Quenched Large Deviations for Multiscale Diffusion Processes in Random Environments

We consider multiple time scales systems of stochastic differential equations with small noise in random environments. We prove a quenched large deviations principle with explicit characterization of the action functional. The random medium is assumed to be stationary and ergodic. In the course of the proof we also prove related quenched ergodic theorems for controlled diffusion processes in random environments that are of independent interest. The proof relies entirely on probabilistic arguments, allowing to obtain detailed information on how the rare event occurs. We derive a control, equivalently a change of measure, that leads to the large deviations lower bound. This information on the change of measure can motivate the design of asymptotically efficient Monte Carlo importance sampling schemes for multiscale systems in random environments.

The system (1.1) can be interpreted as a system of slow and fast motion, X and Y components respectively, with separated scales. We study the regime where the homogenization parameter goes faster to zero than the strength of the noise does. The goal of this paper is to obtain the quenched large deviations principle associated to the component X, that is associated with the slow motion. The case of large deviations for such systems in periodic media was studied in [30], see also [1,6,10]. In [30] (see also [7]), it was assumed that the coefficients are periodic with respect to the y−variable and based on the derived large deviations principle, asymptotically efficient importance sampling Monte Carlo methods for estimating rare event probabilities were obtained. In the current paper, we focus on quenched (i.e. almost sure with respect to the random environment) large deviations and the situation is more complex when compared to the periodic case since the coefficients are now random fields themselves and the fast motion does not take values in a compact space.
We treat the large deviations problem via the lens of the weak convergence framework, [5]. This framework transforms the large deviations problem to convergence of a stochastic control problem. The current work is certainly related to the literature in random homogenization, see [14,15,16,17,19,21,22,23,24,25,26,28]. Our work is most closely related to [15,19], where stochastic homogenization for Hamilton-Jacobi-Bellman (HJB) equations was studied. The authors in [15,19] consider the case δ = ǫ with the fast motion being Y = X/δ and with the coefficients b = f = 0 in a general Hamiltonian setting. In both papers the authors briefly discuss large deviations for diffusions (i.e., when the Hamiltonian is quadratic) and the action functional is given as the Legendre-Fenchel transform of the effective Hamiltonian and the case studied there is δ = ǫ. Moreover, in [18,31] the large deviations principle for systems like (1.1) is considered in the case ǫ = δ with the coefficients b = f = 0. In [18,31] the coefficients are deterministic (i.e., not random fields as in our case) and stability type conditions for the fast process Y are assumed in order to guarantee ergodicity. Lastly, related annealed homogenization results (i.e. on average and not almost sure with respect to the medium) for uncontrolled multiscale diffusions as in (1.1) in the case ǫ = 1, δ ↓ 0 and Y = X/δ have been recently obtained in [28].
In contrast to most of the aforementioned literature, in this paper, we study the case ǫ/δ ↑ ∞. Thus, we also need to consider the additional effect of the macroscopic problem (i.e., what is called cell problem in the periodic homogenization literature) due to the highly oscillating term ǫ δ T 0 b (Y ǫ t , γ) dt. Moreover, because the homogenization parameter goes faster to zero that the strength of the noise does, we are able to derive an explicit characterization of the quenched large deviations principle, Theorem 3.5. The explicit form of the derived large deviations action functional and of the control achieving the large deviations bound gives useful information which can be used to design provably efficient importance sampling schemes for estimation of related rare event probabilities, see [7,30] for related results in the periodic set-up. In the present paper however, we only study the related large deviations problem and we leave the importance sampling problem for future work. Lastly, as it will be also mentioned below, in the course of the proof, we obtain quenched (i.e., almost sure with respect to the random environment) ergodic theorems for uncontrolled and controlled random diffusion processes that may be of independent interest, Theorem 3.3 and Appendix A. Related models where the regime of interest is ǫ/δ ↑ ∞ have been considered in [1,6,7,9,10,12,30,32].
The rest of the paper is organized as follows. In Section 2 we set-up notation, state our assumptions and review known results from the literature on random homogenization that will be useful for our purposes. In Section 3 we state our main results. Sections 4, 5 and 6 contain the proofs of the main results of the paper, i.e., quenched homogenization results for pairs of controlled diffusions and occupation measures in random environments and the large deviations principle with the explicit characterization of the action functional. The Appendix A contains the proofs of the necessary quenched ergodic theorems.

