Exit time asymptotics for small noise stochastic delay differential equations

Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE) with coefficients that depend on the history of the process over a finite delay interval. We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for a broad class of infinite-dimensional small noise stochastic equations.

, neuronal models [12,50,51] and biochemical models of gene regulation [1,4,38]. In many applications, stable equilibrium behavior or periodic oscillatory behavior is critical to a well functioning system and there is a sizable literature on conditions for the stability of equilibrium points [18,26,28,46,47] and periodic orbits [11,29,30,31,35,36,37,54,55,59,60,61,62,63,64,65,66] of deterministic delay differential equations (DDEs). Frequently, small noise is present and it is of interest to understand its effect on the dynamics, especially near stable equilibrium points and periodic orbits (of the corresponding deterministic system). While solutions of the deterministic system that start near a stable equilibrium point or periodic orbit will remain near the equilibrium point or periodic orbit for all time, solutions of the small noise stochastic system will eventually exit any bounded domain that contains the equilibrium point or periodic orbit (provided the noise coefficient is uniformly nondegenerate, see Assumption 3.4 below). The main focus of this work is to estimate the time it takes solutions of the small noise stochastic system to exit certain bounded domains that contain stable equilibrium points or periodic orbits, from the perspective of large deviations. We anticipate that some of the methods we use to study this problem may be useful in the analysis of exit time problems for other small noise stochastic dynamical systems. We focus on the following multidimensional small noise stochastic delay differential equation (SDDE) written in integral form: Here τ > 0 is the length of the finite delay interval, X ε is a continuous vector-valued process on [−τ, ∞), X ε s = {X ε s (u) = X ε (s + u), u ∈ [−τ, 0]} is a continuous process on [−τ, 0] that tracks the history of X ε over the delay interval, b and σ are continuous functions of these path segments, W is a standard multidimensional Brownian motion, the stochastic integral with respect to W is the Itô integral, and ε is a small positive parameter discounting the noise coefficient. (The presence of the square root in (1.1) is a matter of notational preference. Alternatively, one could scale the stochastic integral by ε, in which case ε 2 would appear in place of ε as the scaling constant in our main results, Theorems 3.8 and 3.9.) The SDDE (1.1) can be thought of as a small noise perturbation of the following deterministic DDE obtained by setting ε = 0 in (1.1): (1.2) x(t) = x(0) + where d is the dimension of the vector-valued process X ε and the vector-valued path x. Given an equilibrium point or periodic orbit in C of the DDE (1.2), which we denote by O (see Definition 2.11 below), a bounded domain D in C that contains O, and a solution X ε of the SDDE (1.1), let ρ ε = inf{t ≥ 0 : X ε t ∈ D} denote the first time X ε exits D. If O is stable, D is attracted to O in a manner we will make precise, X ε starts sufficiently close to O, and σ satisfies a uniform nondegeneracy condition (see Assumption 3.4 below), then the expected exit time E[ρ ε ] will grow exponentially as ε converges to zero. Our main result is to obtain upper and lower bounds on the exponential rate at which E[ρ ε ] grows (see Theorem 3.9 below). In addition, we show that when O is an equilibrium point in C, and the domain D is the uniform ball centered at O, then the upper and lower bounds coincide (see Lemma 3.13 below).
