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Tails of exit times from unstable equilibria on the line

For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.


Introduction
In a recent paper [BPG17], we studied tails of diffusion exit times from neighborhoods of unstable critical points, in the limit of vanishing noise, in one dimension. The typical exit time τ ε in this setting (see the detailed description of the setting below) is of the order of 1 λ log 1 ε , where λ > 0 is the local expansion coefficient of the linearization of the system near the critical point, and ε ↓ 0 is the noise magnitude. The main result of [BPG17] is that the following polynomial asymptotics holds for a class of initial conditions near the critical point: P τ ε > α λ log 1 ε = cε α−1 (1 + o(1)), ε ↓ 0, (1.1) for α > 1, with an explicit dependence of the factor c on the initial condition and the parameters of the model. This result is a part of an ongoing effort to understand the long-term properties of multidimensional diffusions in the context of noisy heteroclinic networks, including the limiting behavior of invariant distributions associated with such systems in the vanishing noise limit. The typical behavior in such settings is understood for time scales logarithmic in ε −1 , see [Bak10], [Bak11], [AB11]. To see what happens in the long run though, one has to quantify rare events responsible for transitions that are atypical for the logarithmic time scale. Our work in progress shows that these rare events punctuating the long-term dynamics near noisy heteroclinic networks occur exactly due to atypically long stays near unstable critical points, so that the resulting picture is similar to that of metastability but with polynomial transition rates in place of exponential ones. Thus the result of the form (1.1) and its ramifications will be crucial for that program. However, the technique we used in [BPG17] to analyze densities of auxiliary random variables, was based on heavy tools from Malliavin calculus. That approach somewhat obscures the reason why this result is true and does not seem to be tractable when applied to the study of the analogous exit problem in the neighborhood of a hyperbolic saddle in R d , d > 1, i.e. when both attracting and repelling directions are present.
In the present note, our goal is to give a new proof of this result that (a) is based on a more precise description of the dynamics at small scales, (b) uses more elementary tools of stochastic calculus, and (c) has a strong potential to be applicable in higher dimensions. In fact, we prove a slightly more general result on probabilities of the form P τ ε > α λ log 1 ε + t , α > 1, for all t ∈ R instead of t = 0 considered in [BPG17]. It turns out that, asymptotically, the dependence on t is exponential, which implies that for any T ∈ R, τ ε − α λ log 1 ε −T conditioned on τ ε − α λ log 1 ε > T converges in distribution to an exponential random variable. This phenomenon is a manifestation of loss of memory in the system under conditioning and it is consistent with the fact that τ ε − 1 λ log 1 ε converges in distribution to a random variable with exponentially decaying right tails.
An important ingredient in this note is a conditional equidistribution result (Lemma 3.2) that states that the distribution of the diffusion, conditioned on no exit from a small interval, converges to the uniform distribution. Thus our new approach is closer in the spirit to the quasi-stationary distribution considerations, see [CV16], but the existing general theory does not provide answers for us since in our situation both the system and the domain depend on ε, and the time scales we are interested in are too short for the t → ∞ limit to be a good approximation while taking ε ↓ 0.
Let us be more precise now. We consider the family of stochastic differential equations (1.2) on a bounded interval I = [q − , q + ] ⊆ R, where the drift is given by a vector field b ∈ C 2 (R) and the random perturbation is given via a standard Brownian motion W with respect to a filtration (F t ) t≥0 defined on some probability space (Ω, F, P) under the usual conditions. The noise magnitude is given by a small parameter ε > 0 in front of the diffusion coefficient σ, which is assumed to be Lipschitz and satisfy σ(0) > 0. Although we are interested only in the evolution within I, we can assume that b and σ are globally Lipschitz without changing the setting. Standard results on stochastic differential equations (see, e.g., [KS91, Chapter 5]) imply that for any starting location X ε (0) ∈ I, the equation (1.2) has a unique strong solution up to the exit time from I, τ ε I = inf{t ≥ 0 : X ε (t) ∈ ∂I}. Let (S t ) t∈R be the flow generated by the vector field b, i.e., x(t) = S t x 0 is the solution of the autonomous ordinary differential equatioṅ We assume that there is a unique repelling zero of the vector field b on R, which, without loss of generality, we place at the origin. In other words, we assume that b(0) = 0 and, for some λ > 0 and η ∈ C 2 (I), Note that since the origin is the only zero of b in the closed interval I, this assumption implies that for all x = 0, there is a uniquely defined finite time T (x) such that S T (x) ∈ ∂I. Under the condition (1.3), the map f : I → R defined by is an order preserving C 2 -diffeomorphism (see [Eiz84]). In particular, f (q − ) < 0 < f (q + ). Under the above assumptions, a version of the following theorem was proved in [BPG17]. In its statement and throughout the paper we use (1.5) Theorem 1.1. Consider X ε defined by (1.2) with initial condition X ε (0) = εx and let K(ε) be any function that satisfies lim Then, for all α > 1 and all t ∈ R, In particular, for any T ∈ R, where ⇒ stands for weak convergence, and exp λ is the exponential distribution with rate λ > 0, i.e., exp λ [t, ∞) = e −λt for t ≥ 0.
