We develop a thorough mathematical analysis of the effective Mori–Zwanzig (EMZ) equation governing the dynamics of noise-averaged observables in stochastic differential equations driven by multiplicative Gaussian white noise. Building upon recent work on hypoelliptic operators, we prove that the EMZ memory kernel and fluctuation terms converge exponentially fast in time to a unique equilibrium state that admits an explicit representation. We apply the new theoretical results to the Langevin dynamics of a high-dimensional particle system with smooth interaction potential.

The Mori–Zwanzig (MZ) formulation is a technique originally developed in statistical mechanics31,57 to formally integrate out phase variables in nonlinear dynamical systems by means of a projection operator. One of the main features of such formulation is that it allows us to systematically derive exact generalized Langevin equations (GLEs)5,9,48,58 for quantities of interest, e.g., macroscopic observables, based on microscopic equations of motion. Such GLEs can be found in a variety of applications, including particle dynamics,16,17,23,28,43,51 fluid dynamics,18,34,35 and, more generally, systems described by nonlinear partial differential equations (PDEs).2,4,29,30,39–41,44,46,47 Computing the solution to the MZ equation is usually a daunting task. One of the main difficulties is the approximation of the memory integral (convolution term) and the fluctuation term, which encode the interaction between the so-called orthogonal dynamics and the dynamics of the quantity of interest. The orthogonal dynamics is essentially a high-dimensional flow governed by an integro-differential equation that is hard to solve. The mathematical properties of the orthogonal dynamics and hence the properties of the MZ memory integral and the MZ fluctuation term are not well understood. Kupferman, Givon, and Hald19 proved the existence and uniqueness of the orthogonal dynamics for deterministic dynamical systems and Mori’s projection operators. More recently, we proved uniform boundedness of the orthogonal dynamics propagator for Hamiltonian systems using semigroup estimates.15,52

The main objective of this paper is to generalize the MZ formulation to stochastic differential equations (SDEs)16,22 driven by multiplicative Gaussian white noise. In particular, we aim at developing a thorough mathematical analysis of the so-called effective Mori–Zwanzig (EMZ) equation governing the dynamics of noise-averaged observables, i.e., smooth functions of the stochastic flow generated by the SDE which are averaged over the probability measure of the random noise. To this end, we build upon recent work of Eckmann et al.,12–14 Hèrau and Nier21, and Helffer and Nier20 on the spectral properties of backward Kolmogorov operators and show that the generator of EMZ orthogonal dynamics has a discrete spectrum that lies within the cusp-shaped region of the complex plane. This allows us to rigorously prove exponential relaxation to a unique equilibrium state for both the EMZ memory kernel and the EMZ fluctuation term.

This paper is organized as follows: In Sec. II, we develop a self-consistent MZ formulation for stochastic differential equations driven by multiplicative Gaussian white noise and derive the effective Mori Zwanzig equation governing the dynamics of noise-averaged observables. In Sec. III, we study the theoretical properties of the EMZ equation. To this end, we first review Hörmander’s theory of linear hypoelliptic operators and then show how such theory can used to prove exponential convergence of the EMZ orthogonal dynamics propagator to a unique equilibrium state. In Sec. IV, we apply our theoretical results to the Langevin dynamics of high-dimensional particle systems with smooth interaction potentials that grow at most polynomially fast at infinity. The main findings are summarized in Sec. V.

Let us consider a d-dimensional stochastic differential equation on a smooth manifold M,

dx(t)dt=F(x(t))+σ(x(t))ξ(t),x(0)=x0ρ0(x),
(1)

where F:MRd and σ:MRd×m are smooth functions, ξ(t) is m-dimensional Gaussian white noise with independent components, and x0 is a random initial state characterized in terms of a probability density function ρ0(x). The solution (1) is a d-dimensional stochastic (Brownian) flow on the manifold M.26 As is well known, if F:MRd and σ:MRd×m are of class Ck+1 (k ≥ 0) with uniformly bounded derivatives, then the solution to (1) is global and that the corresponding flow is a stochastic flow of diffemorphisms of class Ck (see Refs. 3, 49, and 50). This means that the stochastic flow is differentiable k times (with continuous derivative), with respect to the initial condition for all t. Define the vector-valued phase space function (quantity of interest)

u:MRMxu(x)(quantity of interest).
(2)

By evaluating u(x) along the stochastic flow generated by the SDE (1) and averaging over the Gaussian white noise, we obtain

Eξ(t)[u(x(t))|x0]=F(t,0)u(x0).
(3)

The evolution operator F(t,0) is a Markovian semigroup36 generated by the following (backward) Kolmogorov operator:25,38,42

K(x0)=k=1dFk(x0)x0k+12j=1mi,k=1dσij(x0)σkj(x0)x0ix0k,
(4)

which corresponds to the Itô interpretation of the SDE (1). Formally, we will write

F(t,0)=etK.
(5)

