A LOCAL SENSITIVITY ANALYSIS FOR THE KINETIC KURAMOTO EQUATION WITH RANDOM INPUTS

We present a local sensivity analysis for the kinetic Kuramoto equation with random inputs in a large coupling regime. In our proposed random kinetic Kuramoto equation (in short, RKKE), the random inputs are encoded in the coupling strength. For the deterministic case, it is well known that the kinetic Kuramoto equation exhibits asymptotic phase concentration for well-prepared initial data in the large coupling regime. To see a response of the system to the random inputs, we provide propagation of regularity, localin-time stability estimates for the variations of the random kinetic density function in random parameter space. For identical oscillators with the same natural frequencies, we introduce a Lyapunov functional measuring the phase concentration, and provide a local sensitivity analysis for the functional.

1. Introduction.Collective behaviors of oscillatory complex systems are ubiquitous in our nature, e.g., flashing of fireflies, chorusing of crickets, synchronous firing of cardiac pacemaker and metabolic synchrony in yeast cell suspension [1,7,15,32,33] etc. Aforementioned collective patterns come down to synchronization phenomena.The jargon "synchronization" represents the adjustment of rhythms in an ensemble of weakly coupled oscillators.Compared to long human history, a rigorous treatment for synchronization started only several decades ago in the pioneering works by Kuramoto and Winfree in [25,26,37].They introduced simple, continuous dynamical systems for weakly coupled oscillators, and explained where ν i is an intrinsic natural frequency of the i-th oscillator whose pdf is g = g(ν), and nonnegative random field κ ij = κ ij (z) measures the random coupling strength between i and j-th oscillators.Throughout the paper, we assume that κ ij is symmetric in i and j: For the deterministic case where all randomness were quenched, i.e., emergent dynamics of (1) has been extensively studied in [2,6,10,11,12,13,14,20,23,28,29,30,34,35,36] where the complete synchronization and stability conditions were proposed.The pathwise well-posedness of (1) can be done using the standard Cauchy-Lipschitz theory.In authors' recent work [18], a local sensitivity analysis for (1) has been addressed.In this paper, we are interested in the correpsonding mean-field equation which can effectively describe the dynamics of system (1) with N 1.Let T = R/(2πZ), and f = f (t, θ, ν, z) be an one-oscillator probability density function on the extended phase space T × R × Ω at time t.Then, the density function f satisfies the RKKE [1,27]: When the extrinsic randomness in κ(z) and initial data f 0 (z) are quenched, wellposedness and dynamic features of the kinetic Kuramoto equation have been studied in [4,5,8,27].
In this paper, we address a local sensitivity analysis for (2) to see the effect of random parameter in f = f (t, θ, ν, z) which is one of topics in uncertainty quantification (UQ).More precisely, we study the propagation of regularity of f in the random space, and provide concentration and stability estimates of z-variations {∂ α z f }.The UQ for mean-field flocking models were first addressed in [3,9] where particle based gPC methods were discussed.On the other hand, systematic local sensitivity analysis for the Cucker-smale and Kuramoto models have been addressed in authors' series of recent works [16,17,18].Thus, the current work is a continuation of this systematic research on the local sensitivity analysis for flocking and synchronization models.
The main results of this paper are two-fold.First, we present pathwise wellposedness and stability estimate of the RKKE by establishing a priori estimates (see Theorem 3.2, Theorem 3.3 and Theorem 3.4): for T ∈ (0, ∞), z ∈ Ω, l ≤ k, where f and f are solution processes to (2) corresponding to initial data f 0 and f 0 , respectively.Second, we consider identical oscillator with ϕ(ν) = δ 0 .In this case, we can write f (t, ν, θ, z) = ρ(t, θ, z)ϕ(ν) and ρ satisfies Now, we introduce a Lyapunov functional L measuring the phase concentration: Note that the zero convergence of L[ρ] as t → ∞ implies the asymptotic formation of phase concentration in probability sense.This can be seen easily from Chebyshev inequality as follows: This implies Under suitable conditions on initial data and system parameters, we will show that there exists random functions C(z) and Λ(z) such that (see Theorem 4.3 for details).However, higher-order sensitivity analysis for L[|∂ α z ρ|] (t, z) with α 1 might lead to the exponential growth (see Remark 4), a phenomenon not observed in earlier works in this direction.