Assumptions, notation and review of useful known results
In this section we setup notation and pose the main assumptions of the paper. In this section, and for the convenience of the reader, we also review well known results from the literature on random homogenization that will be useful for our purposes. The content of this section is classical. 2 We start by describing the properties of the random medium. Let (Γ, G, ν) be the probability space of the random medium and as in [13], a group of measure-preserving transformations {τ y , y ∈ R d } acting ergodically on Γ.
Definition 2.1. We assume that the following hold.
(i) τ y preserves the measure, namely ∀y ∈ R d and ∀A ∈ G we have ν(τ y A) = ν(A).
Forφ ∈ L 2 (Γ) (i.e., a square integrable function in Γ), we define the operator T yφ (y) =φ(τ y γ). It is known, e.g. [21], that T y forms a strongly continuous group of unitary maps in L 2 (Γ). Moreover, if the limit exists, the infinitesimal generator D i of T y in the direction i is defined by and is a closed and densely defined generator. Next, forφ ∈ L 2 (Γ), we define φ(y, γ) =φ(τ y γ). This definition guarantees that φ will be a stationary and ergodic random field on R d−m . Similarly, for a measurable functionφ : R m ×Γ → R m we consider the (locally) stationary random field (x, y) →φ(x, τ y γ) = φ(x, y, γ). We follow this procedure to define the random fields b, c, σ, f, g, τ 1 , τ 2 that play the role of the coefficients of (1.1).
The main assumption for the coefficients of (1.1) is as follows.
It is known that under Condition 2.2, there exists a filtered probability space (Ω, F, F t , P) such that for every given initial point (x 0 , y 0 ) ∈ R m × R d−m , for every γ ∈ Γ and for every ǫ, δ > 0 there exists a strong Markov process (X ǫ t , Y ǫ t , t ≥ 0) satisfying (1.1). However, if we define a probability measure P = ν ⊗ P on the product space Γ × Ω, then when considered on the probability space is not a Markov process. From the previous discussion it is ease to see that the periodic case is a special case of the previous setup. Indeed, we can consider the periodic case with period 1, Γ to be the unit torus and ν to be Lebesgue measure on Γ. For every γ ∈ Γ, the shift operators τ y γ = (y + γ) mod 1 and we have φ(y, γ) =φ(y + γ) for a periodic functionφ with period 1.
For every γ ∈ Γ, we define next the operator and we let Y γ t to be the corresponding Markov process. It follows from [25,23,21], that we can associate the canonical process on Γ defined by the environment γ, which is a Markov process on Γ with continuous transition probability densities with respect to d-dimensional Lebesgue measure, e.g., [21]. In particular, we let We denote the infinitesimal generator of the Markov process γ t bỹ Following [23], we assume the following condition on the structure of the operator defined in Definition 2.3.
Condition 2.4. We can write the operatorL in the following generalized divergence form We assume that m(γ) is bounded from below and from above with probability 1, that there exist smoothd i,j (γ) such thatβ j = j D jdi,j with |d i,j | ≤ M for some M < ∞ and where the space Sobolev space H 1 is the Hilbert subspace of H = L 2 (Γ, G, ν) equipped with the inner product Notice that a trivial example that satisfies Condition 2.4 is the gradient case. Let f (y, γ) = −∇Q(y, γ) andτ 1 (γ) = √ 2D = constant andτ 2 (γ) = 0. Then, we have thatm(γ) = exp[−Q(γ)/D] andβ j = 0 for all 1 ≤ j ≤ d. Moreover, ifm = 1 andd i,j are constants then the operator is of divergence form.
Next, we recall some classical results from random homogenization.
Then π is the unique ergodic invariant measure for the environment process {γ t } t≥0 .
We will denote by E ν and by E π the expectation operator with respect to the measures ν and π respectively.
This implies that Dχ ρ ∈ L 2 (π) and that it has a L 2 (π) strong limit, i.e., there exists aξ ∈ L 2 (π) such that In addition, sinceb is bounded under Condition 2.2,χ ρ is also bounded. This follows because the resolvent operator R ρ corresponding to the operator ρI − L is associated to a L ∞ (Γ) contraction semigroup, see Section 2.2 of [21].

Main results
In this section we present the statement of the main results of the paper. In preparation for stating the large deviations theorem, we first recall the concept of a Laplace principle.