To derive the effective Mori–Zwanzig (EMZ) equation governing the time evolution of the averaged observable (3), we introduce a projection operator P and the complementary projection Q=IP. By following the formal procedure outlined in Refs. 11, 54, and 55, we obtain

tetKu(0)=etKPKu(0)+etQKQQKu(0)+0tesKPKe(ts)QKQQKu(0)ds.
(6)

Applying the projection operator P to (6) yields

tPetKu(0)=PetKPKu(0)+0tPesKPKe(ts)QKQQKu(0)ds.
(7)

Note that both the EMZ Eq. (6) and its projected form (7) have the same structure as the classical MZ equation for deterministic (autonomous) systems.52,54,55

Let us consider the weighted Hilbert space H=L2(M,ρ), where ρ is a positive weight function in M. For instance, ρ can be the probability density function of the random initial state x0. Let

h,gρ=Mh(x)g(x)ρ(x)dx,h,gH,
(8)

be the inner product in H. We introduce the following projection operator:

Ph=i,j=1MGij1ui(0),hρuj(0),hH,
(9)

where Gij=ui(0),uj(0)ρ and ui(0) = ui(x) (i = 1, …, M) are M linearly independent functions. With P defined as in (9), we can write the EMZ equation (6) and its projected version (7) as

dq(t)dt=Ωq(t)+0tK(ts)q(s)ds+f(t),
(10)
ddtPq(t)=ΩPq(t)+0tK(ts)Pq(s)ds,
(11)

where q(t)=E[u(x(t))|x0] (column vector) and

Gij=ui(0),uj(0)ρ(Gram matrix),
(12a)
Ωij=k=1MGjk1uk(0),Kui(0)ρ(streaming matrix),
(12b)
Kij(t)=k=1MGjk1uk(0),KetQKQQKui(0)ρ(memory kernel),
(12c)
fi(t)=etQKQQKui(0)(fluctuation term).
(12d)

In Eqs. (12a)–(12d), we have uj(0) = qj(0) = uj(x0) (j = 1, …, M). In addition, the Kolmogorov operator K is not skew-symmetric relative to ⟨,⟩ρ, and therefore, it is not possible (in general) to represent the memory kernel K(t) as a function of the auto-correlation of f(t) using the second fluctuation–dissipation theorem.53,55

Let us consider the Ornstein–Uhlenbeck process defined by the solution to the Itô stochastic differential equation

ddtx=θx+σξ(t),
(13)

where σ and θ are positive parameters and ξ(t) is Gaussian white noise with the correlation function ⟨ξ(t), ξ(s)⟩ = δ(ts). As is well-known, the Ornstein–Uhlenbeck process is ergodic and it admits a stationary (equilibrium) Gaussian distribution ρeq=N(0,σ2/2θ). Let x(0) be a random initial state with probability density function ρ0 = ρeq. The conditional mean and conditional covariance function of the process x(t) are given by

Eξ(t)[x(t)|x(0)]=x(0)eθt,
(14)
Eξ(t)[x(t)x(s)|x(0)]=x(0)2eθ(t+s)+σ22θeθ|ts|eθ(t+s).
(15)

Averaging over the random initial state yields

Ex(0)[Eξ(t)[x(t)|x(0)]]=0,
(16)
Ex(0)[Eξ(t)[x(t)x(s)|x(0)]]=σ22θeθ|ts|.
(17)

At this point, we define the projection operators

P1()=ρeqx(0)ρeqx(0),P2()=,x(0)ρeqx(0),x(0)ρeqx(0).
(18)

The Kolmogorov backward operator associated with (13) is

K(x)=θxx+σ222x2.
(19)

By using the identity

ddtEx(0)[Eξ(t)[x(t)x(0)|x(0)]]=ddtEx(0)[Eξ(t)[x(t)|x(0)]x(0)]=ddtF(t,0)x(0),x(0)ρeq,
(20)

it is straightforward to verify that the EMZ equation (10) with P=P1 and the EMZ Eq. (11) with P=P2 can be written, respectively, as

ddtM(t)=θM(t),ddtC(t)=θC(t).
(21)

Here, M(t)=Eξ(t)[x(t)|x(0)] is the conditional mean of x(t), while C(t)=Ex(0)[Eξ(t)[x(t)x(0)|x(0)]] is the autocorrelation function of x(t). Clearly, Eq. (21) are the exact evolution equations governing M(t) and C(t). In fact, their solutions coincide with (14) and (17), respectively. Note that M(t) is a stochastic process [x(0) is random], while C(t) is a deterministic function.