In order to obtain exit time asymptotics, we first prove a sample path large deviation principle (LDP) for solutions of the SDDE that is uniform over (initial conditions in) bounded sets (see Theorem 3.8 below). A sample path LDP (for a fixed initial condition) provides upper and lower bounds on the exponential rate of decay, as the noise vanishes, for the probability the solution of the SDDE lies in a measurable set, and the rate of decay is expressed in terms of the large deviations rate function. Such a sample path LDP for an SDDE with additive noise was first established by Langevin, Oliva and de Oilveira [34], and for an SDDE with multiplicative noise by Mohammed and Zhang [41] (see also, the subsequent work by Mo and Luo [39]). A uniform sample path LDP over bounded sets provides upper and lower bounds on the rate of decay that hold uniformly over initial conditions in a bounded set, and the uniformity over bounded sets is crucially used to prove the exit time asymptotics. In the finite-dimensional stochastic differential equation setting, bounded sets are relatively compact, and the uniform sample path LDP over compact sets follows from the sample path LDP for a fixed initial condition using standard techniques. For general stochastic equations with multiplicative noise whose state spaces are not locally compact, the techniques for establishing uniform sample path LDPs over compact sets do not readily extend to proving uniform sample path LDPs over bounded sets. (For a stochastic equation with additive noise, the contraction principle can be used to prove a uniform LDP over bounded sets; see, e.g., the proof of [13,Theorem 12.15].) Nevertheless, uniform sample path LDPs over bounded sets have been shown for certain stochastic partial differential equations (SPDEs) with multiplicative noise and used to obtain exit time asymptotics [9,10,53]. Moreover, Budhiraja, Dupuis and Salins [8] recently established a uniform sample path LDP over bounded sets for a broad class of SPDEs. Their approach uses a variational representation for (expectations of) exponential functionals of solutions of an SPDE along with weak convergence methods to prove a uniform LDP over bounded sets for a modified version of the SPDE, which is shown to imply a uniform LDP over bounded sets for the original SPDE. One limitation is their approach relies on compactness of an associated semigroup for all times t > 0. In our SDDE setting, similar compactness type conditions generally only hold for times t ≥ τ , where we recall that τ > 0 is the length of the delay interval, and so the approach in [8] is not readily adapted to the SDDE setting. In this work we take a new approach, outlined below, which also uses the variational representation for exponential functionals of solutions, but does not rely on weak convergence methods. Since variational representations have been shown for exponential functionals of solutions for broad classes of stochastic equations, including those driven by finite-dimensional Brownian motions [3], infinite-dimensional Brownian motions [5,6] and Poisson random measures [7], we anticipate that the overall approach introduced here may be useful for proving uniform sample path LDPs over bounded sets for solutions of a variety of infinite-dimensional stochastic equations.
To begin with, we impose a uniform Lipschitz continuity condition on the coefficients (see Assumption 2.1 below) that ensures strong existence and uniqueness of solutions of the SDDE. (While we impose a uniform Lipschitz condition on the coefficients throughout this work, we explain in Remark 3.11 that our main exit time asymptotics result, Theorem 3.9, holds under a local Lipschitz condition on the coefficients.) This, along with the variational representation for exponential functionals of Brownian motion obtained in [3, Theorem 3.1] yields a variational representation for exponential functionals of solutions of the SDDE (see Lemma 5.2 below). With the variational representation in hand, we prove a uniform Laplace principle over bounded sets (see Theorem 5.10 below). The Laplace principle (with fixed initial condition) establishes asymptotics of exponential functionals of the solution in terms of the large deviations rate function and has been shown by Varadhan [56] and Dupuis and Ellis [19,Theorem 1.2.3] to be equivalent to the LDP (with fixed initial condition). Variational representations for exponential functionals of strong solutions (of broad classes of stochastic equations) have been used extensively in the weak convergence approach to prove Laplace principles and uniform Laplace principles over compact sets (see, e.g., [3,5,6,7,19]). Our proof of the uniform Laplace principle over bounded sets contains some important distinctions from the weak convergence proof of the uniform Laplace principle over compact sets. In the weak convergence approach one first establishes tightness of a family of random variables (over ε > 0 sufficiently small and initial conditions in a compact set) that appear in the variational representation and then characterizes the limit of any convergent subsequence as satisfying the Laplace principle upper and lower bounds. (See, for example, the proof of [3,Theorem 4.3]. The proof is for a fixed initial condition; however, the tightness arguments can be readily adapted to allow for initial conditions in a compact set.) In our SDDE setting, since bounded sets generally are not relatively compact, the family of random variables that appear in the variational representation is not necessarily tight. Instead, we leverage the fact that we are working with strong solutions, so we can build our family of small noise processes on a common probability space with a common driving Brownian motion. Then, using the Lipschitz continuity of the coefficients and standard stochastic estimates, we prove a key convergence result for the family of random variables that appear in the variational representation (see Lemma 5.8 below). The convergence result is used to prove the uniform Laplace principle over bounded sets, which is shown to imply a uniform LDP over bounded sets.