Definition 3.1. Let {X ǫ , ǫ > 0} be a family of random variables taking values in a Polish space S and let I be a rate function on S. We say that {X ǫ , ǫ > 0} satisfies the Laplace principle with rate function I if for every bounded and continuous function h : S → R If the level sets of the rate function (equivalently action functional) are compact, then the Laplace principle is equivalent to the corresponding large deviations principle with the same rate function (Theorems 2.2.1 and 2.2.3 in [5]).
In order to establish the quenched Laplace principle, we make use of the representation theorem for functionals of the form E e − 1 ǫ h(X ǫ,γ ) in terms of a stochastic control problem. Such representations were first derived in [4].
Let A be the set of all F s −progressively measurable n-dimensional processes u .
In the present case, let Z(·) = (W (·), B(·)) and n = 2k. Then, for the given γ ∈ Γ we have the representation where the pair (X ǫ ,Ȳ ǫ ) is the unique strong solution to This representation implies that in order to derive the Laplace principle for {X ǫ }, it is enough to study the limit of the right hand side of the variational representation (3.2). The first step in doing so is to consider the weak limit of the slow motionX ǫ of the controlled couple (3.3).
Fix γ ∈ Γ and let us define for notational convenience Z = R κ and Y = R d−m . Due to the involved controls, it is convenient to introduce the following occupation measure. Let ∆ = ∆(ǫ) ↓ 0 as ǫ ↓ 0 that will be chosen later on and is used to exploit a time-scale separation. Let A 1 , A 2 , B, Θ be Borel sets of Z, Z, Y, [0, T ] respectively. Let u ǫ i ∈ A i , i = 1, 2 and let (X ǫ ,Ȳ ǫ ) solve (3.3) with u ǫ i in place of u i . We associate with (X ǫ ,Ȳ ǫ ) and u ǫ i a family of occupation measures P ǫ,∆,γ defined by Next, we introduce the notion of a viable pair, see also [6]. Such a notion will allow us to characterize the limiting behavior of the pair X ǫ,γ , P ǫ,∆,γ .
whereξ is the L 2 limit of Dχ ρ as ρ ↓ 0 that is defined in Section 2. Consider the operatorL defined in Definition 2.3. We say that a pair (ψ, is viable with respect to (λ,L) and we write (ψ, P) ∈ V, if the following hold.
• The function ψ is absolutely continuous and P is square integrable in the sense that and for a given P, there is a unique well defined ψ satisfying (3.4).
For notational convenience later on, let us also definẽ Now, that we have defined the notion of a viable pair we are ready to present the law of large numbers results for controlled pairs X ǫ,γ , P ǫ,∆,γ . Theorem 3.3. Assume Conditions 2.2 and 2.4. Fix the initial point (x 0 , y 0 ) ∈ R m ×Y and consider a family {u ǫ = (u ǫ 1 , u ǫ 2 ), ǫ > 0} of controls (that may depend on γ) in A satisfying a.s. with respect to γ ∈ Γ, the bound A.20 and Then the family {(X ǫ,γ , P ǫ,∆,γ ), ǫ > 0} is tight almost surely with respect to γ ∈ Γ. Given any subsequence of {(X ǫ , P ǫ,∆ ), ǫ > 0}, there exists a subsubsequence that converges in distribution with limit (X, P) almost surely with respect to γ ∈ Γ. With probability 1, the limit point (X, P) ∈ V, according to Definition 3.2.
Next, we are ready to state the quenched Laplace principle for {X ǫ , ǫ > 0}.
, ǫ > 0} be, for fixed γ ∈ Γ, the unique strong solution to (1.1) and assume that ǫ/δ ↑ ∞. We assume that Conditions 2.2 and 2.4 hold. Define with the convention that the infimum over the empty set is ∞. Then, we have (i) The level sets of S are compact. In particular, for each s < ∞, the set almost surely with respect to γ ∈ Γ.
In other words, under the imposed assumptions, {X ǫ,γ , ǫ > 0} satisfies the quenched large deviations principle with action functional S.
Actually, it turns out that in this case we can compute of the quenched action functional in closed form.
, ǫ > 0} be, for fixed γ ∈ Γ, the unique strong solution to (1.1). Under Conditions 2.2 and 2.4, {X ǫ,γ , ǫ > 0} satisfies, almost surely with respect to γ ∈ Γ, the large deviations principle with rate function Notice that the coefficients r(x) and q(x) that enter into the action functional are those obtained if we had first taken to (1.1) δ ↓ 0 with ǫ fixed and then consider the large deviations for the homogenized system. This is in accordance to intuition since in the case ǫ/δ ↑ ∞, δ goes to zero faster than ǫ. This implies that homogenization should occur first as it indeed does and then large deviations start playing a role.