In this section, we develop an in-depth mathematical analysis of the effective Mori–Zwanzig Eq. (6) using Hörmander’s theory.20,21,33 In particular, we build upon the result of Hérau and Nier,21 Eckmann et al.,12–14 and Helffer and Nier20 on linear hypoelliptic operators to prove that the generator of the EMZ orthogonal dynamics, i.e., QKQ, satisfies a hypoelliptic estimate. Consequently, the propagator etQKQ converges exponentially fast (in time) to statistical equilibrium. This implies that both the EMZ memory kernel (12c) and fluctuation term (12d) converge exponentially fast to an equilibrium state. One of the key results of such analysis is the fact that the spectrum of QKQ lies within a cusp-shaped region of the complex half-plane. For consistency with the literature on hypoelliptic operators, we will use the negative of K and QKQ as semigroup generators and write the semigroups appearing in EMZ Eq. (6) as etK and etQKQ. Clearly, if K and QKQ are dissipative, then K and QKQ are accretive. Unless otherwise stated, throughout this section, we consider scalar quantities of interest, i.e., we set M = 1 in Eq. (2).

The Kolmogorov operator (4) is a Hörmander-type operator, which can be written in the general form

K(x)=i=1mXi*(x)Xi(x)+X0(x)+f(x),
(22)

where Xi(x) (0 ≤ im) denotes a first-order partial differential operator in the variable xi with space-dependent coefficients, Xi*(x) is the formal adjoint of Xi(x) in L2(Rn), and f(x) is a function that has at most polynomial growth at infinity. To derive useful spectral estimates for K, it is convenient to first provide some definitions

Definition 1.
Let N be a real number. Define
Pol0N=fC(Rn)supxRn(1+x)N|αf(x)|Cα,
where α is a multi-index of arbitrary order. Note that Pol0N is the set of infinitely differentiable functions growing at most polynomially as ‖x‖ → . Similarly, we define the space of kth order differential operators with coefficients growing at most polynomially with x as
PolkN=G:C(Rn)C(Rn)G(x)=G0(x)+j=1ni=1kGji(x)ji,GjiPol0N.

It is straightforward to verify that if XPolkN and YPollM, then the operator commutator [X,Y]=XYYX is in Polk+l1N+M.

Definition 2.
The family of operators {A1,,Am} defined as
Ai(x)=j=1nAij(x)j,i=1,,m,
(23)
is called non-degenerate if there are two constants N and C such that
y2C(1+x2)Ni=1mAi(x),y2x,yRn,
where Ai(x),y=j=1nAij(x)yj.

It was recently shown by Eckmann and Hairer12,13 that K is hypoelliptic if the Lie algebra generated by the operators {X0,,Xm} in (22) is non-degenerate. The main result can be summarized as follows:

Proposition 1

(Ref.13 ). Let{X0,,Xm}andfin(22)satisfy the following conditions:

  1. XjPol1Nfor allj = 0, …, mandfPol0N;

  2. There exists a finite integerMsuch that the family of operators consisting of{Xi}i=0m,{[Xi,Xj]}i,j=1m,{[Xi,[Xj,Xk]]}i,j,k=1m, and so on up to the commutators of rankMis non-degenerate;

Then, the operatorKdefined in(22)andt+Kare both hypoelliptic.

Conditions 1 and 2 in Proposition 1 are called poly-Hörmander conditions. Eckmann et al.12,14 also proved the hypoellipticity of the operator t+K* for a specific heat condition model, which guarantees smoothness (in time) of the transition probability governed by the Kolmogorov forward equation. Hereafter, we review additional important properties of the Kolmogorov operator K. As a differential operator with C tempered coefficients (i.e., with all derivatives polynomially bounded), K and its formal adjoint K* are defined in the Schwartz space S(Rn), which is dense in Lp(Rn) (1 ≤ p < ). On the other hand, since K and K* are both closable operators, all estimates we obtain in this section hold naturally in S(Rn), which can be extended to L2(Rn). Hence, we do need to distinguish between K and its closed extension in L2(Rn). We now introduce a family of weighted Sobolev spaces

Sα,β={uS(Rn):ΛαΛ̄βuL2(Rn)α,βR},
(24)

where S(Rn) is the space of tempered distributions in Rn. The operator Λ̄β is the product operator defined as Λ̄β(1+x2)β/2, while Λα is a pseudo-differential operator (see Refs. 13, 14, and 21) that reduces to

Λ2=1Δ
(25)

for α = 2. The weighted Sobolev space (24) is equipped with the scalar product

h,gα,β=ΛαΛ̄βh,ΛαΛ̄βgL2,

which induces the Sobolev norm ‖·‖α,β. Throughout this paper, ‖·‖ denotes the standard L2 norm. With the above definitions, it is possible to prove the following important estimate on spectrum of the Kolmogorov operator K:

Theorem 1
(Ref.13 ). LetKPol2Nbe an operator of the form(22)satisfying conditions 1 and 2 in Proposition 1. Suppose that the closure ofKis a maximal-accretive operator inL2(Rn)and that for everyϵ > 0, there are two constantsδ > 0 andC > 0 such that
uδ,δC(u0,ϵ+Ku)
(26)
for alluS(Rn). If, in addition, there exist two constantsδ > 0 andD > 0 such that
u0,ϵD(u+Ku),
(27)
thenKhas compact resolvent when considered as an operator acting onL2(Rn), whose spectrumσ(K)is contained in the following cusp-shaped regionSKof the complex plane (seeFig. 1 ):
SK={zC:Rez0,|z+1|<(8C1)M/2(1+Rez)M}
(28)
for some positive constantC1andMN.

