Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations

We consider a class of dissipative stochastic differential equations (SDE's) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE's to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and for improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation-dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural~networks and robustness of their estimates.


Introduction
In many scientific applications a systematic determination of the response of a complex nonlinear dynamical system to time-dependent perturbations is of key importance; topical examples in high-dimensional, non-autonomous and/or stochastic settings include climate models (e.g., [1,58,61,33,35,17]), statistical physics and non-equilibrium thermodynamics (e.g., [49,83,84,45,74]), and even neural networks (e.g., [19,75,23]). The sought response is usually quantified in terms of a change in an 'observable' expressed as a statistical/ensemble average of some functional defined on the trajectories of the underlying dynamical system. The classical theory of linear response (e.g., [82,58]) is concerned with capturing changes in observables to sufficiently small perturbations of the original dynamics close to its statistical equilibrium. It turns out that in such a setting the response can be expressed, with some caveats, through formulas linking the external perturbations to spontaneous fluctuations and dissipation in the unperturbed time-asymptotic dynamics (e.g., [83,84,55]). The classical fluctuation-dissipation theorem (FDT) is of fundamental importance in statistical physics (e.g., [50,5,27]), and it roughly states that for systems of identical particles in statistical equilibrium, the average response to small external perturbations can be calculated through the knowledge of suitable correlation functions of the unperturbed time-asymptotic dynamics; see, for example, [51,15] for some of the many applications of the FDT in the statistical physics setting.
The validity of the linear response and fluctuation-dissipation relationships for more general dynamical systems encountered, for example, in climate modelling (e.g., [58]) is an important topic which is particularly relevant for uncertainty quantification in reduced-order predictions and reduced model tuning (e.g., [33,60,17,62]). In an early influential work Leith [54] suggested that if the climate dynamics satisfied a suitable FDT, the climate response to small external forcing could be calculated by estimating suitable statistics in the unperturbed climate 1 . Climate dynamics is modelled as a forced dissipative chaotic or stochastic dynamical system which is arguably rather far from the statistical physics' setting for FDT. Nevertheless, Leith's conjecture stimulated a lot of activity in generating new theoretical formulations (e.g., [39,61]) and in designing approximate algorithms for FDT to study the climate response (e.g., [1,2,3,58,61,33,35,36,37,38,63]). However, despite numerous applications in autonomous and non-autonomous settings, there is little rigorous evidence supporting the validity of the linear response and FDT in the non-autonomous setting beyond the formal derivation of FDT for time-dependent stochastic systems [61].
The goal here is to provide a more rigorous justification of the linear response theory for a class of forced dissipative stochastic differential equations (SDE's) with time-periodic coefficients which induce time-periodic probability measures. Our objective is twofold: (i) Establish sufficient conditions for the existence and ergodicity (in an appropriate sense) of time-periodic measures associated with time-asymptotic dynamics for a class of 'dissipative' SDE's (defined later in (4.13)) with time-periodic coefficients in finite dimensions.
(ii) Analyse the linear response of such SDE's in the time-periodic regime to small perturbations, and express the change in the statistical observables based on time-periodic ergodic measures via fluctuation-dissipation type relations.
The results derived in the sequel will concern SDE's whose time-periodic measures are supported on certain stable random periodic solutions. In principle, the results discussed in the context of the linear response apply to a wider class of dynamical systems generating time-periodic measures; however, establishing conditions for the existence and ergodicity of such measures in a more general setting (for SDE's or otherwise) is not trivial and is beyond the scope of this work.
Time-periodic probability measures associated with the time-asymptotic dynamics are arguably ubiquitous in many mathematical models. In particular, seasonal and diurnal cycles in climate models due to time-periodic forcing or retarded self-interactions in neural networks provide some of the obvious candidates, and highlight the need for developing the linear response framework in the time-periodic setting. It is worth stressing that the need for rigorous formulation of the linear response and FDT for dissipative stochastic dynamical systems (in line with, e.g., [80,58,59,76,77,63]) is justified by contemporary approaches to the simulation and reduced-order modelling of high-dimensional, multi-scale dynamical phenomena. For example, comprehensive models for climate change prediction or molecular dynamics simulations involve stochastic components (e.g., [76,57,77,59,92,4,29]) to mimic the effects of unresolved dynamics, while reduced-order models typically involve stochastic noise terms (e.g., [22,56,48,71,16]. Here, similar to [39,61], the presence of noise leads to improved regularity of the problem which simplifies key aspects of the analysis compared to deterministic, dissipative nonlinear systems (e.g., [81,11,12,10,34]). As a consequence, we are able to focus on systems that have other important features of realistic dynamics, namely a lack of ellipticity, non-compactness of state space, and a lack of global Lipschitz continuity of the coefficients in the underlying SDE. The results established below apply to a broad class of nonlinear functionals which include common quantities of interest, such as the mean and covariance of subsets of variables.

General setup
Our framework relies on the theory of Markovian 2 random dynamical systems (RDS), which provides a geometric link between stochastic analysis and dynamical systems. This relationship was established through the discovery (e.g., [53,8]) that for sufficiently regular coefficients b, σ the stochastic differential equation (SDE) generates a stochastic flow {φ(t, s, · , · ) : s, t ∈ I ⊆ R, s t} of homeomorphisms on R d such that X s,x t (ω) = φ(t, s, ω, x) P -a.s. for x = X s (ω), ω ∈ Ω in the Wiener space (Ω, F, P) with W t an m-dimensional Brownian motion. For b = b(x), σ = σ(x), the SDE will be called autonomous, and non-autonomous otherwise. It turns out (e.g., [8,7]) that, for an autonomous SDE (in the above sense) with sufficiently regular coefficients there exists essentially a one-to-one correspondence between the SDE and an RDS; a rough but convenient interpretation (skipping the filtration) is that in the autonomous case there exists an RDS generating the SDE, which in turn generates the stochastic flow and vice-versa. One of the key concepts relevant for the analysis of the long-time behaviour of RDS is the extension of the notion of ergodicity to the random setting (e.g. [8,14,13,25,43,67,69,68]). These important results are established in the regime of (random) stationary measures and (random) stationary processes, in the case when the source of time-dependence is only due to the noise process (i.e., b = b(x), σ = σ(x) in (2.1) and the SDE is autonomous in the jargon established above). Over the last decade significant progress has been made in the study of the long-time behaviour of SDE's generated by time-dependent vector fields (e.g., [30,31,32,87,88,89,91,21]). Based on the insight from the latter results, we shall study the ergodicity of SDE's with time-periodic coefficients in order to establish fluctuation-dissipation formulas through the linear response in the random periodic regime. Our strategy is to first prove the existence of a unique time-periodic measure under certain 'dissipative' assumptions on the SDE via a version of Lyapunov second method and coupling. The standard Lyapunov second method is a well-known and powerful technique for the investigation of stability of solutions of nonlinear dynamical systems in finite and infinite dimensions. An extension of this method to an RDS generated by an autonomous SDE is essentially due to Hasḿinskii (e.g., [43]); subsequent extensions include applications to SDE's with random switching (e.g., [66]) and to the case of non-trivial random stationary solutions and random attractors by Schmalfuss [85]. Importantly, this method involves the study of random invariant sets (under the considered dynamics) without the need for the explicit knowledge of solutions of the underlying SDE, and it is based solely on the vector fields encoded in the coefficients of the SDE even when the drift term, i.e., b in (2.1), is only locally Lipschitz continuous. However, in the present non-autonomous, time-periodic setup, the lack of stationarity and the unavoidable skew-product structure of the underlying dynamics pose additional challenges when dealing with ergodicity of time-asymptotic probability measures. The main issue which prevents one from using the 'classical' methods (e.g., [26,43]) for proving ergodicity the random periodic regime stems from the fact that these probability measures are defined on the skew-product fibre bundle in the space of measures on the time-extended state space and that they are not mixing. Here, this complication is overcome by employing an extension of the Krylov-Bogolyubov procedure (e.g., [8, §1.5]) which allows for dealing with ergodicity of probability measures on appropriate Poincaré sections in the narrow topology generated by the dual of an appropriate discrete-time transition semigroup, and then 'linking' the results via the continuous-time transition semigroup induced by the SDE dynamics on the space of skew-product probability measures. In the present case, the properties of the time-periodic measures established with the help of the Lyapunov's second method for dissipative SDE's allows us to dispense with explicit assumptions on the ergodicity in the Poincaré sections which is otherwise required.
The rest of the article is organised as follows. In the remainder of this section we outline the frequently used notation. In Section 3, we recap some basic results and definitions, including the notion of a Random Dynamical System (RDS) generated by an SDE in finite dimensions, and we outline the notion of a random periodic process. In Section 4, we first prove the existence of stable random time-periodic solutions for a class of dissipative SDE's with time-periodic coefficients, and the existence of the associated time-periodic measures ( §4.2); sufficient conditions for ergodicity of such measures (in an appropriate sense) are established in §4. 3. Section 5 deals with the linear response theory in the above setting. The derivation of the linear response formula in the timeperiodic setting is followed by the derivation of two classes of fluctuation-dissipation relationships: the first one applies to perturbations of dynamics with time-periodic ergodic probability measures required only to exist in the unperturbed dynamics; the second fluctuation dissipation formula involves simpler formulas but it requires persistence of time-periodicity in the perturbed measures.
2.1. Function spaces. Below, we outline function spaces which are used in the sequel.
Let (X , d) be a complete separable metric space. We consider either X = R d or X = R × R d , or a flat cylinder X = [0, τ ) × R d , 0 < τ < ∞, [0, τ ) R mod τ , which arises when 'lifting' the dynamics generated by a non-autonomous SDE. In this section, we use X for all these spaces to unify the notation. Throughout, we set N 0 := {0, 1, 2, . . . } and N 1 := {1, 2, . . . }, while P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > ([0, τ )), P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > (R d ) are the spaces of Borel probability measures on, respectievely, [0, τ ) and R d .
The map x → φ(t, s, ω, x) is a stochastic flow of C l -diffeormorphisms, if it is a homeomorphism and φ(t, s, ω, x) is l-times continuously differentiable with respect to x ∈ R d for all s, t ∈ I ⊆ R and the derivatives are continuous in (s, t, x) ∈ I × I × R d . The stochastic flow is referred to as 'forward' for s t, and as 'backward' for t s. In the sequel, we will confine the discussion to (Ω, F, P) which is the Wiener space defined in §2.1.
Definition 3.2 (Filtration generated by a stochastic flow). Given a probability space (Ω, F, P), let F t s ⊆ F be the smallest S-algebra on Ω generated by and containing all null sets of F. The two-parameter filtration {F t s : s t} is the filtration generated by the forward stochastic flow φ(t, s, · , · ) : s, t ∈ I ⊆ R, s t and the filtered probability space is denoted by Ω, F, (F t s ) s t , P .