Proof of Theorem 3.3
In this section we prove Theorem 3.3. Tightness is established in Subsection 4.1, whereas the identification of the limit point is done in Subsection 4.2.
Proof. (i). Let us first prove the first part of the Lemma. It is clear that we can write Let us denote by P ǫ,∆,γ 1,t (A 1 × A 2 ) and by P ǫ,∆,γ 2,t (B) the first and second marginals of P ǫ,∆,γ It is clear that tightness of {P ǫ,∆,γ , ǫ > 0} is a consequence of tightness of {P ǫ,∆,γ 1,t , ǫ > 0} and of {P ǫ,∆,γ 2,t , ǫ > 0}. Let us first consider tightness of {P ǫ,∆,γ 1,t , ǫ > 0}. For this purpose, we claim that the function is a tightness function, i.e., it is bounded from below and its level sets R k = {r ∈ P(R 2k × [0, T ]) : g(r) ≤ k} are relatively compact for each k < ∞. Notice that the second marginal of every Hence, R k is tight and thus relatively compact as a subset of P.
by continuity ofλ ρ on the first argument, stationarity and the uniform integrability obtained in Lemma 4.1.
The first term on the right hand side of (4.12) goes to zero in probability, almost surely with respect to γ ∈ Γ, due to continuous dependence of G 0,γ x,y,z 1 ,z 2 φ(y, γ) on x ∈ R m , tightness ofX ǫ,γ , stationarity and δ/ǫ ↓ 0.

Compactness of level sets and quenched lower and upper bounds
Compactness of level sets of the rate function is standard and will not be repeated here (e.g., Subsection 4.2. of [6] or [11]).
Let us now prove the quenched lower bound. First we remark that we can restrict attention to controls that satisfy Conditions A.19 and A.20, which are required in order for Lemma A.6 to be true. For this purpose we have the following lemma, whose proof is deferred to the end of this section. Given such controls, we construct the controlled pair (X ǫ,γ , P ǫ,∆,γ ) based on such a family of controls. Then, Theorem 3.3 implies tightness of the pair (X ǫ,γ , P ǫ,∆,γ ), ǫ, ∆ > 0 . Let us denote by (X,P) ∈ V an accumulation point of the controlled pair in distribution, almost surely with respect to γ ∈ Γ. Then, by Fatou's lemma we conclude the proof of the lower bound. Indeed which concludes the proof of the Laplace principle lower bound. It remains to prove the quenched upper bound for the Laplace principle. To do so, we fix a bounded and continuous function h : C ([0, T ]; R m ) → R, and we show that The idea is to fix a nearly optimizer of the right hand side of the last display and construct the control which attains the given upper bound. Fix η > 0 and consider Boundedness of h implies that S(ψ) < ∞ which means that ψ is absolutely continuous. Since the local rate function L o (x, v) (6.2) is continuous and bounded as a function of (x, v) ∈ R m , standard mollification arguments (Lemmas 6.5.3 and 6.5.5 in [5]) allow to assume thatψ is piecewise constant. Next, we define the elements of L 2 (Γ) u 1,ρ (t, x, γ) = (σ(x, γ) + Dχ ρ (γ)τ 1 (γ)) T q −1 (x)(ψ t − r(x)) andũ 2,ρ (t, x, γ) = (Dχ ρ (γ)τ 2 (γ)) T q −1 (x)(ψ t − r(x)) and the associated stationary fields u 1,ρ (t, x, y, γ) =ũ 1,ρ (t, x, τ y γ) and u 2,ρ (t, x, y, γ) =ũ 2,ρ (t, x, τ y γ). We recall thatχ ρ satisfies the auxiliary problem in (2.3). Let us consider now the solution X ǫ t ,Ȳ ǫ Then, replacing c(x, y, γ) by c(t, x, y, γ) = c(x, y, γ) + σ(y, γ)u 1,ρ (t, x, y, γ), and g(x, y, γ) by g(t, x, y, γ) = g(x, y, γ) + τ 1 (y, γ)u 1,ρ (t, x, y, γ) + τ 2 (y, γ)u 2,ρ (t, x, y, γ) Theorem A.6 implies that (5.1)X ǫ →X in law, almost surely with respect to γ ∈ Γ, as ǫ ↓ 0 where we have that w.p. 1 the limit is Moreover, by Theorem A.6 we have that for any η > 0, there exists a N η with ν [ Therefore, noticing that for each fixed x ∈ R m and almost every t ∈ [0, T ] we finally obtain The first line follows from the representation (3.2) and the second line from the choice of the particular control. The third line follows from he convergence of the X ǫ and of the cost functional using the continuity of h. Then, the fourth line follows from the factX t = ψ t . Since the last statement is true for every η > 0 the proof of the upper bound is done.