Lastly, we use the uniform LDP over bounded sets to prove our main exit time asymptotics result (Theorem 3.9). While this proof is structurally similar to the proof of [17,Theorem 5.7.11], there are several nontrivial modifications to the proof that are due to the fact that the version of the uniform LDP over bounded sets we obtain takes a slightly different form from the uniform LDP over compact sets. Furthermore, the SDDE is degenerate in the sense that the natural state space is infinite-dimensional while the driving Brownian motion is finite-dimensional. This degeneracy restricts the set of paths in C that a solution of the SDDE can follow when exiting a domain in C and leads to unresolved challenges in proving the upper and lower bounds for the exit time coincide (see Remark 3.12 below). However, in the case the domain is a uniform ball centered at an equilibrium point, we prove that the upper and lower bounds coincide (see Lemma 3.13 below).

1.2.
Prior and related work. The study of exit time asymptotics for finite-dimensional SDEs is a classical subject in the theory of sample path large deviations, beginning with the work of Freidlin and Wentzell [57,58], which culminated in the books [24,25]. There have been numerous other works related to exit time asymptotics for SDEs, including [14,15,16,20,21,22,23,32]. In [13,Chapter 12], da Prato and Zabczyk detail a general approach for estimating exit time asymptotics for a class of small noise SPDEs with additive noise. As mentioned above, in [9,10,53] the authors obtain exit time asymptotics for a variety of SPDEs with multiplicative noise and in [8] the authors develop a general approach for proving a uniform LDP over bounded sets for a broad class of SPDEs with multiplicative noise and compact semigroups.
There has been limited work on exit time asymptotics for SDDEs, especially those with multiplicative noise. Langevin, Oliva and de Oilveira [34] consider exit time asymptotics for SDDEs with additive noise and analyze the quasipotential (see definition (3.8) below) associated with an asymptotically stable equilibrium point of the corresponding DDE. The proof of the exit time asymptotics in the case of additive noise relies on the contraction principle to prove a uniform LDP over bounded sets, and the method does not extend to the case of multiplicative noise. As stated above, Mohammed and Zhang [41] prove a sample path LDP for time-inhomogeneous SDDEs with multiplicative noise and fixed initial condition in the case that b and σ depend only on time, the current state and the delayed state, i.e., b(X ε s ) = f (s, X ε (s), X ε (s − τ )) and σ(X ε s ) = g(s, X ε (s), X ε (s − τ )) for suitable functions f and g (see also, the work of Mo and Luo [39]). We extend their result (in the case of time-homogeneous coefficients) by proving a uniform LDP over bounded sets and also allowing the coefficients to depend on the entire history of the process over the delay interval, not just the current state and delayed state. Lastly, we mention the work of Azencott, Geiger and Ott [2] who consider a linear SDDE with additive noise as a local approximation of a nonlinear SDDE and develop methods for efficient numerical computation of the rate function.
1.3. Outline. The remainder of this work is organized as follows. Precise definitions for a solution of the small noise SDDE and a solution of the related DDE are given in Section 2.
The definition of the rate function and our main results on the uniform sample path LDP over bounded sets and exit time asymptotics for the SDDE are presented in Section 3. Some useful properties of the rate function, including compactness of level sets, are proved in Section 4. The proof of the uniform sample path LDP over bounded sets is given in Section 5. The proof of the exit time asymptotics for the SDDE is given in Section 6.
1.4. Notation. Let R = (−∞, ∞) denote the real numbers. For r ∈ R, we say r is positive (resp. negative, nonnegative, nonpositive) if r > 0 (resp. r < 0, r ≥ 0, r ≤ 0). For r, s ∈ R, we let r ∧ s = min(r, s) and r ∨ s = max(r, s). For a closed interval I in R and a positive integer d, let C(I, R d ) denote the space of continuous functions from I into R d . We endow C(I, R d ) with the topology of uniform convergence on compact intervals in I. This is a Polish space. Given x ∈ C(I, R d ) and a compact interval J ⊂ I, we define the finite supremum norm of x over J by For a closed interval I in R, a real number p ≥ 1 and a positive integer m, let L p (I, R m ) denote the Banach space of Lebesgue measurable functions f from I to R m with finite L p -norm: where functions that are equal almost everywhere are identified. For T > 0 we say that a sequence For T, N > 0, we let When equipped with the weak topology, L 2 N ([0, T ], R m ) is metrizable as a compact Polish space (see [33,Theorem III.1]).