Definition 3.3 (Transition kernel).
Consider the stochastic flow φ(t, s, · , · ) : s, t ∈ I; s t induced by the SDE (2.1) for some fixed s ∈ I. Given the Borel-measurable space R d , B(R d ) , the transition probability kernel P (s, x; t, · ) induced by solutions of (2.1) is defined by The transition kernel satisfies the Chapman-Kolmogorov equation for any s, t, u ∈ I, s u t, and for all x ∈ R d , A ∈ B(R d ).
Definition 3.4 (Transition evolution and its dual). Given the forward stochastic flow φ(t, s, · , · ) : s, t ∈ I; s t and the transition kernel (3.1) induced by the solutions of (2.1), the operator P s,t : where we use the shorthand notation E ϕ(φ(t, s, x)) := Ω ϕ φ(t, s, ω, x) P(dω). The action of transition evolutions to arbitrary measurable functions is extended in a standard way.
For any probability measure µ s ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 Consequently, with the help of (3.2), we have for any s, u, t ∈ I, s u t, and for all A ∈ B(R d ) Theorem 3.5 (Stochastic flows generated by solutions of SDE's). Suppose that the coefficients of the SDE (2.1) are such that b( · , x), σ( · , x) are continuous for all x ∈ R d , and for all , denote the columns of σ. If the initial condition X s in (2.1) is independent of the S-algebra generated by W t−s ( · ), t s, and E |X s | 2 < ∞, there exist unique global solutions of (2.1) which generate a forward stochastic flow of homeomorphisms
The canonical filtration on (Ω, F, P) for the RDS is generated by (θ t ) t∈R .
Definition 3.8. (Canonical DS for processes with stationary increments). Consider a probability space (Ω, F, P ξ ) with the measure P ξ on (Ω, F) induced by the law of a stochastic process with continuous time ξ = (ξ t ) t∈R , ξ t : Ω → R d . A process ξ is said to have stationary increments if for any t 0 · · · t n , n ∈ N 1 , the distribution of (ξ t 1 +t − ξ t 0 +t , . . . , ξ tn+t − ξ t n−1 +t ) is independent of t ∈ R; i.e., θ(t)P ξ = P ξ for all t ∈ R, (3.9) where (θ t ) t∈R is a semigroup of time shifts. The corresponding measurable dynamical system Θ := (Ω, F, P ξ , (θ t ) t∈R ) is called the canonical dynamical system for the process with stationary increments; see, e.g. [8,Appendix A.3] for details. Proposition 3.9 (Canonical DS for Brownian motion/Wiener process). For the Wiener probability space (Ω, F, P) defined in §2.1, the canonical dynamical system Θ = Ω, F, P, (θ t ) t∈R for a stochastic process with stationary increments is given by 3 Strictly, Lt in (3.7) coincides with the generator of (2.1) on f ∈ C 1,2 c (R × R d , R + ) but it is well-defined for f ∈ C 1,2 (R × R d , R + ) and we refer to Lt as the generator throughout; the same applies to L (2) t in (3.8). 4 Here, the notation θP = P means that P({ω ∈ Ω : θtω ∈ A}) = P({ω ∈ Ω : ω ∈ A}), ∀ A ∈ F, t ∈ I; i.e., the semigroup (θt)t∈I, θt : Ω → Ω, preserves the measure P; we restrict the definition of the RDS to I = R. so that the set Ω = C 0 (R, R m ) is invariant w.r.t. the shifts (θ t ) t∈R . The canonical stochastic process W t (ω) = ω(t), t ∈ R, with stationary independent increments is the Wiener process/Brownian motion (with two-sided time) which satisfies identically Proof: See [8,Appendix A.3] for an outline or, e.g., [72].
Remark 3.10. In the sequel it will be more convenient to use (3.11) in the alternative form Assuming suitable regularity of the coefficients of autonomous SDE's, such as those in Theorem 3.5, together with adoption of two-sided stochastic calculus, the solutions of autonomous SDE's generate 5 an RDS over Θ (e.g., [8,7,28,44,53]). We will consider the non-autonomous dynamics of the SDE (2.1) with time-periodic coefficients as an RDS on a suitably extended space.
3.1. Time-periodic setting. In the sequel, we consider non-autonomous SDE's (2.1) on R d with time-periodic coefficients; i.e., b(t + τ, · ) = b(t, · ), σ(t + τ, · ) = σ(t, · ), 0 < τ < ∞, satisfying the conditions in Theorem 3.5 so that (2.1) has global solutions generating the forward stochastic flow φ(t + s, s, · , · ) : s ∈ R, t ∈ R + such that, for all s ∈ R, t ∈ R + , The above property follows from the time-periodicity of the coefficients and the uniqueness of solutions of (2.1). The relationship in (3.13) is essential for constructing an RDS on [0, τ )×R d from solutions of (2.1) with time-periodic coefficients, which is important for asserting the existence and ergodicity of time-periodic measures supported on random time-periodic paths defined below.
Definition 3.11 (Random periodic path of a stochastic flow [30,31,91]). A random periodic path of period 0 < τ < ∞ generated by a stochastic flow φ(t+s, s, · , · ) : s ∈ R, t ∈ R + is a measurable function S : R × Ω → R d such that for any s ∈ R the following holds S(τ + s, ω) = S(s, θ τ ω) and φ(t + s, s, ω, S(s, ω)) = S(t + s, ω) P -a.s. ∀ t ∈ R + . (3.14) Definition 3.12 (Random periodic path of RDS [32,91]). A random periodic path of period 0 < τ < ∞ generated by an RDS, Φ : R + ×Ω×R d → R d , is a measurable function S : R×Ω → R d such that for any s ∈ R and almost all ω ∈ Ω the following holds , be a globally Lipschitz vector field, and consider the deterministic flow {ψ(t, · ) : t ∈ R + }, defined via ψ(t, · ) ≡ φ(t, 0, · ) and generated by the autonomous ODE Assume that there exists a periodic solution Y : R → R d of the ODE (3.16) of period 0 < τ < ∞, Y(τ + s) = Y(s) and ψ(t, Y(s)) = Y(t + s), s ∈ R, t ∈ R + . 5 The generation of an RDS from an SDE requires a 'perfection of the crude cocycle' associated with the SDE (see, Consider the stochastic process X t (ω) = Y(t) + Z t (ω), where Z t solves the following SDE If Z(t, ω) is a random τ -periodic solution of (3.17), then S(t, ω) = Y(t) + Z(t, ω) is a random τ -periodic solution of the autonomous SDE: Example 3.14 (Stochastic FitzHugh-Nagumo model with periodic current). Consider the following SDE with nontrivial random periodic solutions (see [31]) which has less restrictive conditions on the drift than those considered in the sequel: t is a two-sided Wiener process on R. Let X s,x t (ω) = φ(t, s, ω, x), s t, be the solution of (3.18) represented via Now, consider the projections P − : R 2 → E − , P + : R 2 → E + , where the linear subspaces are The process S(t, ω) defined by is a random 2π/τ -periodic solution of the flow generated by the SDE (3.18); see, e.g., [31,21].