FIG. 1.

Sketch of the cusp-shaped region of the complex plane enclosing the spectrum of K and QKQ. The curve γext represents the boundary of the cusp. We denote by SK the curve defined by the union of γext and γ̄, while SK is defined by the union γext (up to intersection with γint) and γint.

FIG. 1.

Sketch of the cusp-shaped region of the complex plane enclosing the spectrum of K and QKQ. The curve γext represents the boundary of the cusp. We denote by SK the curve defined by the union of γext and γ̄, while SK is defined by the union γext (up to intersection with γint) and γint.

Close modal

We remark that in Ref. 13, the cusp SK is defined as SK={zC:Rez0,|Imz|<(8C1)M/2(1+Rez)M}. Clearly, SK in Eq. (28) is also a valid cusp since it can be derived directly from (29) (see, e.g., the Proof of Theorem 4.3 in Ref. 13).

One of the key estimates used by Eckmann and Hairer in the Proof of Theorem 1 is

14|z+1|2/Mu2C1[1+Rez]2u2+(Kz)u2,Rez0.
(29)

In a series of papers, Hèrau, Nier, and Helffer20,21 proved that the Kolmogorov operator K corresponding to classical Langevin dynamics generates a semigroup etK that decays exponentially fast to an equilibrium state. Hereafter, we show that similar results can be obtained for Kolmogorov operators in the more general form (22).

Theorem 2.
Suppose thatKsatisfies all conditions in Theorem 1. If the spectrumσ(K)ofKinL2(Rn)is such that
σ(K)iR={0},
(30)
then for any0<α<min(Reσ(K)/{0}), there exists a positive constantC = C(α) such that the estimate
etKu0π0u0Ceαtu0
(31)
holds for allu0L2(Rn)and for allt > 0, whereπ0is the spectral projection onto the kernel ofK.

Proof.
The Kolmogorov operator K is closed, maximal-accretive, and densely defined in L2(Rn). Hence, by the Lumer–Phillips theorem, the semigroup etK is a contraction in L2(Rn). It was shown in Refs. 13 and 21 that the core of K is the Schwartz space and that the hypoelliptic estimate (29) holds for any uL2(Rn). According to Theorem 1, K only has a discrete spectrum, i.e., σ(K)=σdis(K). Condition (30) requires that λ = 0 is the only eigenvalue on the imaginary axis iR. This condition, together with the von-Neumann theorem (see the Proof of Theorem 6.1 in Ref. 20), allows us to obtain a weakly convergent Dunford integral37 representation of the semigroup etK given by
etKu0π0u0=12πiSKetz(zK)1u0dz,
(32)
where SK=γintγext is the union of the two curves shown in Fig. 1 and (zK)1 is the resolvent of K. Weak convergence is relative to the inner product
(etKπ0)u0,ϕ=12πiSKetz(zK)1u0,ϕdz
(33)
for u0L2(Rn) and ϕD(K*). Equation (32) allows us to formulate the semigroup estimation problem as an estimation problem involving an integral in the complex plane. In particular, to derive the upper bound (31), we just need an upper bound for the norm of resolvent (zK)1. To derive such bound, we note that for all zSK, where SK is the cusp (28), and for Re z ≥ 0, we have |z + 1|2/M ≥ (8C1)(1 + Re z)2. A substitution of this inequality into (29) yields, for all uL2(Rn),
18|z+1|2/Mu2C1(Kz)u2,Rez0,zSK.
Hence, (Kz)18C1|z+1|1/M. Next, we rewrite the Dunford integral (32) as
12πiSKetz(zK)1u0dz=12πiγintetz(zK)1u0dz+12πiγextetz(zK)1u0dz.
(34)
Since (Kz)1 is a compact linear operator, we have that for any 0<α<min(Reσ(K)/{0}), there exists a constant Cα > 0 such that (Kα)uCαu. On the other hand, K is also a real operator, which implies that for all complex numbers z=(α+iy)σ(K), we have
(K(α+iy))u2=(Kα)u2+y2u2(Cα2+y2)u2,
i.e.,
(K(α+iy))1u1Cα2+y2u.
(35)
This suggests that the resolvent (Kz)1 is uniformly bounded by 1/Cα along the line γint, which leads to
12πiγintetz(zK)1u0dzCeαtu0.
(36)
The boundary γext is defined by all complex numbers z = x + iy such that |z+1|=(8C1)M/2(1+Rez)M. In addition, if zSK, then the norm of the resolvent is bounded by (Kz)18C1|z+1|1/M=(x+1)1. Combining these two inequalities yields
12πiγextetz(zK)1u0dzCu0γextetx(1+x)1dzCu0αetxdxCu0eαtt,t>0.
(37)
At this point, we recall that etK is a dissipative semigroup and that π0 is a projection operator into the kernel of K. This allows us to write etKu0π0u0=etK(u0π0u0)u0π0u0. By combining this inequality with (32), (34), (36), and (37), we see that there exists a constant C = C(α) such that
etKu0π0u0Ceαtu0.
(38)
This completes the proof.□