The rest of this paper is organized as follows.In Section 2, we briefly introduce the random kinetic Kuramoto equation and discuss its basic properties.In Section 3, we study a pathwise well-posedness of the RKKE by providing a priori estimates such as boundedness of H l π (L ∞ θ,ν ) in any finite time interval, and provide the stability estimates for the RKKE.In Section 4, we perform a local sensitivity analysis for a Lyapunov functional (3).Finally, Section 5 is devoted to a brief summary of our main results and some remaining issues to be explored in future.In Appendices A and B we provide the proof for Theorem 3.2 and Lemma 4.2, respectively.

Gallery of Notation:
Throughout the paper, we use the following notation: Let π : Ω → R + ∪ {0} be a nonnegative p.d.f.function, and let y = y(z) be a scalar-valued random function defined on Ω.Then, we define the expected value as and a weighted L 2 -space: with an inner product and norm: Let h = h(θ, ν, z) be scalar-valued random function defined on the extended phase space T × R × Ω.For such h, we define a Sobolev norm W k,∞ θ,ν and a mixed norm Moreover, as long as there is no confusion, we suppress π and Ω dependence in 2. Preliminaries.In this section, we briefly introduce the RKKE and study its basic properties.Let θ i = θ i (t, z) be a random phase process of the i-th Kuramoto oscillator whose dynamics is governed by the following system of random ordinary differential equations (ODEs): Here, the coupling matrix (κ ij (z)) is assumed to be a symmetric random matrix.In literature [1,13,21] on the Kuramoto model, the randomness in natural frequencies is assumed to be time-independent and quenched so that ν i is a constant parameter.It will be interesting to see how random natural frequency ν i and random coupling strength κ ij (z) interplay in the synchronization process of (4).This uncertain quantification (UQ) question for (4) was addressed in authors' recent work [18].Next, we consider a situation where the number of oscillators tend to infinity and the coupling strengths κ ij (z) are uniform and identical: For the derivation of the mean-field model associated with (1) and ( 5), we refer to [27,31] for details.It is more convenient to rewrite system (4) with ( 5) as a dynamical system on the extended phase space T × R: Next, we return to the pathwise mean-field limit of (6) as N → ∞.The formal mean-field limit equation can be easily identified using the standard BBGKY hierarchy based on the formal weak limit of marginal distribution functions and molecular chaos assumption (we refer to [22,24] for a brief introduction of BBGKY hierarchy).Note that for a frozen z ∈ Ω, the vector field generated by system ( 6) is bounded and Lipschitz continuous, so it does satisfy Neunzert's framework in [31] based on particle-in-cell method and measure-valued solutions.In fact, this has been rigorously done in [27] in any finite-time interval for any initial data.Recently, for an augmented Kuramoto model uniform-in-time mean-field limit is also derived by combining uniform stability analysis and finite-in-time mean-field limit in [21].Thus, for each z ∈ Ω, we can perform the same argument as the deterministic case to derive the kinetic equation with random inputs: subject to initial data: Here the initial datum f 0 is assumed to satisfy the following constraints: Lemma 2.1.Let f be a smooth solution to (7), ( 8) and (9) satisfying additional periodic boundary condition: Then, for each z ∈ Ω and t > 0, we have Proof.We multiply 1 and ν to (7) 1 and integrate the resulting relation over T × R to obtain the desired estimates: Since the ν-variable is not a dynamic variable as can be seen from ( 6) 2 , thus the ν-support of f will not be changed along the random Kuramoto flow.For a later use, we present the above argument in the following lemma.
Lemma 2.2.Let f be a C 1 -regular process to (7), and suppose that the initial process f 0 (z) has a compact support in ν for each z ∈ Ω: there exists a positive random function Then for each z ∈ Ω and t > 0, we have Before we leave this section, we state a Grönwall-type lemma to be used in later sections.
where α, β and C are non-negative constants α = β.Then y satisfies Proof.We multiply ( 10) by e αt and integrate it over (0, t] to give This yields the desired estimate. 3. Propagation of Sobolev regularity and stability estimate.In this section, we present a pathwise well-posedness of (2) and propagation of For each l ∈ N ∪ {0}, it is easy to check that the z-variations {∂ l z f } satisfy a hierarchical system: , l ≥ 1.