Remark 1.1. We say that a function satisfying (1.6) grows subpolynomially at 0. We say that a function c(ε) decays subpolynomially at 0 if 1/c(ε) grows subpolynomially at 0. For brevity, we will usually omit the reference to 0.
Remark 1.2. The theorem is stated for initial conditions that are at most of the order of ε away from the origin up to a subpolynomial factor. The case of initial conditions of the order of ε β for β < 1 is less interesting since then the tails of exit times decay as stretched exponentials of ε −1 instead of the power decay given by (1.7) (see Proposition 2.1).
The brief outline of our approach to the proof of this theorem is as follows. It is convenient to work in coordinates given by the function f defined in (1.4) where the drift is linear. We study the dynamics of the linear process in two separate phases: (1) in a neighborhood of the critical point of radius ε β for β ∈ (0, 1); (2) between leaving this small neighborhood and reaching the boundary of f (I).
In the second stage, the drift dominates the noise, and the process closely follows the corresponding deterministic trajectory one obtains by setting ε = 0. The outcome of the first stage, i.e., the exit from [−ε β , ε β ], is determined though by a delicate interplay between the noise and the drift in an even smaller neighborhood of the origin (β can be chosen arbitrarily close to one). We study this regime by introducing an auxiliary process Z ε (t) with constant diffusion coefficient approximating Y ε (t) = f (X ε (t)) pathwise at least over time intervals that are not too large and for which Theorem 1.1 is easier to establish. While Z ε (t) and Y ε (t) are not, in general, close on longer timescales, we prove a useful approximation result under conditioning on not having exited the interval under consideration. This allows us to finish the proof using an iterative scheme.
The plan of the paper is as follows. In Section 2, we perform the aforementioned change of variables to linearize the drift and prove Theorem 1.1 using an intermediate result on the exit from a small neighborhood of the origin. In Section 3, we introduce an auxiliary process, which is fully linear and thus allows us to derive certain properties of the exit problem through explicit calculations. In Section 4, we prove an approximation result which allows us to transfer these properties from the fully linear process to the case where only the drift is linear as long as the timescales involved are not too large. Finally, in Section 5, we use an iterative scheme to lift this limitation thereby finishing the proof of the intermediate result.
Acknowledgment. Yuri Bakhtin gratefully acknowledges partial support from NSF via grant DMS-1811444.
2 Proof of Theorem 1.1 As outlined above, we study the system first in a small neighborhood of the origin and then after the process has escaped this small neighborhood.
Let us start with the first part. The diffeomorphism f : I → R introduced in (1.4) and its inverse g = f −1 provide a conjugation between the flow (S t ) and a linear flow: Note that the integrand in (1.4) is quadratic when x is close to zero and thus we have f (0) = 0 and f ′ (0) = 1. Outside of I, we define f so that f ′ and f ′′ are bounded. Let Y ε (t) = f (X ε (t)) for times prior to the escape from I. Itô's formula and (2.1) then imply that this process satisfies the stochastic differential equation for t < τ ε I , whereσ(y) = f ′ (g(y))σ(g(y)) and h(y) = f ′′ (g(y))σ 2 (g(y)). Due to boundedness of f ′ and f ′′ ,σ and h are also bounded.
By Duhamel's formula, Y ε satisfies the integral equation Due to our conventions on f ′ , f ′′ outside of I, the processes U ε (t) and V ε (t) are defined for all t ≥ 0. Moreover, the boundedness of h immediately implies the boundedness of V ε (t): where · ∞ is the sup-norm on [0, ∞). The boundedness ofσ yields a similar conclusion about the quadratic variation of U ε (t). Hence, the existence of constants c 1 , c 2 , N 0 > 0 such that is implied by the following exponential martingale inequality (see, e.g., Problem 12.10 in [Bas11]): Let us take β ∈ (0, 1) and set V = g [−ε β , ε β ] ⊆ I. The following result describes the tail behavior of τ ε V , the exit time from V. In particular, it says that, in the ε ↓ 0 asymptotics, the choice of the exit direction is distributed symmetrically independently of the exit time.
We give the proof of Theorem 2.1 in Section 5. After exit from V, the deterministic dynamics dominates the evolution, which means that the exit time will be close to the time it takes for X 0 (t) to exit I starting at g ±ε β . This is captured by the following standard large deviation estimates.
Then for every β ′ ∈ (0, β) and subpolynomially decaying function c(ε) > 0, we have Proof: We start by showing that with overwhelming probability the exit happens through the endpoint that is on the same side as the starting point. Indeed, (2.3)and (2.4) imply , and (2.9) follows.