In the following corollary, we derive an upper bound for the norm of the derivatives of the semigroup etK:

Corollary 2.1.
Suppose thatKsatisfies all conditions listed in Theorem 2. Then, for anyt > 0, thenth order time derivative of the semigroupetKsatisfies
etKKnu0π0K+Btnnu0,
(39)
where
B(t)=Ceαt1+1t+1t2+1tM,
(40)
Cis a positive constant,αandπ0are defined in Theorem 2, andMis the constant defining the cusp(28).

Proof.
By combining the resolvent identity (zK)1K=z(zK)1I with the Cauchy integral representation theorem and the Dunford integral representation (32), we obtain, for all t > 0,
etKKπ0K=12πiSKetz(zK)1Kdz=12πiSKzetz(zK)1dz.
(41)
As before, we split the integral along SK into the sum of two integrals [see Eq. (34)],
etKKπ0K=12πiγintzetz(zK)1dz+γextzetz(zK)1dz.
(42)
If z = x + iy is in γint, then we have that |z| is bounded by constant. By using the uniform boundedness of the resolvent (35), we obtain
12πiγintetzz(zK)1dzCeαt.
(43)
To derive an upper bound for the second integral in (42), we note that if z = x + iy is in γext, then |z|<|z+1|=(8C1)M/2(1+x)M and (zK)18C1|z+1|1/M=(1+x)1. A substitution of these estimates into the second integral at the right-hand side of (42) yields, for all t > 0,
12πiγextzetz(zK)1dzCαetx(1+x)M1dxCeαt1t+1t2+1tMCeαt1+1t+1t2+1tMB(t).
(44)
Combining (43) and (44), we conclude that the Dunford integral (41) is bounded by B(t). Since K has a compact resolvent, if there is any zero eigenvallue, then it must have finite algebraic multiplicity (Theorem 6.29, p. 187 in Ref. 24). This implies that the projection operator π0 is a finite rank operator that admits the canonical form (in L2)
π0=i=1nαi,viui.
(45)
On the other hand, since
π0Kf=i=1nαiKf,viui=i=1nαif,K*viui,
where K* is the L2-adjoint of K, we have
π0Ki=1n|αi|K*viui.
(46)
Hence, π0K is a bounded operator. By using the Dunford integral representation over SK, it is straightforward to show that etKK is also a bounded operator for t > 0. Combining these results with the triangle inequality, we have that for any fixed t > 0 and any nN,
etK/nKπ0KetK/nKπ0KBtn.
(47)
Finally, by using the operator identity etKKn=(etK/nK)n, we obtain
etKKnetK/nKnBtn+π0Knt>0,
(48)
which completes the proof.□

Inequality (39) suggests that the flow defined by the semigroup etK has bounded derivatives in time. We emphasize that estimate (39) is not sufficient to prove the convergence of the formal power series expansion of etK since

limnetKKnn!0.
(49)

In this section, we analyze the semigroup etQKQ generated by the operator QKQ, where K is the Kolmogorov operator (4) and P and Q=IP are projection operators in L2(Rn). Such semigroup appears in the EMZ memory and fluctuation terms [see Eqs. (10), (12c), and (12d)]. In principle, the projection operator P and therefore the complementary projection Q can be chosen arbitrarily.6,52 Here, we restrict our analysis to finite-rank symmetric projections in L2(Rn). Mori’s projection (9) is one of such projections.

Theorem 3.
LetP:L2(Rn)L2(Rn)be a finite-rank, symmetric projection operator. IfKsatisfies all conditions listed in Theorem 1, then the operatorQKQis also maximal accretive and has a compact resolvent. Moreover, the spectrum ofQKQlies within the cusp
SQKQ={zC|Rez0,|z+1|<(8CQ)MQ/2(1+Rez)MQ}
(50)
for some positive constantsCQand integerMQ.