Throughout this work we let τ > 0 denote a fixed delay. For T ≥ 0 we let d T (·, ·) denote the metric on C([−τ, T ], R d ) induced by the uniform norm · [−τ,T ] . As noted in Section 1.1 above, when T = 0 we use the abbreviation C = C([−τ, 0], R d ), which is the natural state space for solutions of (1.1) and (1.2). For a subset A ⊂ C and µ > 0 let Given a closed interval I of the form [−τ, ∞) or [−τ, T ] for some T > 0, a path x ∈ C(I, R d ) and a nonnegative time t ∈ I, define x t ∈ C by x t (s) = x(t + s) for s ∈ [−τ, 0]. We emphasize that x(t) lies in R d and x t lies in C.
By a filtered probability space, we mean a quadruple (Ω, F , {F t , t ≥ 0}, P ), where F is a σ-algebra on the outcome space Ω, P is a probability measure on the measurable space (Ω, F ), and {F t , t ≥ 0} is a filtration of sub-σ-algebras of F such that (Ω, F , P ) is a complete probability space, and for each t ≥ 0, F t contains all P -null subsets of F and F t = ∩ s>t F s . We let E denote expectation under P . Given two σ-finite probability measures P and Q on a measurable space (Ω, F ), the notation P ∼ Q will mean that P and Q are mutually absolutely continuous, i.e., for any A ∈ F , P (A) = 0 if and only if Q(A) = 0. By a continuous process we mean a process with all continuous sample paths.

Delay differential equations
In this section we introduce two delay equations -a small noise stochastic equation and a corresponding deterministic equation. Recall that we are fixing τ > 0, which will be referred to as the delay. In addition, we fix positive integers d and m, recall that C = C([−τ, 0], R d ) and fix functions b : C → R d and σ : C → M d×m satisfying the following uniform Lipschitz continuity condition.
Assumption 2.1. There exists κ 1 > 0 such that Remark 2.2. A simple consequence of Assumption 2.1 is that there exists κ 2 > 0 such that We impose a uniform Lipschitz continuity condition to ensure existence and uniqueness of strong solutions of the SDDE. In general, a local Lipschitz continuity condition with a linear growth condition is sufficient; however, for convenience we impose the uniform condition. In Remark 3.11 below, we note that our main result Theorem 3.9 on the exit time asymptotics for the SDDE is readily extended to the case of locally Lipschitz coefficients.
2.1. Small noise stochastic delay differential equation. Throughout this section we fix ε > 0. The natural initial condition is a C-valued random element ξ on (Ω, F 0 , P ).
Proposition 2.4. Given an m-dimensional Brownian motion W = {W (t), t ≥ 0} on a filtered probability space (Ω, F , {F t , t ≥ 0}, P ) and a C-valued random element ξ on (Ω, F 0 , P ), there exists a unique solution X ε of the SDDE with initial condition X ε 0 = ξ and driving Brownian motion W . Furthermore, the process satisfies the strong Markov property.
Remark 2.5. Here uniqueness means that any two solutions of the SDDE on a filtered probability space (Ω, F , {F t , t ≥ 0}, P ) with common initial condition ξ and driving Brownian motion W are indistinguishable.
Proof. See, e.g., [ As a consequence of Proposition 2.4 we have the following corollary on the existence of a measurable function that takes a Brownian motion to the solution of the SDDE. The existence of such a function is important for our proof of the sample path LDP, as shown in Section 5.1.
Corollary 2.6. For φ ∈ C and T > 0, there exists a Borel measurable function Proof. Given φ ∈ C and T > 0, the existence of Λ ε φ,T follows from the fact that, by Proposition 2.4, there exists a unique solution of the SDDE on any filtered probability space (Ω, F , {F t , t ≥ 0}, P ) that supports an m-dimensional Brownian motion. In particular, by taking the canonical set up where (Ω, F , P ) is m-dimensional Wiener space, W = {W (ω, t) = ω(t), ω ∈ Ω, t ≥ 0} is the coordinate process and {F t , t ≥ 0} is the P -augmented filtration generated by W , the existence of the measurable map follows via a standard method. For a detailed outline of this method, we refer the reader to [49,Chapter V.10].
Throughout the remainder of this work we fix an m-dimensional Brownian motion W on a filtered probability space (Ω, F , {F t , t ≥ 0}, P ).