We conclude this section with the proof of Lemma 5.1.
Proof of Lemma 5.1. First, we explain why Condition A.19 can be assumed without loss of generality.
Without loss of generality, we can consider a function h(x) that is bounded and uniformly Lipschitz continuous in R m . Namely, there exists a constant L h such that We recall that the representation is valid in γ by γ basis. Fix a > 0. Then for every ǫ > 0, there exists a control u ǫ ∈ A such that So, letting M 0 = h ∞ = sup x∈R m |h(x)| we easily see that such a control u ǫ should satisfy sup ǫ>0,γ∈Γ Given that the latter bound has been established, the claim that in proving the Laplace principle lower bound one can assume Condition A.19 without loss of generality, follows by the last display and the representation (5.3) as in the proof of Theorem 4.4 of [3]. In particular, it follows by the arguments in [3] that if the last display holds, then it is enough to assume that for given a > 0 the controls satisfy the bound a which proves that in proving the Laplace principle lower bound one can assume Condition A.19 without loss of generality.
Second, we explain why Condition A.20 can be assumed without loss of generality. It is clear by the representation (5.3) that the trivial bound holds where the control u ǫ (·) = 0 is used to evaluate the right hand side. Thus, we only need to consider controls that satisfy which by the Lipschitz assumption on h, implies that Let us next define the processesX ǫ t = 1 δX ǫ It is easy to see thatX ǫ t satisfies the SDEX andX ǫ t satisfies the same SDE with the control u ǫ (·) = 0. So, we basically have that For notational convenience, we define Since for x > 0, the function x 2 is increasing, the latter inequality, followed by Jensen's inequality give us The next step is to derive an upper bound of m ǫ T in terms of |ν ǫ T | 2 . Writing down the differencê X ǫ T −X ǫ T , squaring, taking expectation and using Lipschitz continuity of the functions b, c, σ and boundedness of σ we obtain the inequality where the constants C 0 , C 1 depends only on the Lipschitz constants of b, c, κ and on the sup norm of κ. Defining for notational convenience Gronwall lemma, gives us Let us now rewrite and upper bound |a ǫ T | 2 . We notice that, Hölder inequality followed by Young's inequality give us Putting these estimates together, we obtain Therefore, by choosing δ/ǫ sufficiently small such that Since β ǫ T and θ ǫ go uniformly in γ ∈ Γ to zero at the speed O( δ ǫ 2 ) as ǫ ↓ 0, we get that |ν ǫ T | 2 ≤ C(δ/ǫ) 2 , where the constant C, depends on T , but not on ǫ, δ or γ. This concludes the argument of why Condition A.20 can be assumed without loss of generality.
6. Proof of Theorem 3.5 In this section we prove that the explicit expression of the large deviation's action functional is given by Theorem 3.5.
Due to Theorem 3.4, we only need to prove that the rate function given in (3.7) can be written in the form of Theorem 3.5. First, we notice that one can write (3.7) in terms of a local rate function, in the form This follows directly by the definition of a viable pair (Definition 3.2). We call this representation the "relaxed" formulation since the control is characterized as a distribution on Z × Z rather than an element of Z × Z. However, as we shall demonstrate below, the structure of the problem allows us to rewrite the relaxed formulation of the local rate function in terms of an ordinary formulation of an equivalent local rate function, where the control is indeed given as an element of Z × Z. In preparation for this representation, we notice that any element P ∈ P (Z × Z × Γ) can be written of a stochastic kernel on Z × Z given Γ and a probability measure on Γ, namely P(dz 1 dz 2 dγ) = η(dz 1 dz 2 |γ)π(dγ).
Hence, by the definition of viability, we obtain for everyf ∈ D(L) that ΓLf (γ)π(dγ) = 0 where we used the independence ofL on z to eliminate the stochastic kernel η. Then Proposition 2.5 guarantees that π takes the form π(dγ) .