Time-periodic ergodic measures for dissipative SDE's
In this section we consider a class of non-autonomous SDE's (2.1) which generate stable random periodic paths. First, in §4.2 we prove the existence of a unique stable random periodic solution for a class of 'dissipative' 6 SDE's with time-periodic coefficients, and we assert the existence of time-periodic measures induced by such dynamics (Theorem 4.7). Ergodicity (in an appropriate sense, and under typical regularity conditions) of these time-periodic measures are established in Theorem 4.11 of §4.3. We conclude with an example of a periodically forced stochastic Lorenz model, which is then used in §5 to illustrate the utility of fluctuation-dissipation formulas for timeperiodic measures when considering the linear response of the dynamics to small perturbations. 4.1. Preliminaries, definitions, and assumptions. First, we recall the notion of a timeperiodic probability measure which will be needed throughout the reminder of this paper. Definition 4.1 (Time-periodic probability measure [32]). A measure-valued map given by t → µ s+t ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > (R d ) and induced by the family (P * s,s+t ) t∈R + , s ∈ R, defined in (3.4) is referred to as a time-periodic probability measure of period 0 < τ < ∞, if the following holds for any s ∈ R µ s+t = P * s, s+t µ s and µ s+τ = µ s ∀ t ∈ R + .
Then, the family (µ s+t ) s∈R,t∈R + consists of τ -periodic probability measures on R d .
The above results will be generalised to the dynamics in the extended state space in §4. representation on [0, τ )×R d ; this fact allows to prove ergodicity (in an appropriate sense) of timeperiodic measures supported on the random periodic paths on the fibre bundle 9 P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > [0, τ ) ×P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > R d .
To this end, consider the solutions of the SDE (2.1) satisfying the conditions of Theorem 3.5 and assume that the coefficients of (2.1) are time-periodic with period 0 < τ < ∞; we recast the solutions of (2.1) as an extended processX t (ω) = t, X s,x t (ω) T in the skew-product representation whereW t−s (ω) = 0, W t−s (ω) , ω ∈ Ω, and W t−s is the m-dimensional Brownian motion for the two-sided time (see §2.1 or [8]), andb : The dynamics in (4.2) or (4.3) can be represented in a more convenient form for the subsequent derivations by setting t → t + s, so that whereW t+s =W t+s (θ −s ω) is the Brownian motion satisfying (3.12). Finally, given the form of the coefficientsb,σ, it is convenient to consider the dynamics induced by (4.4) on the flat cylinder The RDS associated with the lifted dynamics (4.4) is generated in the skew-product representation (see, e.g., [24,8] The cocycle property 10 ofΦ in (4.5), i.e.,Φ(t+r, ω, · ) =Φ t, θ r ω,Φ(r, ω, · ) for all r, t ∈ R + , and a.a. ω ∈ Ω, can be verified by recalling that t+r mod τ = t+r−kτ, where k = t+r τ , and utilising (3.13). Note that, unless (2.1) is autonomous, {φ(t + s, s, θ −s ω, · ) : s ∈ R, t ∈ R + } does not have the cocycle property, and hence it does not generate an RDS on R d . The RDS representation of the non-autonomous dynamics of the SDE (2.1) will be useful in §4.3 when considering the ergodicity of measures supported on random periodic paths, and in the discussion of the linear response in §5.3.
for allÃ ≡ J × A ∈ B [0, τ ) × B R d and the transition kernel P defined in (3.1). 9 See, e.g., [24] for a detailed description of such structures on spaces of probability measures. 10 To be more accurate, the so-called 'crude' cocycle property can be easily verified from the flow induced by the SDE, and the crude cocycle needs to be 'perfected' in order to generate an RDS over the DS for the Brownian motion (see, e.g., [ The transition evolution (P t ) t∈R + induced byΦ and its dual (P * t ) t∈R + are given by 11 µ t+r (Ã) = P * tμ r (Ã) := [0,τ )×R dP (x; t,Ã)μ r (dx) ∀μ r ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > [0, τ )) ⊗ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > (R d , r ∈ R + , (4.8) with the short-hand notationμ r (dx) = δ (r mod τ ) (s)ds ⊗ µ r (dx) for skew product probability measures in the fibre bundle ; see, e.g., [24] for more details on skew-product fibre bundles on spaces of probability measures. Extension of (4.7) to arbitrary functions M [0, τ ) × R d can be carried out in a standard way.
Lemma 4.3. The transition evolutions (P t ) t∈R + and (P * t ) t∈R + have a semigroup structure. In particular, forμ t = δ (t mod τ ) ⊗ µ t in the fibre bundle P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 Every such τ -periodic measure is invariant under the discrete dynamics induced by (P * nτ ) n∈N 0 , i.e., Proof: The first claim is a direct consequence of (4.8), and the proof follows either by using the cocycle property ofΦ in the first line of (4.6) or by utilising the Chapman-Kolmogorov equation (3.2) for P in the last line of (4.6).
Regarding the second claim, consider measures supported on the random periodic pathS SinceS is a random periodic path of the RDSΦ, we have for allÃ for all r, t ∈ R + by the general properties the random periodic path (3.15); this could also be obtained directly from (4.8) by using the invariance ofS under the action ofΦ. Moreover, by the property (3.15). Thusμ t is a τ -periodic measure for the RDS Φ (t, · , · ) : t ∈ R + on [0, τ ) × R d which is supported on the random periodic pathS. The last two claims are simple consequences of the properties established above and the skew-product structure of probability measures supported on random periodic paths.
In the following sections, after outlining some general assumptions, we will investigate the existence and uniqueness of stable random periodic paths of the RDS Φ (t, · , · ) : t ∈ R + , and we will prove the ergodicity of probability measures associated with the dynamics of the skew-product lift (4.4) of the dynamics in (2.1) under some standard regularity assumptions.