Notation 2.7. Given φ ∈ C we write X ε,φ to denote the unique solution of the SDDE with initial condition X ε,φ 0 = φ and driving Brownian motion W .
Notation 2.9. Given φ ∈ C, we let x φ denote the unique solution of the DDE with initial condition φ.
Remark 2.10. It follows from (1.2), the continuity of the function t → x t from [0, ∞) to C and the continuity of b that any solution x of the DDE is continuously differentiable on (0, ∞) and its derivative satisfies Given a periodic solution x * with period p > 0, observe that x * is also periodic with period kp for any positive integer k. Thus, the period is not unique; however, the orbit O in C is unique.
Remark 2.14. Suppose ν * ∈ R d is an equilibrium point of (1.2). Then b(φ * ) = 0, where the constant function φ * ∈ C is given by φ * (·) ≡ ν * . In addition, for any p > 0, x * (·) ≡ ν * is a periodic solution of (1.2) with period p and orbit O = {φ * }, We close this section with an example of a stable equilibrium point of a one-dimensional DDE and a stable periodic orbit of a one-dimensional DDE. We also provide examples of domains that are attracted to their respective orbits.
Example 2.17. Consider the following one-dimensional linear DDE (in differential form): where B > A ≥ 0. Then zero is an equilibrium point of the DDE and (2.6) has characteristic equation Let θ 0 be the unique solution in [π/2, π) to cos θ 0 = −A/B, and define If τ < τ 0 , then every solution of the characteristic equation has negative real part and it follows that the orbit O = {φ * }, where φ * ∈ C is given by φ * (·) ≡ 0, is stable and every bounded domain D in C that contains O is uniformly attracted to O (see, e.g., [52,Theorem 4.3]).
Example 2.19. Consider the following one-dimensional nonlinear DDE: as r → ±∞. By [64, Theorem 1], there exists τ 1 > 0 such that for all τ ≥ τ 1 there exists a unique (up to time translation) slowly oscillating periodic solution x * of (2.8), its orbit O is stable and for every δ > 0 sufficiently small, the domain B(O, δ) is uniformly attracted to O.

Main results
In this section we summarize our main results on the small noise asymptotics for solutions of the SDDE.
3.1. The rate function. In this section we introduce the rate function and provide conditions under which the rate function can be explicitly evaluated. In Section 4 we prove some useful properties of the rate function, including compactness of level sets. Given For φ ∈ C and T > 0, define the rate function where we recall that x φ denotes the unique solution of the DDE with initial condition φ.
In general, the variational form (3.2) of the rate function is difficult to explicitly evaluate. However, when m = d and the following uniform ellipticity condition holds, we can explicitly evaluate the variational form.
Remark 3.5. Under Assumption 3.4, since a is continuous and uniformly elliptic, it follows from standard arguments that a −1 is well-defined, continuous and uniformly bounded on C. Thus, if m = d, then σ −1 is well-defined and given by σ −1 = σ ′ a −1 . otherwise.
Here Λ : C × R d → R + is the continuous function defined by However, the function s → Λ(x s ,ẋ(s)) need not be integrable, in which case we adopt the convention that J T (x) is infinite.
The proof of Lemma 3.6 is given in Section 4.3.

Uniform large deviation principle. Throughout this section we fix
Theorem 3.8. Suppose Assumption 2.1 holds. Let K ⊂ C be a bounded subset and T > 0. Then the following hold:

For all open sets
The proof of Theorem 3.8 is given in Section 5.3.

3.3.
Exit time asymptotics. Let x * be a periodic solution of (1.2) with period p > 0 and let where we recall that x φ denotes the solution of the DDE with initial condition φ ∈ C. Define For ε > 0 and φ ∈ C, define the {F t }-stopping time and for all α > 0, Proof. The theorem follows immediately from Lemmas 6.9 and 6.10. Remark 3.11. Assumptions 2.1 and 3.4 impose a uniform Lipschitz continuity condition on the coefficients and a uniform nondegeneracy condition on the diffusion coefficient a = σσ ′ (on all of C). However, since Theorem 3.9 is only concerned with the process X ε,φ up until its first exit time from the bounded domain D, the result readily extends to the case that the coefficients are uniformly Lipschitz continuous on D and the diffusion coefficient a is uniformly nondegenerate on D.