(11) Note that the L.H.S. for f and its z-variations ∂ l z f have the same transport structure, while the R.H.S. for (11) 2 has a lower-order z-variation terms.Thus, the characteristics for ∂ l z f will be the same as that of f and hence, independent of l.More precisely, for a given (θ, ν) ∈ T × R and z ∈ Ω, we define a random forward characteristics (θ(t, z), ν(t, z)) := (θ(t; 0, θ, ν, z), ν(t; 0, θ, ν, z)) as a solution to (11): Now, we define the ν-support of the process ∂ l z f and its diameter as follows: For l = 0, we set Note that the ν-support of the process ∂ l z f is a subset of the ν-support of f .Then, since the ν-support of f does not change along the dynamics of (11) 1 by Lemma 2.2, we have In this subsection, we study the propa- whose dynamics are given by the hierarchical system (11).For notational simplicity, we set , and we denote a generic non-negative random function by C(z, T ) which depends on T and z and it may differ from line to line.
Proposition 1.For k ∈ N and z ∈ Ω, let f 0 := f 0 (z) be the initial process satisfying Then, for T ∈ (0, ∞), there exists a unique W k,∞ -regular solution process where C(z, T ) is a nonnegative random function C(z, T ).
Proof.The existence and uniqueness of the solution can be found in [27].So we only provide a priori estimate for the solution process f .We apply Next, we split our estimate into two parts Let z ∈ Ω be fixed.
• Case A (α + β = 0): Note that We use the method of characteristics to obtain Since • Case B (1 ≤ α + β ≤ k): Next, we consider the cases: Case B-1 (β = 0 case): In this case, we use the same argument as in Case A to see that On the other hand, since The R.H.S. of ( 19) can be estimated as Now, we integrate relation (19) along the characteristics and use (20) to get Case B-2 (β ≥ 1 case): It follows from (15) that (22) can be simplified as We integrate the above relation along the characteristics and use the estimate to obtain Therefore, we obtain relations ( 21) and ( 24), sum the resulting relation over all 1 ≤ α + β ≤ k and add (18) to obtain a Gronwall's inequality: Then Grönwall's lemma yields the desired estimate (14).
Remark 2. Note that the Sobolev embedding theorem yields that for k > 2, W k,∞ θ,νsolution is C 1 (T × R) along the path.Thus, it is a classical solution to (2) along the path.Now, we provide a lemma regarding the estimates of the frequency ω[f ].Lemma 3.1.For l ≤ k ∈ N and T ∈ (0, ∞), suppose that the two solution processes Then, for t ∈ [0, T ) and z ∈ Ω there exists a nonnegative random function C(z) such that Proof.We first recall the relation: where we used k l ≤ 2 k and C(z) is given by (ii) Similar to (i), we have where C(z) is given by (iii) The third estimate follows from the defining relation of ω[f ] in (2).Now, we are ready to provide well-posedness of the process ∂ l z f for every l ∈ N. Theorem 3.2.For k ∈ N and T ∈ (0, ∞), suppose that the initial process f 0 satisfies the following conditions: for each z ∈ Ω and l = 0, Proof.Since the proof is lengthy and tedious, we postpone its detailed proof in Appendix A.Here we briefly explain why one has a lower W k−l,∞ -regularity for higher z-variation ∂ l z f .In Proposition 1, we have f (t, z) W k,∞ ≤ C(z, T ).Then, it follows from (11) that Since R.H.S. of the above relation has a term (∂ θ f ), the above relation yields ∞ -estimate.Thus, we can get at most W k−2,∞ estimate for ∂ 2 z f .Inductively, we can get W k−l,∞ -estimate for ∂ l z f .This is why we have lower-order regularity for ∂ l z f .Next, we provide the boundedness of the solution process in H l π (L ∞ θ,ν )-norm.Theorem 3.3.For k ∈ N and T ∈ (0, ∞), suppose that the initial process f 0 and coupling strength satisfy the following conditions: Then, for T ∈ (0, ∞) we have The proof is almost similar to that of Proposition 1.Thus, we briefly outline the proof here.By the same argument as in the proof of Proposition 1, we have We use Grönwall's lemma to obtain Finally, we square both sides in (25), multiply by π(z) and integrate over Ω to obtain the desired estimate:

3.2.
Local-in-time stability estimate.In this subsection, we provide a localin-time W k,∞ -stability estimate for the RKKE.More precisely, we derive pathwise stability estimate of ( 2) with respect to initial data.