4.1.2.
Assumptions. Throughout, we assume that the SDE (2.1) with time-periodic coefficients of period 0 < τ < ∞ satisfies the conditions of Theorem 3.5, so that (2.1) has global solutions.
In order to establish the existence of stable random periodic paths in §4.2, we will require the following assumption: satisfy the following: (i) There exist λ ∈ L 1 (R; dt), and a constant C 1, such that for some 1 < p < ∞ and all where L (2) is the two-point generator defined in (3.8) and associated with the SDE (2.1).
As pointed out later (Remark 4.12 in §4.3), this assumption is not strictly required for proving ergodicity of τ -periodic probability measures. However, without showing the existence of random periodic paths (in this case, stable random periodic paths), the existence of the skew-product τ -periodic measures would have to be assumed a priori alongside the ergodicity ofμ t for all fixed t ∈ [0, τ ) with respect to the discrete transition evolution (P * nτ ) n∈N 0 , as done in [32].
(a) An important class of coefficients satisfying Assumption 4.4, which yield global solutions of (2.1) are specified in Appendix A. In particular, we might take b(t, · ) ∈C 1,δ (R d ) and Here, ·, · denotes the dot product on R d and · hs denotes the Hilbert-Schmidt norm (aka Frobenius norm) defined by A 2 hs = trace(AA T ). Condition (4.12) is satisfied for (4.13) when (see 12 This condition can be replaced by a stronger but a more concrete constraint on the global existence of the p-th and it also leads to the global existence of the p -th absolute moment of the law of the associated SDE; tighter bounds can be obtained for p = 2, 3 as shown in Propositoin A.2 in Appendix A. Condition (4.12) is reminiscent of the Haśminskii-type regularity condition [43] for the existence and uniqueness of global solutions of SDE's; sufficient conditions for verification of Haśminskii's conditions require the existence of real-valued functions Coefficients satisfying (4.13) also satisfy (4.15), since for some L b ∈ C ∞ R, R + we have (b) Construction of the Lyapunov function V satisfying Assumption 4.4 is often not straightforward. However, one can construct (e.g., [43,46,64]) a polynomial Lyapunov function growing at infinity as |x| 2N , N ∈ N 1 , for a broad class of SDE's whose coefficients Many important classes of SDE's driven Levy processes (including the Brownian motion) satisfy the dissipative conditions (4.16) -(4.17); see [43,46,64] for more details.
In order to study the ergodicity of τ -periodic measures, we will require variants of the following standard conditions (e.g., [42]) to be satisfied: (i) Relative compactness property of the transition kernel P in (3.3).
(ii) Irreducibility of the transition kernel.
(iii) Strong Feller property 13 of the transition evolution (P s,t ) t s (3.3).
Thus, we will require the following version of the Hörmander condition (e.g., [73,65]) in §4.3 in addition to Assumption 4.4: Assumption 4.6. Denote by σ k , 1 k m, the columns of σ in (2.1), and assume that the following are satisfied for all t ∈ R: and [F, G ] is the Lie bracket between the vector fields F and G defined by which are supported on a unique random periodic pathS of the RDS {Φ(t, · , · ) : t ∈ R + } on [0, τ ) × R d and defined in (4.5) .
Proof. First, for ξ ∈ L p (Ω, F s −∞ , P), 1 < p < ∞, where F s −∞ := r s F s r , we show that {φ(t, s, ω, ξ) : s, t ∈ R, s t} converges to a random process S(t, ω) ∈ R d almost surely as s → −∞, and that S(t, ω) is bounded and independent of ξ. Next, we show that t → S(t, ω) is a unique stable random periodic path of period 0 < τ < ∞ for {φ(t, s, ω, · ) : s, t ∈ R, s t}. Finally, we conclude that the law of the random periodic pathS(t, ω) = t mod τ, S(t, ω) generates a τ -periodic measure for the skew-product RDS generated byΦ on the flat cylinder Existence of random periodic paths for the stochastic flow φ. Set ξ, η ∈ R d to be random variables on the filtered probability space Then, by Itô formula (e.g., Theorem 4.2.4 in [53] or Theorem 8.1 in [52]) and Assumption 4.4 we have for s t Thus, by the first part of (4.10) and Gronwall's inequality, we arrive at (4.20) Finally, given the bound (4.20), for r < s < t, we have and, utilising the above with Assumption 4.4(iii), yields lim sup Thus, for ξ ∈ L p (Ω, F s −∞ , P), 1 < p < ∞, the above bound implies that the L p limit of the flow {φ(t, s, · , ξ) : s t} exists as s → −∞. Note that this limit is independent of the initial condition ξ by (4.12). We denote this limit by the random process S : Then, by Chebyshev's first inequality (aka Markov's inequality; e.g., [6]), for any ε > 0, we have which implies that the convergence is also in probability. Thus, there exists a subsequence To simplify notation, we write Note that for ξ ∈ L p (Ω, F s −∞ , P) with the norm · p := (E| · | p ) 1/p we have by condition (4.12) of Assumption 4.4. Consequently, for any t ∈ R, we have Next, we show that t → S(t, ω) is a random periodic path of period 0 < τ < ∞ for the stochastic flow {φ(t, s, · , · ) : s t} using its τ -periodic property (see equation (3.13) with appropriately changed variables); namely Then, by the continuity of (t, s, x) → φ(t, s, · , x) and the flow property, we have The equalities (4.25) and (4.26) imply that S(t, ω) is a random periodic path (3.14) of period 0 < τ < ∞ of the stochastic flow φ(t + s, s, · , · ) : s ∈ R, t ∈ R + on R d .
Finally, let (μ t ) t∈R + ,μ t ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 It follows from (4.27) -(4.28) and Lemma 4.3 that the probability measureμ t is τ -periodic under the action of the transition evolution (P * t ) t∈R + which is induced by the RDS {Φ(t, · , · ) : t ∈ R + } on [0, τ ) × R d . The skew-product structure of these measures in P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > [0, τ ) ⊗ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > R d arises from Lemma 4.3, or directly from (4.6), so that for any