Remark 3.12. One would like to show that V = V . For a general orbit O in C and bounded domain D in C that contains O, it is not clear if this equality holds. This is due to the degeneracy that arises because the state space C is infinite-dimensional while the driving Brownian motion is finite-dimensional. In particular, given an element φ on the boundary of a "regular" domain D in C (i.e., D is equal to the interior of its closure), it is possible that the solution of the SDDE with initial condition φ will almost surely remain in the domain for a positive amount of time, which is in contrast to the finite-dimensional stochastic differential equations setting. For example, suppose d = 1, D = {ψ ∈ C : sup s∈[−τ,0] |ψ(s)| < 1} is the unit ball about the zero function in C and φ(t) = t/τ for all t ∈ [−τ, 0]. Then φ lies in the boundary of D and it is readily seen (due to the continuity of sample paths) that, for any ε > 0, the solution X ε,φ of the SDDE almost surely remains in D for a positive amount of time .
and define V and V as in (3.10) and (3.11), respectively. Then V = V .
The proof of Lemma 3.13 is given in Section 6.4.
Then for all t ∈ [0, T ], (4.13) x Proof. It follows from the definition of the rate function in (3.2) that x 0 = φ and, given α > 0, we can choose u ∈ U T (x) so that (3.1) holds and (4.14) By (3.1), the fact that x φ satisfies (1.2) with x φ in place of x, the fact that x 0 = x φ 0 = φ, two applications of the Cauchy-Schwarz inequality, (4.14), the Lipschitz continuity of b (Assumption 2.1) and the bound (2.2), we have By Gronwall's inequality, Since α > 0 was arbitrary, this implies (4.13).
Proof. Let M 1 > 0 be such that An application of Gronwall's inequality yields ). Again using (3.1), two applications of the Cauchy-Schwartz inequality, (1.5), (2.2) and (4.17), we have, for 0 ≤ s < t ≤ T , where Since L 2 N ([0, T ], R m ) is a compact Polish space when equipped with the weak topology, there exists a subsequence {n k } ∞ k=1 , and an element u of L 2 N ([0, T ], R m ) such that u n k → u in the weak topology as k → ∞. Letting k → ∞ in (4.18) (with n k in place of n), we see that (3.1) holds. Thus, u ∈ U T (x). By (3.3) and the facts that u ∈ L 2 N ([0, T ], R m ) and N = 2M + α, we have Since α > 0 was arbitrary, this proves x ∈ K M and thus K M is closed.

Evaluation of the variational form of the rate function.
Proof of Lemma 3.6. Fix T > 0 and x ∈ C([−τ, T ], R d ). We first show that if I T (x) < ∞, then I T (x) = J T (x). Suppose I T (x) < ∞ and let u ∈ U T (x). It follows from (3.1) that x is absolutely continuous and at the almost every t ∈ [0, T ] that x is differentiable, the derivative of x satisfies Therefore, ifẋ ∈ L 1 ([0, T ], R d ) is such that (3.5) holds, it follows thatẋ(t) satisfies, for almost every t ∈ [0, T ],ẋ (t) = b(x t ) + σ(x t )u(t).
Since this holds for every u ∈ U T (x), by the definition of the rate function in (3.2), we have where the square integrability of u follows from the fact that J T (x) is finite. Rearranging and substituting into (3.5) we see that (3.1) holds. Thus, u ∈ U T (x) and so I T (x) < ∞.

Uniform large deviation principle
In this section we prove Theorem 3.8. Throughout this section we fix T > 0. With some abuse of notation we write W = {W (t), t ∈ [0, T ]} to denote the restriction of the Brownian motion to the interval [0, T ] and, for ε > 0 and φ ∈ C, we write X ε,φ = {X ε,φ (t), t ∈ [−τ, T ]} to denote the restriction of the solution of the SDDE to the interval [−τ, T ]. We let {F W t , t ∈ [0, T ]} denote the P -augmented filtration generated by W , i.e., where N = {A ∈ F : P (A) = 0} denotes the P -null sets in F . We say that an m-dimensional Let α > 0 be arbitrary and v α ∈ L 2 ([0, T ], R m ) be such that with the convention that the infimum over the empty set is equal to T , and define the process It follows from (5.5), (5.3) and the bound on g that By Chebyshev's inequality and our choice of N , Thus, using the fact that |v α,N | ≤ |v α | holds pointwise, the bound on g, (5.5) and (5.6), we have Since v α,N ∈ L 2 N ([0, T ], R m ) and α > 0 was arbitrary, this completes the proof of (5.4).