Appendix A. Quenched ergodic theorems
In this appendix we prove quenched ergodic theorems that are required for the proof of Theorem 3.3. For notational convenience and without loss of generality, we mostly consider a process Y driven by a single Brownian motion with diffusion coefficient κ(y, γ) such that κκ T = τ 1 τ T 1 + τ 2 τ T 2 .
Lemma A.1. Consider the process Y ǫ,y 0 ,γ t satisfying the SDE Consider also a functionΨ ∈ L 2 (Γ) ∩ L 1 (π) and define Ψ(y, γ) =Ψ(τ y γ). Assume that Ψ : DenoteΨ . = ΓΨ (γ)π(dγ). Then for any sequence h(ǫ) that is bounded from above and such that δ 2 /[ǫh(ǫ)] ↓ 0 (note that in particular h(ǫ) could be a constant), there is a set N of full π−measure such that for every γ ∈ N and also that π(dγ) is the invariant ergodic probability measure for the environment process γ t = τŶ y 0 ,γ t γ (Proposition 2.5). Suppose that δ 2 /[ǫh(ǫ)] ↓ 0. By the ergodic theorem, there is a set N of full π−measure such that for any γ ∈ N It follows from Egoroff's theorem that for every η > 0 there is a set For any η ∈ (0, 1), there exists a set N η with π(N η ) ≥ 1 − η and a sequence {h(ǫ), ǫ > 0} such that the following are satisfied: For notational purposes we will write that h(ǫ) ∈ H Proof of Lemma A.3. We start with the following decomposition Let us first treat the second term on the right hand side of (A.5). By the ergodic theorem, Lemma A.1, and Egoroff's theorem we know that there exists a set N η with π(N η ) ≥ 1 − η such that we have that the second term on the right hand side of (A.5) goes to zero. Next, we treat the first term on the right hand side of (A.5). Since, the function θ γ (u) is decreasing, we get that Therefore, the first term on the right hand side of (A.5) goes to zero, if we can choose h(ǫ), such that sup γ∈Nη θ γ h(ǫ) δ 2 /ǫ /h(ǫ) ↓ 0. This is a little bit tricky here because the argument of θ depends on h(ǫ). However, this can be done as follows. Fix β ∈ (0, 1) (e.g., β = 1/2) and choose h(ǫ) ≥ δ 2 /ǫ 1−β . Then, the monotonicity of f , implies that This proves that we can choose h(ǫ) such that the first term of the right hand of (A.5) goes to zero. The claim follows, by noticing that the previous computations imply that we can choose h(ǫ) that may go to zero, but slowly enough, such that both the first and the second term on the right hand side of (A.5) go to zero. Let us consider a functionΨ : [0, T ] × R m × Γ such thatΨ(t, x, ·) ∈ L 2 (Γ) ∩ L 1 (π) and define Ψ(t, x, y, γ) =Ψ(t, x, τ y γ). We assume that the function Ψ : [0, T ] × R m × R d−m × Γ → R is measurable, piecewise constant in t and uniformly continuous in x with respect to (t, y).
DenoteΨ(t, x) . = ΓΨ (t, x, γ)π(dγ) for all (t, x) ∈ [0, T ] × R m . Fix η > 0. Then there exists a set N η such that π(N η ) ≥ 1 − η and h(ǫ) ∈ H Slightly abusing notation, we denote by Y ǫ,y 0 ,γ t andŶ y 0 ,γ t the processes corresponding to Y ǫ,x,y 0 ,γ t andŶ ǫ,x,y 0 ,γ t with c(t, x, y) = 0. Lemma A.3 guarantees that the statement of the Lemma is true for Y ǫ,y 0 ,γ t , namely that there exists a set N η such that π(N η ) ≥ 1 − η and h(ǫ) ∈ H The fact that the convergence is also locally uniform with respect to the parameter x ∈ R m follows by the uniform continuity of Ψ in x, which implies that in Lemma A.3, we can choose the sequence h(ǫ) so that the convergence holds uniformly with respect to x in each bounded region, see for example Theorem II.3.11 in [29].
To translate this statement to what we need we use Girsanov's theorem on the absolutely continuous change of measures on the space of trajectories in C([0, T ]; R d−m ). Let φ(s, x, y, γ) = −κ −1 (y, γ)g(s, x, y, γ) and define the quantity