Ergodicity of time-periodic measures.
In this section, we turn to establishing ergodicity of the τ -periodic measures (μ t ) t∈R + ,μ t ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > R d , generated by the Markovian 14 RDS Φ (t, · , · ) : t ∈ R + which was constructed in (4.5) in the skew-product representation on the flat cylinder [0, τ ) × R d from the lifted flow of solutions of the SDE (2.1) with time-periodic coefficients. The existence of τ -periodic measures supported on stable random periodic paths was established in Theorem 4.7. The lack of stationarity and the unavoidable skew-product structure of the underlying dynamics pose additional challenges when dealing with ergodicity of P * t -invariant measures, as outlined below. The main theorem of this section (Theorem 4.11) is preceded by some preparatory results and definitions.
is ergodic with respect to the transition semigroup (P * t ) t∈R + in (4.8). One can check by the linearity ofμ 0 →P * tμ 0 and Fubini's theorem thatμ is an invariant measure for the transition semigroup ( where t →S(t, ω) = t mod τ, S(t, ω) , t ∈ R + , is a random periodic path (3.15) of an RDS generated by the lifted dynamics of the SDE (2.1) viaΦ in (4.5), and m 1 is the Lebesgue measure on R. Thus, given the invariance ofμ under the action of the transition semigroup (P * t ) t∈R + in (4.8), and the τ -periodicity ofμ t (see Definition 4.1), one has for anyÃ ∈ B [0, τ ) × B R d and any u ∈ R + . This implies that the expected time spent by the random periodic path t →S(t, ω) in any setÃ ∈ B [0, τ ) × B R d over a time interval of exactly one period is independent of the starting point.
Verification of ergodicity (in the sense of Definition 4.8) of τ -periodic measures (μ t ) t∈R + supported on the random periodic paths ofΦ requires one to assert that the time-averaged skewproduct measureμ in the fibre bundle P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n the need to deal with the random periodic, skew-product nature of the underlying dynamics, and it prevents a direct application of the classical tools for asserting ergodicity in the (asymptotically) stationary case. In particular, it is well-known (e.g., [78,Theorem 3.2.4]) that the following are equivalent 15 : (i) A probability measureμ is weakly mixing.
Thus, given the form of the transition kernelP in (4.6) and the underlying skew-product structure, it is clear that one cannot establish the mixing property in the random periodic regime 16 . Thus, this key condition in Doob's Theorem [26] does not hold in the random periodic regime which, alongside the lack of irreducibility of the transition kernel, renders the Hasminskii's Theorem [43] for asserting regularity of the transition kernel (needed in Doob's Theorem) inapplicable. Instead, theP * t -ergodicity ofμ can be verified by means of a proposition which was proved in [32, Lemma 2.18]; we repeat its statement below with a concise proof to make this section self-contained. The main benefit of utilising the proposition below when dealing withμ is that it essentially relies on ergodicity of τ -periodic measuresμ t for any fixed t ∈ [0, τ ) with respect to the discrete dynamics induced by (P * nτ ) n∈N 0 ; the subsequent use of the semigroup property of (P * t ) t∈R + allows one to show the ergodicity ofμ. Importantly, theP * nτ -ergodicity ofμ t on the respective Poincaré sections with a fixed t ∈ [0, τ ) turns the problem into a stationary one which can be dealt with using the standard methods. The result below provides an extension of the classical Krylov-Bogolyubov procedure (see, e.g., [8, §1.5]). Proof. Recall from (e.g., [8]) thatμ is ergodic ifP t IÃ = IÃ,μ -a.e.Ã ∈ B([0, τ )) × B(R d ) implies that eitherμ(Ã) = 0 orμ(Ã) = 1. First, we assume that (4.30) holds for anyÃ ∈ B([0, τ ))×B(R d ) withP (x; t,Ã) =P t IÃ(x) = IÃ(x). Then, it follows from (4.30) that This implies that IÃ(x) is a constant forμ -a.e.x ∈ [0, τ ) × R d . Thus, eitherμ(Ã) = 0 or µ(Ã) = 1. Conversely, assume thatμ is ergodic, then for anyÃ and (4.30) follows from (4.31) and from the Cauchy-Schwartz inequality. 15 These statements are not restricted to the skew-product representation of time-periodic measures. 16 As before, we exclude the stationary regime from the random periodic regime by requiring that fundamental period 0 < τ < ∞; see Definition 4.1.
Consequently, the subsequent verification of the ergodicity of theP * t -invariant measureμ on B([0, τ )) × B(R d ), relies (explicitly or otherwise) on the semigroup property and periodicity of the transition semigroup (P * t ) t∈R + , and on proving the strong Feller property of the transition evolution (P s,t ) t s in (3.3). Recall that the transition evolution (P s,t ) t s has the strong Feller property (i.e., P s,t ϕ ∈ C ∞ (R d ) for any ϕ ∈ M ∞ (R d )) if and only if (i) (P s,t ) t s is Feller; i.e., P s,t : (ii) For any ϕ ∈ C ∞ (R d ) the family (P s,t ϕ) t s is equicontinuous.
The first condition follows from the existence of the stochastic flow (see, e.g., [53,43]); thus, we only derive the second item in Proposition 4.10 below.
Proposition 4.10. Suppose that Assumption 4.6 holds. Then, for any t ∈ [s, s + T ), there exist 0 < C T < ∞ such that, for any x, y ∈ R d and any ϕ ∈ C ∞ (R d ), we have Proof. The proof consists of a tedious but relatively straightforward extension of results which are well known in the autonomous case; for detailed derivations, involving some Malliavin calculus estimates; see Theorem B.10 in Appendix B.2.
Given the above setting, we have the following main result of this section: Remark 4.12. The requirement in the above theorem that Assumption 4.4 holds is inherited from the conditions required in Theorem 4.7 for the existence of stable random periodic paths on which the τ -periodic skew-product measures (μ t ) t∈R + are supported; hence, the only additional condition in Theorem 4.11 is introduced by imposing Assumption 4.6 which is required in Proposition 4.10 to assert the strong Feller property of (P s,t ) t s . If one dropped Assumption 4.4, the existence of τ -periodic skew-product probability measures would have to be assumed a priori alongside the ergodicity ofμ t for all fixed t ∈ [0, τ ) w.r.t. the discrete transition evolution (P nτ ) n∈N 0 , as done in [32]. In the present case, the properties of the τ -periodic measures derived explicitly in the previous section allow us to dispense with such assumptions.
Proof of Theorem 4.11. The proof is relatively long and we divide it into four steps. Throughout, we skip the dependence on ω ∈ Ω in all quantities involving expectations w.r.t. P.
Step I: First, we show that for a random periodic path S : R × Ω → R d of the stochastic flow φ on R d , and η ∈ L p (Ω, F s −∞ , P), 1 < p < ∞, there exists 0 <C < ∞ such that To see this, note that from the definition of the random periodic path of a stochastic flow (3.14) we have S(s + nτ, ω) = φ(s + nτ, s, ω, S(s, ω)) P -a.s., so that by Assumption 4.4(i) and the fact that S(s) ∈ L p (Ω, F s −∞ , P), 1 < p < ∞, which was shown in the proof of Theorem 4.7.
Step II: We show that for 1 < p < ∞ there exists 0 < C τ < ∞ such that for n ∈ N 0 To see this, we note that from the definition of the periodic measure µ s , we have that i.e., µ s is invariant under the action of the dual of the discrete transition evolution (P * s,s+nτ ) n∈N 0 . Thus, for ψ ∈ Lip ∞ (R d ), we have for 1 < p < ∞, where we applied Hölder's inequality and estimate (4.32) in the last two lines respectively. Now, let ϕ ∈ C ∞ (R d ) be given. Setting ψ = P s+nτ, s+τ +nτ ϕ = P s,s+τ ϕ in (4.35), which holds due to (3.13), and using the invariance of µ s under the transition evolution (P * s,s+nτ ) n∈N 0 , we obtain by Proposition 4.10 that where C τ = C τC , and C τ is a constant appearing in Proposition 4.10.
Step III: Let A ⊂ R d be a closed set, take ϕ = I A , and consider the sequence (ϕ m ) m∈N 1 of functions defined by Next, for s ∈ [0, τ ), we have which implies that P (s, · ; s + nτ, A) = P s,s+nτ I A ∈ C ∞ (R d ) and, since µ s is invariant under (P * s,s+nτ ) n∈N 0 , (4.36) leads to By the covering lemma (e.g., [6]), the inequality (4.37) holds for any A ∈ B(R d ), and thus for By condition (4.11) of Assumption 4.4, there exists 0 < β < 1, 0 < K < ∞, such that It then follows that Step IV: In this final step, with the help of Step III, we show the convergence of Krylov-Bogolyubov scheme for the τ -periodic skew-product measures (μ t ) t∈R + on the cylinder [0, τ )×R d .
Remark 4.13. The invariance of the τ -periodic probability measures under the discrete evolution (P * nτ ) n∈N 0 on their respective Poincaré sections was pointed out in Lemma 4.3. It can be shown, as a consequence of [32,Theorem 4.11], that such τ -periodic probability measures are ergodic w.r.t. the discrete evolution (P * nτ ) n∈N 0 on their respective Poincaré sections; given that we require Assumption 4.6 to be satisfied, these measures are supported on all of R d . This fact will be useful in §5 concerned with ergodic averages in the context of the linear response.

Linear response in the random time-periodic regime
In this section, we derive a general formula for the linear response function which characterises the change of a statistical observable in response to small perturbations of SDE dynamics with a time-periodic ergodic probability measure. The results presented below build on and extend the derivations obtained for time-dependent stochastic systems in [61]. First, we derive the linear response formula associated with perturbations of dynamics with time-periodic measures, and we represent it via formulas exploiting the asymptotic statistical properties of the unperturbed dynamics; in line with terminology from statistical physics, these are termed fluctuation-dissipation formulas. In Theorem 5.14, we derive the fluctuation-dissipation formulas in the case when only the unperturbed dynamics has a time-periodic ergodic probability measure. In Theorem 5.16 we consider the linear response associated with perturbations of dynamics with a time-periodic ergodic probability measure under stronger conditions when perturbed dynamics also has a timeperiodic ergodic measure. During the revision of the manuscript, we become aware of related results derived independently in [20] for non-autonomous SDE's; those results are complementary to ours since they are confined to a finite time interval in the non-autonomous case with a restricted class of perturbations, and they do not deal with perturbations of time-periodic ergodic probability measures. We conclude with some examples of the linear response for the periodically forced stochastic Lorenz model used earlier in Example 4.14. In principle, the results discussed below apply to a wider class of SDE's generating time-periodic measures under less stringent conditions than those in Assumption 4.4; however, establishing the existence and ergodicity of such measures in a more general setting is not trivial and it is beyond the scope of this work.
Finally, shifting the time t → t + s allows one to represent the dynamics in (5.3) for t ∈ R + as dX α t+s =b(α(t + s),X α t+s )dt +σ(α(t + s),X α t+s )dW t+s ,X α s =x, (5.4) whereW t+s =W t+s (θ −s ω) due to (3.12). In what follows, we will always assume that the SDE with α = 0 satisfies the conditions of Theorem 4.7.
The lifted processΦ α : Note that if the flow φ α is induced by the solutions of (5.1) with the coefficientsb α(t), t, x andσ α(t), t, x which are time periodic for all α ∈ A,Φ α can be represented on a flat cylinder [0,τ ) × R d , [0,τ ) R modτ , 0 <τ < ∞, and the results of §4 hold; one obvious case is for α = 0 when, by construction, the coefficients are time periodic with period τ . Ifτ = τ for any α ∈ A, both the unperturbed (α = 0) and perturbed (α = 0) dynamics can be considered on the same cylinder. We will consider such a case in the last theorem of this section (Theorem 5.16).