5.3.
Proof of the uniform LDP. The proof of Theorem 3.8 uses the uniform Laplace principle over bounded sets proved in Theorem 5.10 and follows a similar outline to the proof of Theorem 1.2.3 in [19], which establishes the equivalence between the LDP and the Laplace principle. However, the proof contains some nontrivial differences that arise because we prove the LDP holds uniformly over bounded sets.
Proof of Theorem 3.8. Fix a bounded subset K in C and T > 0. We first prove part 1. Let F be a closed subset of C([−τ, T ], R d ). Define the lower semicontinuous function f : For j ≥ 1 define Then f j is bounded and Lipschitz continuous for each j ≥ 1, and f j converges to f pointwise from below as j → ∞. Along with (5.31), this implies that for all φ ∈ K and each j ≥ 1, Thus, by Theorem 5.10, for each j ≥ 1, We are left to show that If lim η→0 inf φ∈K I φ T (F η ) = 0, then (5.34) automatically holds since I φ T and f j are nonnegative. We assume that lim η→0 inf φ∈K I φ It suffices to show that First, consider the case that lim η→0 inf φ∈K I φ T (F η ) < ∞. For a proof by contradiction, suppose there exists η > 0, α > 0, a subsequence {j k } ∞ k=1 and a sequence {x k } ∞ k=1 in (F η ) c such that for each k ≥ 1, x k 0 ∈ K and (5.36) By the uniform bound in the last display and the definition of f j in (5.32), we have d T (x k , F ) → 0 as k → ∞. Thus, due to the definition of F η in (3.6), x k ∈ F η for all k sufficiently large. However, this implies I T (x k ) ≥ inf φ∈K I φ T (F η ), which contradicts (5.36) (since f j is nonnegative). With this contradiction thus obtained, it follows that (5.35) holds when lim η→0 inf φ∈K I φ T (F η ) < ∞. Next, consider the case that lim η→0 inf φ∈K I φ T (F η ) = ∞. For a proof by contradiction, suppose there exists η > 0, M > 0, a subsequence {j k } ∞ k=1 and a sequence {x k } ∞ k=1 in (F η ) c such that for each k ≥ 1, x k 0 ∈ K and (5.37) By the uniform bound in the last display and the definition of f j in (5.32), we have d T (x k , F ) → 0 as k → ∞. Thus, due to the definition of F η in (3.6), x k ∈ F η for all k sufficiently large. However, this implies I T (x k ) ≥ inf φ∈K I φ T (F η ) = ∞, which contradicts (5.37). With this contradiction thus obtained, it follows that (5.35) holds when lim η→0 inf φ∈K I φ T (F η ) = ∞. This proves part 1 of the theorem.
We now prove part 2. Let G be an open subset of C([−τ, T ], R d ). If lim η→0 sup φ∈K I φ T (G η ) = ∞, we are done. We assume lim η→0 sup φ∈K I φ T (G η ) < ∞. Let α > 0. Choose η † > 0 such that (5.38) sup Let M > sup φ∈K I φ T (G η † ) and define Then f is nonnegative, bounded above by M , Lipschitz continuous, and satisfies f (x) = 0 for all Therefore, by (1.4), the last display, Theorem 5.10 and the fact that f (x) = 0 for all x ∈ G η † , , it follows from the last display and (5.38) that Since α > 0 was arbitrary, this proves part 2 of the theorem.