Proof.
We first show that if K is closely defined and maximal accretive, and QKQ has the same properties. According to the Lumer–Phillips theorem,15 the adjoint of a maximal-accretive operator is accretive, and therefore,
ReKf,f0fD(K),ReK*f,f0fD(K*).
Here, D(K) and D(K*) denote the domain of the linear operators K and K*, respectively (see, e.g., Ref. 45). On the other hand, if P is a symmetric operator in L2(Rn), then Q=IP is also symmetric. This implies that
ReQKQf,f=ReKQf,Qf0fD(K),Re(QKQ)*f,f=ReK*Qf,Qf0fD(K*),
i.e., QKQ and its adjoint QK*Q are both maximal-accretive. QKQ is also a closable operator defined in D(K). This can be seen by decomposing it as QKQ=KKPPKQ. In fact, if K is a closed operator, then QKQ is closed since KP and PKQ are bounded,24 as we shall see hereafter. By using the Lumer–Philips theorem, we conclude that QKQ is also maximal accretive, and its closure generates a contraction semigroup etQKQ in L2(Rn). Next, we show that if K satisfies the hypoelliptic estimate uδ,δC(u+Ku), then so does QKQ, i.e.,
uδ,δC(u+QKQu).
(51)
By using the triangle inequality, we obtain
uδ,δC(u+Ku)C(u+KPu+QKQu+PKQu).
To prove (51), it is sufficient to show that KP and PKQ are bounded operators in L2(Rn). To this end, we recall that any finite-rank projection admits the canonical representation
P=i=1mλi,ϕiφi,
(52)
where {ϕi}i=1m and {φi}i=1m are elements L2(Rn). This implies that
KPu=i=1mλiu,ϕiKφii=1m|λi|Kφiϕiu=Cu,
(53)
PKQu=i=1mλiKQu,ϕiφi=i=1mλiu,QK*ϕiφii=1m|λi|QK*ϕiφiu=Cu.
(54)
This proves that KP and PKQ are both bounded linear operators. At this point, we notice that if QKQ is accretive, then (QKQ+I) invertible. Moreover, since K is accretive, we have
(QKQ+I)u2=QKQ2+2ReKQu,Qu+u2QKQ2+u2.
This implies that
uδ,δC(u+QKQu)2C(QKQ+I)u(QKQ+I)1uδ,δ2Cu,
i.e., (QKQ+I)1 is a bounded operator from L2 into the weighted Sobolev space Sδ,δ defined in (24). At this point, we recall that Sδ,δ is compactly embedded into L2 (Lemma 3.2, Ref. 13). Hence, (QKQ+I)1 is compact from L2 into L2, and therefore, QKQ has a compact resolvent.24 To prove that the discrete spectrum of QKQ lies within the cusp SQKQ defined in (50), we follow the procedure outlined in Ref. 13. To this end, let KPol2N. Then, for δ = max{2, N}, we have the bound
(K+I)uCuδ,δuSn
and
(QKQ+I)uQ(Ku+KPu)+uC(Ku+u)2C(K+I)uCuδ,δ.
Recall that QKQ:D(QKQ)L2(R2d) is maximally accretive. Therefore, by Lemma 4.5 in Ref. 13, for all δ > 0, we can find an integer MQ>0 and a constant C such that
u,[(QKQ+I)*(QKQ+I)]1/MQCuδ,δ2.
(55)
By using the hypoelliptic estimate (51) and (55), Proposition B.1 in Ref. 20, and the triangle inequality, we obtain
14|z+1|2/MQu2Cuδ,δ2+(QKQz)2CQ([1+Rez]2u2+(QKQz)u2).
This result, together with the compactness of the resolvent of QKQ, implies that if zσ(QKQ) (spectrum of QKQ), then
18|z+1|2/MQu2<14|z+1|2/MQu2CQ(1+Rez)2u2.
This proves that the spectrum of QKQ is contained in the cusp-shaped region SQKQ defined in Eq. (50). If zSQKQ, then we have the resolvent estimate
(QKQz)18CQ|z+1|1/MQ.
(56)

Remark.

The main assumption at the basis of Theorem 3 is that P is a finite-rank symmetric projection. Mori’s projection (9) is one of such projections. If P is of finite-rank, then both KP and PKQ are bounded operators, which yields the hypoelliptic estimate (26). On the other hand, if P is an infinite-rank projection, e.g., Chorin’s projection,7,8,52,58 then KP and PKQ may not be bounded. Whether Theorem 3 holds for infinite-rank projections is an open question.

With the resolvent estimate (56) available, we can now prove the analog of Theorem 2 and Corollary 2.1, with K replaced by QKQ. These results establish exponential relaxation to equilibrium of etQKQ and the regularity of the EMZ orthogonal dynamics induced by etQKQ.

Theorem 4.
Assume thatKsatisfies all conditions listed in Theorem 1. LetP:L2(Rn)L2(Rn)be a symmetric finite-rank projection operator. If the spectrum ofQKQinL2(Rn)satisfies
σ(QKQ)iR{0},
(57)
then for any0<αQ<min(Reσ(QKQ)/{0}), there exists a positive constantC=C(αQ)such that
etQKQu0π0Qu0CeαQtu0
(58)
for allu0L2(Rn)and for allt > 0, whereπ0Qis the spectral projection onto the kernel ofQKQ.