g C x + A U O K A F O u A K d E E P Y C D A A
3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > (R d ); see, e.g., [24] for more details concerning the structure of skew-product fibre bundles of probability measures. The definition in (5.6) can be extend to ϕ ∈ M R × R d in a standard fashion.
The generator of the lifted one-point motion is given byL α = ∂ s + L α (see Definition 3.6). By construction (see, e.g., [61]), one can check that if µ α t ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n In the sequel, we derive fluctuation-dissipation formulas associated with the linear response for time-asymptotic SDE's dynamics in the random time-periodic regime with the ergodic measurē µ ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > [0, τ ) ⊗ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > R d , as in Theorem 4.11. We start with the definition of a linear response function which approximates changes in the statistical observables due to sufficiently small perturbations of the unperturbed dynamics with a time-periodic ergodic probability measure.  R d is the probability measure on the initial condition in (5.4), which is assumed throughout to be given by the τ -periodic ergodic (skew-product) probability measure associated with the 'unperturbed' dynamics with α = 0. If there exists a locally integrable function Rμ 0 ϕ such that the Gateaux derivative ofFμ 0 ϕ ( · , α) at α = 0 satisfies we say that Rμ 0 ϕ is a linear response function due to perturbations of the statistical observable Fμ 0 ϕ .
In other words, Rμ 0 ϕ can be defined if the functional Fμ 0 ϕ ( · , α) is Gateaux differentiable at α = 0 in the direction of ϑ, and the Gateaux derivative is linear and continuous in the neighbourhood of α = 0. The formula (5.10) can be interpreted as an O(ε) approximation of the change of the statistical observable Fμ 0 ϕ in response to a sufficiently small perturbation εϑ(t) around α = 0. The explicit time dependence in the perturbation ϑ(t) enables one to consider the linear response to small time-dependent changes in the coefficients of (5.4) relative to those in the original dynamics for α = 0; for example, one can consider changes in the "climatological" forcing (e.g., [1,3,58,61,33,35,36,37,38,63]).
Throughout the remainder of this section, we impose the following regularity conditions which reduce to Assumption 4.4 when α = 0, and which imply the smoothing property (e.g., [79]) of the transition evolutions (P α t ) t∈R + (in the x-component of the extended state space R × R d ): Assumption 5.2. Assume that there exists a proper interval A ⊆ R containing α = 0, and that the following conditions are satisfied for all s t, s ∈ R, and for all α ∈ A: (ii) D n ασk (α, t, · ) ∈C ∞ (R d ), n ∈ N 0 , and t →σ k (t, · , · ) is differentiable on A × R × R d , and for any multi-index β, andσ k , 1 k m the columns ofσ.
, · · · , 1 i, j, k m , and [F, G ](α, t, x) is the Lie bracket between the vector fields F and G defined by Remark 5.3. Assumption 5.2, which is a version of the Hörmander condition, implies the existence of a smooth density of the time-marginal probability measure on (R d , B(R d )) induced by the law of the solutions of (5.1); i.e., µ α t (dx) = ρ α t (x)dx, ρ α t ∈ C ∞ ∞ (R d ) ∩ L 1 + (R d ) for all s t. In order to simplify the subsequent derivations, we will abuse notation and set the following when dealing with the skew-product probability measuresμ α t ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4

n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A
3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > R) ⊗ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4

n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A
3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > (R d ,μ α t = δ t ⊗ µ α t , of the lifted process on R×R d associated with the SDE (5.4). This intuitive convention is consistent with the convention introduced in (4.6) for transition evolutions.

Preparatory lemmas.
We start with the following standard and preparatory results which utilise relatively well-known results from [79,86], and are aimed at representingP α t ϕ(x) −P t ϕ(x) in the form amenable to further analysis in the context of the linear response. The main results are derived in §5.3.
Note that the properties of a(x) and b(x) are fully controlled through Assumption 5.2.
Proposition 5.8. Suppose that the conditions of Lemma 5.5 are satisfied, and consider a function Then, f t,α ∈ C(R × R d ) and f t,α < ∞ for any fixedx ∈ R × R d . If Assumption 5.2 holds for 0 < ∞, and the initial condition in the lifted SDE (5.4) has p max(2, ) finite moments, then there exists a constant C = C(T, k, ϕ) > 0 such that for any fixedx ∈ R × R d one has where {Φ(r, · , · ) : r ∈ R + } is the RDS (4.5) generated by the SDE (5.4) with α = 0 (or (4.4)).
Remark 5.9. Note that for a dissipative dynamics satisfying (4.13) the dissipation coefficient L b 2 might not be large enough to satisfy (4.14) with a given p 2. Thus, not all dissipative dynamics automatically satisfy Proposition 5.8 for all time.
Proof. For (s, x) ∈ R × R d , one can obtain directly from (5.21) that The regularity and growth conditions of the coefficients (b,σ) imposed in Assumption 5.2 ensure the existence of global solutions to (5.4) which are represented viaΦ α (t − r, · ,x), t − r ∈ R + , and generateP α t−r in (5.6). If p 2 moments of the initial condition of (5.4) are finite and E[D β x ϕ(Φ α (t,x))] < ∞, |β| 2, t T , then by the assumption on (b,σ) and Lemma 5.5, we have f t,α ∈ C(R × R d ) and f t,α < ∞ for any fixedx ∈ R × R d .
Regarding (5.24), the polynomial growth ofb combined with the standard calculation utilising Itô's formula guarantees the existence of max(2, ) finite moments of the solution for T < ∞ (Theorem 3.5). Thus, (5.24) follows by the Cauchy-Schwarz inequality applied to E|f t,α | and the finiteness of the moments of a, b for T < ∞.
Considering (5.24) for T → ∞, may require additional dissipative constraints on the drift b, as outlined below. Moreover, we specify two explicit classes of ϕ ∈ C 2 The second part of Lemma 5.5, the term on the right of (5.26) is bounded by C ϕ 2,∞ so that one can set explicitly C = C T C ϕ 2,∞ in (5.24). Given, the existence of global solutions of (5.4) for all time (Assumption 5.2 and Theorem 3.5), the bound (5.24) can be extended to T → ∞ provided that the dissipative conditions (4.13)-(4.14) hold for (b,σ) with p = max(2, ); so that E|b(Φ(r,x))| < ∞, E|a(Φ(r,x))| < ∞, r ∈ R + , in (5.25). More generally, if ϕ ∈ C 2 (R × R d ) and ϕ( · , x) C l (1 + |x| l ) with 0 l < ∞, 0 C l < ∞, the bound (5.24) can be extended to T → ∞ provided that the dissipative conditions (4.13)-(4.14) hold for (b,σ) with p = max(2, , l). Both assertions can be obtained through derivations analogous to Lemma A.1 in Appendix A. 27) as in (5.19) of Lemma 5.6; so thatf t,α (r,x) = f t,α (r,x)+O(α) with f t,α defined in Proposition 5.8. Then, under the same conditions as those in Proposition 5.8,f t,α ∈ C(R × R d ) andf t,α < ∞ for any fixedx ∈ R × R d . Furthermore, for p 2 chosen as in Proposition 5.8 one has which can be extended to T → ∞ in a way analogous to that in Proposition 5.8.
Proof. This is a direct consequence of Proposition 5.8 and the fact that the O(α) terms involve D xP α t−r ϕ(x), D 2 xP α t−r ϕ(x), with coefficients given by D n αb and D n ασ which are controlled through Assumption 5.2 and the polynomial bound on the growth of the derivatives ofb w.r.t. α.

Linear response and fluctuation-dissipation formulas for time-periodic measures.
Here, in §5.3.1 we derive a general expression for the linear response function characterising the change in the statistical observable (5.8) to small perturbations of dynamics of an SDE whose time-asymptotic dynamics is characterised by time-periodic ergodic probability measures (see §4). This is followed in §5.3.2 by deriving a more tractable representation of the response function in terms of fluctuation-dissipation type formulas which allow one to express the change in the statistical observables through statistical characteristics of the unperturbed dynamics.
Proof. First, we set α = εϑ ∈ A and show that under the assumptions of the proposition the following holds for any ϕ ∈ C 2 τ (R × R d ) such that E[D β x ϕ(Φ α (t, · ))] ∈ L 1 (μ 0 ), |β| 2 (see Proposition 5.8 for a sufficient condition for this to hold for all time), withṼ defined in (5.21). To this end, it follows from Lemma 5.6 that for ε > 0 sufficiently small whereΦ(r, ω,x) represents the solution of (5.4) with α = 0 (or the solution of (4.4)). By Proposition 5.8, Corollary 5.10, and the dominated convergence theorem, we have Using Fubini's theorem and Proposition 5.8, we have By the Hörmander Lie bracket condition in Assumption 5.2, and ergodicity of the time-periodic measuresμ r , there exists 0 < ρ r ∈ C ∞ (R d ) ∩ L 1 + (R d ) such that (P * rμ 0 )(dx) =μ r (dx) =ρ r (x)dx, whereρ r (x)dx is understood in the sense of (5.11). Thus, for any ϕ ∈ C 2 Finally, by the definition of the response functional Rμ 0 ϕ we have for ϑ ∈ C 1 whereρ r (x)dx is understood in the sense of (5.11).