To prove (2.8), let us introduce where we used (2.7), (2.9), (2.4), (2.5), and the subpolynomial decay of c(ε). Similarly, and |x for small enough ε, so the right-hand side is a function subpolynomially growing at 0. Let us define θ ± ε = T ± ε − (τ ε I − τ ε V ). Due to (2.9), where the error term (along with all subsequent error terms) is uniform in the in starting points |x| ≤ K(ε). Note that the strong Markov property implies the conditional independence of θ ± ε and τ ε V given Y ε (τ ε V ) = ε β . This, along with (2.8), allows us to give upper and lower estimates of the first term on the right-hand side: where c(ε) is an arbitrary positive function that decays subpolynomially. Now we may apply Theorem 2.1 with C = −t + λ −1 log |f (q ± )| to both sides and conclude the proof.

Linear system with additive noise
In this section, we introduce an auxiliary process, which is a simpler special case of (2.2). Namely, we consider where σ 0 =σ(0) = σ(0) > 0, |z| < K(ε), and K(ε) grows subpolynomially at 0. We will need a precise description of the exit of Z ε from V Z = [−ε β (1 + δ ε ), ε β (1 + δ ε )] for β ∈ (0, 1) and any δ ε > 0 satisfying δ ε ↓ 0 as ε ↓ 0. Let us introduce a stopping time and, for a constant C and a subpolynomially decaying at 0 function c(ε), a deterministic time We will often use C ε = C − c(ε). The first result of this section is a version of Theorem 2.1 for the process Z ε with stronger control of the dependence on the initial point.
Lemma 3.1. For any α > 1 and any subpolynomially decaying function c(ε) in the definition of t ε , there is c > 0 such that Proof: Duhamel's formula gives an explicit solution to (3.1): where N (t) = t 0 e −λs dW (s). Plugging in τ ε for t, we obtain which is equivalent to By the martingale convergence theorem, N (t) converges to a random variable N ∞ as t → ∞ almost surely and in L 1 . We claim that, in addition, there are c 1 , c 2 > 0 such that for any L > 0, (3.5) for some c 1 , c 2 > 0, which proves the claim since ε α−β e λtε = e −λCε . Let us fix γ ∈ (α − 1, α − β) and write The random variables N (τ ε ) and N (τ ε )−N ∞ are independent due to the strong Markov property. So the Gaussian tail of the maximum of the Brownian motion and (3.5) imply The desired asymptotics of H 1 (z, ε) follows from the explicit form of the Gaussian density of the random variable z + σ 0 N ∞ . ✷ The next result is based on the fact that the distribution of Z ε (t ε ) conditioned on non-exit is approximately uniform over [−ε β (1 + δ ε ), ε β (1 + δ ε )]. In fact, this stronger statement is proved as an intermediate step. We first note that the density of any absolutely continuous random variable conditioned on a positive probability event is well-defined.
Let us fix any w 0 and find c 0 > 0 such that for w > w 0 and all ε > 0. Then there is a constant c 0 such that for all |u| ≤ (1 − δ)ε β and ε > 0, (3.14) The restriction on w implies 0 ≤ D ≤ c 1 ε 2 for some c 1 . To maximize ψ(D, ∆), we compute so ψ(D, ∆) grows in D ∈ [0, ∆], and we obtain from (3.15): Combining (3.14) and (3.16), we obtain (3.13) and hence (3.11). Thus, (3.6) is proved. To prove (3.7), we write and notice that the first term on the right-hand side equals while the second term decays much faster than ε 1−β due to the decay assumption on h. ✷ 4 Proof of Theorem 2.1 for α ∈ (1, 1 + β) We start by studying the deviations ∆ ε (t) = Y ε (t) − Z ε (t) as long as both processes Y ε (t) and Z ε (t) are close to the origin. Let us fix β ∈ (0, 1) and introduce the stopping times Lemma 4.1. Suppose α ∈ (1, 1 + β), β ′ ∈ (α − 1, β), L(ε) > 0 is a bounded function, and Then for sufficiently small ε > 0, we have Clearly, I (1) ε is a martingale satisfying I (1) ε t = O ε 2β for t ≤τ ε , so for any c 0 > 0, there is c > 0 such that by Lemma 2.1. Therefore and on the complementary event we have if c 0 is chosen sufficiently small, which finishes the proof. ✷ Based on this approximation result and the calculation for Z ε in the previous section, the following theorem proves Theorem 2.1 for α not too large.
The first step is the following lemma which establishes that the process needs to stay close to the origin to delay the exit.
Proof: Using the strong Markov property and applying Duhamel's principle (2.3) (2.4) to the initial condition y with |y| > εK(ε), we reduce the lemma to the estimate and the desired inequality follows by (2.5) since β + θ N − 1 > 0 due to (5.1). ✷ We now collect some results needed for our iteration scheme.
This completes the induction step and finishes the proof of (5.6).