Corollary 4.1.
Suppose thatPandKsatisfy all conditions listed in Theorem 4. Then, for anyt > 0, thenth order derivative of the semigroupetQKQsatisfies
etQKQ(QKQ)nu0π0(QKQ)+BQtnnu0,
(59)
where the functionBQ(t)has the same form as(40), withαreplaced byαQandMreplaced byMQ.

The proofs of Theorem 4 and Corollary 4.1 closely follow the proofs of Theorem 2 and Corollary 2.1. Therefore, we omit them. The semigroup estimate (58) allows us to prove exponential convergence to the equilibrium state of the EMZ memory kernel and fluctuation force. Specifically, we have the following corollary:

Corollary 4.2.
Consider a scalar observableu(t) = u(x(t)) with the initial conditionu(0) = u0, and letP()=(),u0u0be a one-dimensional Mori’s projection(9). Then, the EMZ memory kernel(12c)converges exponentially fast to the equilibrium stateQK*u0,π0QKu0, with rateαQ. In other words, there exists a positive constantCsuch that
|K(t)QK*u0,π0QKu0|CeαQt.
(60)

Proof.
A substitution of (58) into (12c) and subsequent application of the Cauchy–Schwartz inequality yield
|K(t)QK*u0,π0QKu0|=|u0,KetQKQQKu0QK*u0,π0QKu0|=|QK*u0,etQKQKu0QK*u0,π0QKu0|CQK*u0Ku0eαQt.
(61)

It is straightforward to generalize Corollary 4.2 to matrix-valued memory kernels (12c) and obtain the following exponential convergence result:

K(t)G1CQMCG1DQMeαQt,
(62)

where M denotes any matrix norm and G is the Gram matrix (12a). In addition, the matrix CQ has entries CijQ=QK*ui(0),π0QKuj(0), while DijQ=QKui(0)Kuj(0). The proof of (62) follows immediately from the following inequality:

ui(0),KetQKQQKuj(0)QK*ui(0),π0QKuj(0)=QK*ui(0),etQKQKuj(0)π0QKuj(0)CQKui(0)Kuj(0)eαQt.
(63)

In fact, a substitution of (63) into (12c) yields (62). Similarly, we can prove that the fluctuation term (12d) reaches the equilibrium state exponentially fast in time. If we choose the initial condition as u0=QKu0, then for all j = 1, …, m, we have

fj(t)π0QQKuj(0)=etQKQQKuj(0)π0QQKuj(0)CeαQtQKuj(0).
(64)

Let us now introduce the tensor product space V=i=1mL2(Rn) and the following norm:

r(t)Vr1(t),r2(t),,rm(t)M,
(65)

where ‖·‖ is the standard L2(Rn) norm and M is any matrix norm. Then, from (64), it follows that

f(t)π0QQKu0VCeαQtQKu(0)V.
(66)

All results we obtained so far can be applied to stochastic differential equations of the form (1), provided the MZ projection operator is of finite-rank. In this section, we study in detail the Langevin dynamics of an interacting particle system widely used in statistical mechanics to model liquids and gases27,38 and show that the EMZ memory kernel (12c) and fluctuation term (12d) decay exponentially fast in time to a unique equilibrium state. Such a state is defined by the projector operator π0Q appearing in Theorem 4 and Corollary 4.2. Hereafter, we will determine the exact expression of such projector for a system of interacting identical particles modeled by the following SDE in R2d:

dqdt=1μp,dpdt=V(q)γμp+σξ(t),
(67)

where μ is the mass of each particle, V(q) is the interaction potential, and ξ(t) is a d-dimensional Gaussian white noise process modeling physical Brownian motion. The parameters σ and γ represent, respectively, the amplitude of the fluctuations and the viscous dissipation coefficient. Such parameters are linked by the fluctuation–dissipation relation σ = (2γ/β)1/2, where β is proportional to the inverse of the thermodynamic temperature. The stochastic dynamical system (67) is widely used in statistical mechanics to model the mesoscopic dynamics of liquids and gases. Letting the mass μ in (67) go to zero and setting γ = 1 yield the so-called overdamped Langevin dynamics, i.e., Langevin dynamics where no average acceleration takes place. The (negative) Kolmogorov operator (4) associated with the SDE (67) is given by

K=pμq+qV(q)p+γpμp1βΔp,
(68)

where “·” denotes the standard dot product. If the interaction potential V(q) is strictly positive at infinity, then the Langevin equation (67) admits an unique invariant Gibbs measure given by

ρeq(p,q)=1ZeβH(p,q),
(69)

where

H(p,q)=p222μ+V(q)
(70)

is the Hamiltonian and Z is the partition function. At this point, we introduce the unitary transformation U:L2(R2d)L2(R2d,ρeq) defined by

(Ug)(p,q)=ZeβH(p,q)/2g(p,q),
(71)

where L2(R2d;ρeq) is a weighted Hilbert space endowed with the inner product

h,gρeq=h(p,q)g(p,q)ρeq(p,q)dpdq.
(72)