5.3.2.
Fluctuation-dissipation formulas. Given the general framework for the linear response in the time-periodic regime, we now derive a set of more tractable representations of the response function (5.30) via formulas exploiting the time-asymptotic statistical properties of the unperturbed dynamics (4.4), or (5.1) with α = 0; in line with the terminology inherited from statistical physics, these are termed 'fluctuation-dissipation' formulas. The first set of results in Theorem 5.14 shadows and formalises formulas derived in [61], while the results in a more restrictive Theorem 5.16 concern the linear response in situations when the 'α-perturbations' do not destroy the time periodicity and ergodicity of the dynamics in the sense that the coefficients in (5.1) remain τ -periodic for all α ∈ A. It turns out the two results are related in a specific way.
(iii) The linear response for perturbations of observables based on the ergodic measureμ is , |β| 2, as in Theorem 5.11 (see also Proposition 5.8 for a sufficient condition), andρ r (x)dx is understood in the sense of (5.11); i.e., As regards Part (iii), notice that due to the fact thatP * rμ =μ for any r ∈ [0, τ ), we have and the desired result can be derived by following analogous derivations to those above.
Similarly, byP * t -ergodicity ofμ (see Theorem 4.11) we have for any u ∈ R + (iii) Note that the function B r in (5.30) of Theorem 5.14 is unique almost everywhere. To see this, suppose that there existsB r ∈ L 1 (μ r ) such that This implies that Given that ϕ ∈ C 2 c ([0, τ ) × R d ) is bounded, taking limit as r → t in the above and applying the dominated convergence theorem, we obtain B t =B tμt -a.e. by arbitrariness of ϕ.
Proof. Part (i) is a direct consequence of Theorem 4.7 and Theorem 4.11 given the fact that Assumptions 5.4-5.2 hold for α in a proper interval A containing α = 0. For Part (ii), we proceed as at the beginning of the the proof of Theorem 5.11, except that due to Part (i), both the unperturbed and the perturbed measures are τ -periodic. Thus, for ϑ ∈ C 1 ∞ (R + ; R) and ε > 0 sufficiently small so that εϑ ∈ A, and for all ϕ ∈ E α there exists 0 < C ε,ϕ < ∞ such that where the bound is due to Proposition 5.8. Averaging both sides over t ∈ [0, τ ) we have Thus,μ α is weakly differentiable on E α . Next, by the Hörmander condition in Assumption 5.2, we haveμ α (dx) =ρ α (x)dx so that for any ϑ ∈ C 1 which implies thatρ α is weakly differentiable at α = 0 (withρ α understood in the sense of (5.11) to simplify notation). Furthermore, (5.36) yields ρ α ,L α ϕ = 0, α ∈ A, ϕ ∈ D(L α ) ∩ E α . Next, we set η, −Lϕ := ∂ αρ α , −Lϕ | α=0 , and note thatP t ϕ ∈ D(L α ) ∩ E α , t 0, for any ϕ ∈ D(L α ) ∩ E α (due to Assumption 5.2 and the associated smoothing property of (P s,t ) s t generating (P t ) t∈R + in (4.7); e.g., Proposition 4.10 and [79]). Thus, we have Kμ ϕ,W (t − r, r) = P r WP t−r ϕ ,ρ = P t−r ϕ, Wρ = P t−r ϕ, η , (5.46) where W is given in (5.40), and subsequently SinceP t−r ϕ ∈ D(L α ) ∩ E α for 0 r t, we have by (5.45) Remark 5.17. Note that Theorem 5.14 is more general than Theorem 5.16 in the sense that it only requires time-periodicity of the coefficients of the SDE (5.1) and the existence of timeperiodic probability measure for the unperturbed dynamics (i.e., for α = 0 in (5.1)) but it does not preclude the perturbed dynamics to have time-periodic measures. Thus, a natural question arises as to the connection between the linear response functionsR ϕ in (5.34) of Theorem 5.14 and (5.39) of Theorem 5.16, respectively, in the case when both the unperturbed and the perturbed dynamics (i.e., for α ∈ A in (5.1)) have time-periodic ergodic measures of period τ . The desired connection stems from the fact that under Assumption 5.2 the identity (5.45) leads to for any ϕ ∈ D(L α )∩E α , whereL = ∂ s +L is the generator of the one-point motionx →Φ(t, ω,x), t 0, on the extended state space [0, τ ) × R d . Thus, we obtain B = −Lμ * W, whereLμ * is the L 2 (μ) dual ofL. In fact, the above result also implies that W in (5.40) of Theorem 5.16 is not unique in contrast to B in (5.34) of Theorem 5.14. To see this, assume that there exists which implies that (see (5.46)) , it follows by the smoothing property ofP t−r (under Assumption 5.2) and the dominated convergence theorem that L ϕ, W −W μ = 0.
Since W −W ∈ L 1 (μ), W −W is a.e. constant; hence, W satisfying Theorem 5.16 is not unique.  due to α(t) = εϑ(t), which is chosen for simplicity to be in the space-time factorised form (recall, however, that the perturbation in this framework can take a more general form; see Remark 5.12).
Note that, analogously to (4.40), the periodically forced dynamics in (5.47) satisfies Assumption 5.2, and recall that in appropriate parameter regimes of (4.40) (or in (5.47) with α = 0) there exists a time-periodic ergodic measure µ t ∈ P < l a t e x i t s h a 1 _ b a s e 6 4 = " N i e x P 6 + 2 L C r w 6 M E J Y Z N 9 / D B N u Z c = " > A A A C D n i c b V D N S g M x G M z W v 1 r / q h 6 9 B I v g q e z W Q u 2 t 6 M V j B d s K 2 6 V k 0 2 w b m k 2 W J C u U Z d / B i 1 d 9 C 2 / i 1 V f w J X w G s 9 t F r T o Q G G a + L 5 m M H z G q t G 2 / W 6 W V 1 b X 1 j f J m Z W t 7 Z 3 e v u n / Q V y K W m P S w Y E L e + k g R R j n p a a o Z u Y 0 k Q a H P y M C f X W b + 4 I 5 I R Q W / 0 f O I e C G a c B p Q j L S R 3 G G I 9 F R h m X T T U b V m 1 + 0 c 8 C 9 x C l I D B b q j 6 s d w L H A c E q 4 x Q 0 q 5 j h 1 p L 0 F S U 8 x I W h n G i k Q I z 9 C E u I Z y F B L l J X n k F J 4 Y Z Q w D I c 3 h G u b q z 4 0 E h U r N Q 9 9 M 5 h F / e 5 n 4 n + f G O j j 3 E s q j W B O O F w 8 F M Y N a w O z / c E w l w Z r N D U F Y U p M V 4 i m S C G v T 0 t J N p h Y v k S p Q I q 1 k M A U 5 3 7 2 0 D V r N g r S d r 4 L 6 j b p z V m 9 c N 2 u d i 6 K q M j g C x + A U O K A F O u A K d E E P Y C D A A 3 g E T 9 a 9 9 W y 9 W K + L 0 Z J V 7 B y C J V h v n 4 U A n F E = < / l a t e x i t > (R d ) with a smooth density ρ t with respect to the Lebesgue measure on R 3 for all t ∈ [0, τ ), 0 < τ < ∞.
Assuming that the conditions of Theorem 5.11 hold, the linear response ∆Fμ 0 ϕ,ϑ (t) in (5.29) Rμ 0 ϕ (t − r, r)ϑ(r)dr, ϑ ∈ C 1 ∞ (R + , R), ϑ(0) = 0, (5.48) of the observable ϕ to the perturbation ε F(v)ϑ(t) in (5.47) in the random time-periodic ergodic regime is determined by convolving the response function Rμ 0 ϕ with ϑ, where (see Theorem 5.14) which is solely based on the unperturbed dynamics. In the above expressionμ t (ṽ) =ρ t (ṽ)dṽ, v = (s, v) ∈ [0, τ ) × R 3 , and Kμ 0 ϕ,Br (t − r, r) is the correlation function (5.32) of the random variables ϕ(ṽ t ), and B r (ṽ) = −∂ v (F(v)ρ r (ṽ))/ρ r (ṽ), withμ t (ṽ) =ρ t (ṽ)dṽ understood in the sense of (5.11). Thus, the response function can be written in a form amenable to computations as (see Remark 5.15(ii)) whereṼ * is defined in (5.22), B r is given in (5.33), and (P r,t ) t r is the family of transition evolutions defined in (3.3). Furthermore, (5.50) can be evaluated in a more practical fashion via appropriate ergodic averages, as discussed in Remark 5.15(ii). For simplicity of the numerical illustration we consider the linear response of the expectation of the solutions to (4.40) to the perturbation ε F(v α )ϑ(t) introduced in the drift coefficient of (5.47). Given the dynamics (5.47) and ϕ(v) = v, setting p = 2 is sufficient for Assumption 4.4 and Proposition 5.8 to hold (i.e., Theorems 4.7, 4.11, 5.11, 5.14 will hold for all time if the perturbation maintains dissipativity, which is the case here). In the examples shown in Figures 2, 3 we denote the expectation of the solutions to (5.47) by E[x t ], E[y t ], E[z t ], and we consider the response of the expectation to a spatially uniform perturbation ε F(v α )ϑ(t) with F(v α ) = (f , 0, 0) T and where The simulations were performed for (5.47) with the same parameter values as those in Example 4.14 in the random time-periodic regime, and the perturbation with t 0 = 80τ, ∆T = 1 with the amplitudes set to ε = 0.05 in Figure 2, and ε = 0.25 in Figure 3. The unperturbed initial time-periodic measure at t = 0, i.e.,μ 0 = δ 0 ⊗ µ 0 , was approximated from long-time simulations of an ensemble of solutions to (5.47) with α = 0 at t = τ n, n ∈ N 0 . To simplify the notation, the linear response ∆Fμ 0 ϕ,ϑ (t) in (5.29) of the expectations is denoted by, respectively, ∆E x (t), ∆E y (t), ∆E z (t). The linear response was estimated with the help of the fluctuation-dissipation formula (5.50), where K ϕ,Br (t − r, r) in (5.32) exploits the statistical correlations in the time-asymptotic dynamics of the unperturbed system (4.40) via (5.31). As expected from the theory, the linear response provides a good approximation for a sufficiently small perturbation (Figure 2), and it deteriorates with the increasing amplitude of the perturbation ( Figure 3); the accuracy of the approximation improves still with the decreasing amplitude of the perturbation but we do not show these unsurprising results.
In the simple example illustrated in Figure 4, we consider the response of the expectation of the solution to (5.47) in the stable random time-periodic regime to a spatially uniform time-periodic perturbation ε F(v α )ϑ(t) with ε = 0.1, F(v α ) = (f , 0, 0) T and Finally, it needs to be stressed that the direct numerical evaluation of the correlation function K ϕ,Br (t−r, r) in (5.49) is, in general, very computationally intensive due to the need for estimating the time-dependent density ρ t (v), v ∈ R 3 , t ∈ [0, τ ); more practical implementations rely on  various approximations (e.g., a Gaussian approximation of the underlying density), and they were discussed in [61] in the time-periodic setting, and in [1,2,3,58,61,33,35,36,37,38,63] in the stationary setting. These references also consider much more elaborate examples than what we could consider in this work.
Next, since |x| p−2 (1 + |x| 2 ) It turns out that sharper bounds can be obtained for p = 2, 3; these are derived in Proposition A.2.
Finally, note that for Y s,x t (ω) = φ(t, s, ω, x) − x we obtain Consider the same setup as in Lemma A.1. Suppose further that there exist bounded functions L b 1 (·), L b 2 (·), L σ (·) ∈ C ∞ (R; R + ) such that Then, there exists a stochastic flow {φ(t, s, · , · ) : s t} on R d induced by the global solutions of (2.1), which has a finite second absolute moment for all time. Moreover, if for some κ > 0 then the stochastic flow has a finite third moment for all time.
Remark A.3. It is worth noting that, for L b 1 , L b 2 , L σ constant, and such thatb 3 > 0, the upper bound on the asymptotic moment E|φ(t, s, x)| p for p = 3 is optimised for κ 2 = 12 .17) this fact merely reflects the Jensen's inequality for the second and third absolute moments, i.e., E|X s,x t | 2 (E|X s,x t | 3 ) 2/3 , but it is useful in Example 4.14.
By Itô's formula, we have But Y x t,s (ω) = φ(t, s, ω, x) − x P-a.s., ϕ(x) = g(x)  Next, note for p 2, |x| p (1 + |x| 2 ) p 2 and by the assumption that V (t, x) C|x| p , we obtain E V (t, φ(t, s, x) − x) CE 1 + |φ(t, s, x) − x| 2 In order to prove Theorem 4.10, which is a generalisation of standard results to non-autonomous SDE's, we first outline some basic notions from Malliavin calculus; the actual proof is given in and discussed in Appendix B.2.
B.1. Malliavin calculus estimates. Establishing the strong Feller property of Markov evolutions (P s,t ) t s in our setting requires some estimates rooted in Malliavin calculus. We recall the main concepts and results on the Wiener space (Ω, F, P); see (e.g., [39,42,65,70,73,90]) for a comprehensive treatment. To this end, consider the Hilbert space H = L 2 ([s, ∞); R m ) equiped with the inner product Following the approach due to Malliavin (e.g., [65,73]), we introduce a derivative operator D for a random variable G on the space L ∞− (Ω; E). We say that G ∈ D 1,∞ (E) if there exists DG ∈ L ∞− (Ω; H × E) such that for any η ∈ H, We define the k-th Malliavin derivative by D k G = D(D k−1 G), which is a random variable with values in H ⊗k ×E. For any integer k 1, the Sobolev space D k,p (E) is the completion of D k,∞ (E) under the norm G k,p,E = G k−1,p,E + D k G 1,p,H ×k ×E .
It turns out that D is a closed operator from L p (Ω; E) to L p (Ω; H × E). The ajoint δ of the operator D called the divergence operator is continuous from D 1,p (H × E) to L p (Ω; E) for any p > 1, with the duality relationship given as E DG, u H×E = E G, δ(u) E , (B.1) for any G ∈ D 1,p (H × E) and u ∈ D 1,q (H × E), with 1 p + 1 q = 1. Throughout the remaining part of this section we assume the following notation: -C is a generic constant which may depend on T, the exponent p > 1, the initial point x and fixed element η of the Hilbert space H = L 2 [s, ∞); R m . -(H n ) denotes a class of coefficients b, σ k , 1 k m, where σ k are columns of σ, such that b(t, · ) ∈C n , σ k (t, · ) ∈C n R d .