Proposition 2. For k ∈ N, z ∈ Ω and T ∈ (0, ∞), let f and f be two W k+1,∞processes to (2) with the initial process f 0 and f 0 satisfying the following conditions: for each t ∈ (0, T ) and z ∈ Ω, Then, there exists a positive random function C(z, T ) such that for each z ∈ Ω, Proof.We use a similar argument as in Proposition 1 to derive the estimate where 0 ≤ α + β ≤ k.Finally, we sum the relation ( 26) over all 0 ≤ α + β ≤ k to derive Therefore, we use Grönwall's inequality on (27) to obtain the desired estimate.
As an application of the arguments in Theorem 3.2 and Proposition 2, we get the local-in-time stability estimate of variations ∂ l z f in W k−l,∞ -norm.Theorem 3.4.For k ∈ N and T ∈ (0, ∞), suppose that two initial data f 0 and f 0 satisfy the following conditions: for each z ∈ Ω, and let f := f (t, z) and f := f (t, z) be two W k+1,∞ -regular solution processes to (11) with initial data f 0 (z) and f 0 (z), respectively.Then, there exists a positive random function C(T, z) such that Proof.We basically follow the arguments in Theorem 3.2 and Proposition 2. Thus, we omit the details.

4.
A local sensitivity analysis for phase concentration.In this section, we provide a local sensitivity analysis for the phase concentration that emerges in (2).Since the kinetic equation ( 2) has been derived from the first-order model, it is not easy to see how frequency synchronization emerges from (2).However, for the kinetic Kuramoto equation with g(ν) = δ 0 (ν), we can study the phase synchronization using a Lyapunov functional approach.First, we set: We substitute this ansatz into (2) to obtain an equation for ρ: Recall a Lyapunov functional L measuring the concentration of phases defined in (3): As discussed in Introduction, if L[ρ] goes to zero, then ρ tends to δ θρ,c in probability.
Next, we define several notation regarding the θ-support of ρ: If there is no confusion, we set θ c := θ ρ,c , where ρ is the solution process to (28).At this point, we would like to see the basic properties of the solution process (28).
Proposition 3. Let ρ := ρ(t, z) be a C 1 -regular process to (28).Then for each z ∈ Ω, we have Proof.(i) The conservation of total phase can be followed by the direct integration of (28) using the periodic boundary condition in θ-variable.For the second relation, we multiply θ to (28) 1 to get where the last equality follows from the antisymmetry of the integrand.
Remark 3. Proposition 3 yields that ρ c := T ρdθ and θ ρ,c are constants.Moreover, as long as the initial process is nonnegative, ρ is also nonnegative.Hence, without loss of generality, we may assume Under the above setting, we have Thus, for the local sensitivity analysis of L[ρ], we just need to consider L[∂ l z ρ], which is also bounded by L[|∂ l z ρ|].
Now we provide the contraction property of the θ-support of ρ in the following lemma.
Lemma 4.1.Suppose that the θ-support of the initial process ρ 0 := ρ 0 (z) satisfies the following compactness condition: Then, for C 1 -regular solution process ρ := ρ(t, z) to (28), we have Proof.Consider a forward characteristic θ := θ(s; t, θ, z) which is defined as a solution to the following equation: First we consider characteristic curve starting from the maximal point θ M (t, z) := sup{θ | θ ∈ supp θ ρ(t, z)}.Then, it is easy to see that it is nonincreasing: Similarly, the characteristics curve starting from the minimal point θ m (t, z) Thus, we can deduce from ( 30) and ( 31) that D θ (ρ)(t, z) does not increase at every time t.This implies our desired result.
Proposition 4. Let ρ := ρ(t, z) be a C 1 -regular solution process satisfying the following condition: Then, L[ρ](t, z) decays exponentially fast along the sample path: where R 0 (z) is defined by the following relation: Proof.Under the setting ( 29), the functional L[ρ] becomes Then, it follows from (28) that where we used the change of variable θ ↔ θ * .
On the other hand, by assumption and Lemma 4.1, Since sin x ≥ R 0 (z)x, x ∈ [0, D θ (ρ 0 )], we have the following relation: We use (33) and (34) to yield where we used and Proposition 4.1 (i).Finally, we use Grönwall's lemma on (35) to obtain the desired result.

As a direct corollary of Proposition 4, we have the following exponential decay of E[L[ρ]](t).
