 3 | 1+ǫ + C 2 |z 2 ||z 3 | + C 3 |z 2 | q+1 , where C 1 , C 2 , C 3 > 0 are positive constants (depending on η 0 and ǫ only).
On the other hand, the definition of C(θ c , η) ensures that for all (v 1 , v 2 , v 3 ) ∈ D t . Therefore, if η > 0 is small enough so that η + o(η) η 0 , then according to (35) we have is a function of t alone) for all (v 1 , v 2 , v 3 ) ∈ D t , and the result follows from Lemma 15 below, provided that ǫ has been chosen so small that 1 + ǫ + 1/q 2.
Proof. Let H(n, k) denote the statement that the conclusion of the lemma is true for the pair (n, k). We shall prove that • H(n, 0) holds for all n 1; • For all n 1 and k 1, H(n + 1, k − 1) implies H(n, k).