Definition B.3 (Mean square gradient). Let G(x)
: Ω → R d be a measurable function for all x ∈ R d and i ∈ F . We say that the mean square gradient of G(x) with respect to x exists if there is a linear map A(x) : Ω → R d×d such that for any v ∈ R d , We denote the mean square gradient matrix A(x) by D x G(x).
Theorem B.4 (e.g., [53,65,73]). Assume the condition (H 2 ) holds true. Let φ(t, s, ω, x), t s, be the solution of the SDE (2.1). Then, the mean square gradient of φ(t, s, · , x) with respect to x exists. If we define J t,s = D x φ(t, s, · , x), then Now, let D r φ(t, s) be the solution of the following SDE: D u φ t,s = 0, for s t < u.
Next, we recall a result on the Malliavin differentiability of the derivative flow J t,s , t s. To this end, let's denote by D u the Malliavin derivative with respect to the -th component of the Brownian motion W at time u.
Lemma B.6. Suppose that the condition (H 3 ) holds. Then for all s t s+T, J t,s ∈ D 1,∞ (R d ×R d ) and for any p 2, there exists a positive constant C = C(T, p, x), such that for all j = 1, · · · , m and u ∈ [s, s + T ], E sup s t s+T |D j u J t,s | p C.
Moreover, for any t s + T, X(t, s) ∈ D 2,∞ (R d ) and for any p 2, there exists a positive constant C = C(T, p, x) such that for all j, l = 1, · · · , m and ς, u t E|D j u (D l ς φ(t, s))| p C. Remark B.7. If (H ∞ ) holds true, then φ(t, s) ∈ D ∞ (R d ) and J t,s ∈ D ∞ (R d × R d ).