Exit time asymptotics
In this section we prove Theorem 3.9. Throughout this section we assume m = d, and b and σ satisfy Assumptions 2.1 and 3.4. Recall that V and V are finite by Remark 3.10. Let κ 1 ≥ 1 be such that (2.1) holds and c > 0 be the constant in Assumption 3.4. According to Remark 3.5, there exists M a ≥ 1 such that Let x * be a periodic solution of (1.2) with period p > 0 and let O = {x * t , t ∈ [0, p)} denote its orbit in C. We assume that x * is stable (see Definition 2.15). Let D be a bounded domain in C that contains O. We assume there exists η 0 > 0 such that B(D, η 0 ) is uniformly attracted to O (see Definition 2.16). Given µ > 0, we let 6.1. Preliminary estimates. In preparation for proving Theorem 3.9 we first establish some useful lemmas. Proof. Fix α > 0. Set T 1 = 1 + τ + p and Let φ ∈ B(O, µ) and ψ ∈ O. Then there exist t * ∈ [0, p) and t † ∈ [t * + 1 + τ, t * + 1 + τ + p) such for all t ∈ (1, T ]. By (6.3) and Remark 2.10, x is absolutely continuous on [0, T ] and By Lemma 3.6, the Lipschitz continuity of b (Assumption 2.1), (6.4), the fact that T ≤ T 1 , (6.3) and the fact that d(φ, x * t * ) ≤ µ, The lemma then follows from our choice of µ in (6.2). Proof. Fix α > 0 and let µ, T 1 > 0 be as in Lemma 6.1. By the definition of V in (3.10), there exist ψ ∈ O, T † > 0 and . h) and, by Lemma 4.1, Since x * is stable (see Definition 2.15) and D is an open subset of C that contains O, we can choose µ 0 > 0 sufficiently small such that B(O, µ 0 ) ⊂ D s , where D s is the subset of D defined in (3.9). For ε > 0, φ ∈ D and µ ∈ (0, µ 0 ), let Remark 6.3. Since D is a bounded subset of C, b is bounded on D and the diffusion coefficient a = σσ ′ is uniformly nondegenerate on D, it can be readily deduced that σ ε,φ µ is a.s. finite.
Let µ ∈ (0, µ 0 /2) be as in Lemma 6.5 (with η † in place of η). By Lemma 6.6, we can choose T > 0 such that Define the closed set F ⊂ C([−τ, T ], R d ) by By the uniform LDP upper bound shown in part 2 of Theorem 3.8, where F η is defined as in (3.6) for η > 0. Next, we show that Let x ∈ F η † . By (3.6), there exists y ∈ F such that d T (x, y) ≤ η † . By (6.35), there exists t ∈ [0, T ] such that y t ∈ D c . Thus, It follows from Lemma 4.1, Lemma 6.5 and our choice of η † > 0 that inf φ∈S(O,µ) Since the last display holds for all x ∈ F η † , we see that (6.37) holds. Along with (6.36), this implies By (6.5), Proof. Fix µ > 0. For S > 0 define the closed set For S > 0 and η > 0, define F η S as in (3.6), but with S, F η S and F S in place of T , F η and F , respectively. It follows from (3.6), (6.40) and the triangle inequality that for all S > 0 and η > 0, By the definition of the rate function in (3.2) and (3.6), I φ S (F η S ) is nondecreasing as η → 0. Thus, it suffices to show that for some η > 0, Fix η ∈ (0, µ/4). By (2.2), we can choose M µ > 0 such that Let t † ∈ [t − τ, t], be such that By (6.45), (6.46) and the fact that η ∈ (0, µ/4), we must have t † > 0. Thus, we have the following inequalities, which are explained below: The first inequality follows from (6.46), (1.2) and (3.1). The second inequality is due to (6.45), (6.43), (6.44) and the Cauchy-Schwarz inequality. Rearranging yields where we have used the fact that η ∈ (0, µ/4). Since this holds for all x ∈ F η S satisfying inf φ∈B(O,µ) I φ S (x) < ∞ and all u ∈ U S (x), it follows from the definition of the rate function in (3.2) that Upon letting S → 0 we see that (6.42) holds. This completes the proof of the lemma.

Exit time upper bound.
In this section we prove upper bounds on the exit time ρ ε,φ defined in (3.13).
The first inequality is due to Lemma 3.6, the definition of x S , and the fact that b(φ * ) = 0 since φ * is an equilibrium point of the DDE. The second inequality follows from (6.1), the continuity of b (Assumption 2.1), and the fact that x is differentiable on (S, T ) with derivative equal to dx(t) dt = x † (S) − ν * for all t ∈ (S, T ) by (6.76). The third inequality is due the definition of T , the definition of x in (6.76), and (6.74). The last inequality is due to (6.72). By (6.77) and the last display, V < V + 3α. Since α > 0 was arbitrary, this completes the proof that V ≥ V .