Corollary 1. Suppose that the initial data and coupling strength satisfy the following conditions: where ε is a positive constant.Then, for any C 1 -regular solution process ρ := ρ(t, z) to (28), there exists a positive constant C = C(ε, η) such that Before we consider the local sensitivity analysis about phase concentration, we first provide the following technical lemma.Lemma 4.2.For k > 1, let ρ be a C k -regular solution process to (28) satisfying Then for α < k, we have the following upper and lower bound estimates: where R 0 (z) is defined in (32).
Proof.Since the proof is rather lengthy, we leave its proof to Appendix B. Now, we are ready to provide the local sensitivity analysis for phase concentration.Note that for l ∈ N, ∂ l z ρ satisfies Note that the θ-support of ∂ l z ρ is a subset of the θ-support of ρ, which is contractive.Now, we would like to analyze the first-order derivative of the functional Λ[ρ] with respect to z-variable.
Theorem 4.3.Suppose that initial density ρ 0 satisfies , for each z ∈ Ω, and the coupling strength κ is continuously differentiable and bounded.Let ρ := ρ(t, z) be a C 2 -regular global solution to (28) satisfying a priori condition: Then we have where the non-negative random function D is given by Proof.We differentiate (28) with respect to z to yield We multiply (37) by sgn(∂ z ρ) to yield Then, one can use (38) to obtain Next, we estimate I 11 and I 12 as follows: • Step A (Estimates for I 11 ): By direct calculation, one has (Estimate of I 111 ): We use relation (34) and similar argument as in I 311 of Appendix B to yield (Estimate of I 112 ): Again similar to I 312 in Appendix B, we use Proposition 4 to find In (40), we combine (41) and ( 42) to obtain • Step B (Estimates for I 12 ): Note that (Estimate of I 121 ): In this case, one gets (Estimate of I 122 ): Note that It follows from Proposition 4 and Lemma 4.2 that where J := J[ρ](z) denotes the following random functional: Thus we use the Grönwall-type inequality in Lemma 2.3 on (47) to yield Therefore, we can obtain We combine ( 43) and (48) to yield where the random function D := D(z) was given by D(z) Hence, we use the Grönwall type inequality to obtain This yields our desired result.
Remark 4. In this remark, we discuss our results about local sensitivity analysis for the functional L[ρ].
1. Note that in the proof of Theorem 4.
Since the coefficient 2κ(z) (1 − R 0 (z)) > 0, the above differential inequality does not yield the time-decay of L[|∂ 2 θ ρ|].Thus, it prohibits our local sensitivity analysis for the phase concentration for higher order derivatives.2. On the other hand, it follows from Lemma 4.2 that

Conclusion.
In this paper, we provided a local sensitivity analysis for the kinetic Kuramoto equation with random inputs in a large coupling regime.In the absence of random inputs, it is well known that the kinetic Kuramoto model exhibits a phase concentration phenomena in the large coupling regime.In authors' earlier series of works, we have begun a systematic local sensitivity analysis for the Kuramoto model with random inputs.For the Kuramoto model, we provided a sufficient framework for the local sensitivity analysis on the asymptotic dynamics of solution process.In this work, we have not only shown the well-posedness of the uncertain problem and stability under random perturbation, but also conducted local sensitivity analysis regarding the phase concentration that could be observed in the Kuramoto model for identical oscillators.For this, we have considered a Lyapunov functional measuring the phase concentration and performed a local sensitivity analysis on the functional.In summary, we found two interesting effects due to uncertainties for (2): • (Decreasing (θ, ν)-regularity for higher-order z-regularity estimate for f ): Propagation of high-order regularity in z-variable is measured in low-order Sobolev norm in the sense that for T • (Formation of zero θ-variance and unbounded variance of ∂ α θ ρ): The θ-variances for ∂ α θ ρ with |α| ≤ 1 tend to zero exponentially fast, whereas variances of higher order quantity ∂ α θ ρ with α 1 grows exponentially fast.
Finally, we gather the results in (60) and (61), sum those over all 1 ≤ α + β ≤ k − l and add the result in (53) to obtain Finally, we use Grönwall's lemma to derive the desired estimate.
where we used Cauchy-Schwarz inequality.In (65), we combine (66) and (67) to obtain • Case B (Estimates for I 32 ): Note that which is our desired upper-bound estimate.
(ii) (A lower bound estimate): we will estimate separately I 3n 's similarly to the upper bound case.Thus we estimate first I 31k 's as follows: We combine all estimates for I 3n 's to yield which gives our desired lower-bound estimate.