The linear transformation (71) is an isometric isomorphism between the spaces L2(R2d) and L2(R2d;ρeq). In fact, for any ũL2(R2d), there exists a unique uL2(R2d;ρeq) such that ũ=(eβH/2/Z)u and

ũL2=uLeq2.
(73)

By applying (71) to (68), we construct the transformed Kolmogorov operator K̃=U1KU, which has the explicit expression

K̃=pμq+V(q)p+γβp+β2μpp+β2μp.
(74)

This operator can be written in the canonical form (22) as

K̃=i=1dXi*XiX0,
(75)

provided we set

X0=pμqV(q)p,Xi=γβpi+β2μpi,Xi*=γβpi+β2μpi.
(76)

Note that X0 is skew-symmetric in L2(R2d). In addition, Xi* and Xi can be interpreted as creation and annihilation operators, similarly to a harmonic quantum oscillator.56 The Kolmogorov operator K̃ and its formal adjoint K̃* are both accretive and closable, and with maximally accretive closure in L2(R2d) (see, e.g., Refs. 12, 20, and 21) similar to the Kolmogorov operator K̃=U1KU, we can transform the MZ projection operators P and Q into operators in the “flat” Hilbert space L2(R2d) as P̃=U1PU and Q̃=U1QU. The relationship between L2(R2d), L2(R2d;ρeq), and the operators defined between such spaces can be summarized by the following commutative diagram:

formula

The properties of all operators in L2(R2d) and L2(R2d;ρeq) are essentially the same since U is a bijective isometry. For instance, if P is compact and symmetric, then P̃ is also a compact and symmetric operator.

Next, we apply the analytical results we obtained in Secs. III A and III B to the particle system described by the SDE (67). To this end, we just need to verify whether K̃ is a poly-Hörmander operator, i.e., if the operators {Xi}i=0d appearing in (75) and (76) satisfy the poly-Hörmander conditions in Proposition 1 and the estimate in Theorem 1 (see Sec. III A). This can be achieved by imposing additional conditions on the particle interaction potential V(q) (see Ref. 12, Proposition 3.7). In particular, following Helffer and Nier,20 we assume that V(q) satisfies the following weak ellipticity hypothesis:

Hypothesis 1.

The particle interaction potentialV(q) is of classC(Rd), and for allqRd, it satisfies the following conditions:

  1. αNdsuch that |α| = 1,|qαV(q)|Cα1+V(q)2for some positive constantCα.

  2. There existsMN, andC ≥ 1, such thatC1(1+q2)1/(2M)1+V(q)2C(1+q2)M/2.

Hypothesis 1 holds for any particle interaction potential that grows at most polynomially at infinity, i.e., V(q) ≃ ‖qM as q. With this hypothesis, it is possible to prove the following proposition:

Proposition 2
(Ref.20 ). Consider the Langevin equation(67)with particle interaction potentialV(q) satisfying Hypothesis 1. Then, the operatorK̃defined in(74)has a compact resolvent and a discrete spectrum bounded by the cuspSK. Moreover, there exists a positive constantCsuch that the estimate
etK̃u0π̃0ũ0Ceαtũ0
(77)
holds for allũ0L2(R2d)and for allt > 0, whereπ̃0is the orthogonal projection onto the kernel ofK̃inL2(R2d).

By using the isomorphism (71), we can rewrite Proposition 2 in L2(R2d;ρeq) as

etKu0π0u0Leq2=etK̃ũ0π̃0ũ0L2Ceαtũ0L2=Ceαtu0Leq2,
(78)

where π0=Uπ̃0U1 is the orthogonal projection π0()=E[()]. The inequality (78) is completely equivalent to estimate (31). It is also possible to obtain a prior estimate on the convergence rate α by building a connection between the Kolmogorov operator and the Witten Laplacian (see Refs. 20 and 21 for further details).

Our next task is to derive an estimate for the operator Q̃K̃Q̃ and for the semigroup etQ̃K̃Q̃ generated by the closure of Q̃K̃Q̃. According to Theorem 3, the spectrum of Q̃K̃Q̃ is bounded by the cusp SQ̃K̃Q̃, provided that P is an orthogonal finite-rank projection operator. On the other hand, Theorem 4 establishes exponential convergence of etQ̃K̃Q̃ to equilibrium if Q̃K̃Q̃ satisfies condition (57). It is left to determine the exact form of the spectral projection π̃0Q̃, i.e., the projection onto the kernel of Q̃K̃Q̃ (see Theorem 4), and verify condition (57). To this end, we consider a general Mori-type projection P and its unitarily equivalent version P̃=U1PU,

P()=i=1m,viρeqvi,P̃()=i=1m,viρeq/2vieβH/2,
(79)

where {vj}j=1m={vj(q,p)}j=1m are zero-mean, i.e., viρeq=0, orthonormal basis functions. In (79), we used the shorthand notation

hρeq/2=1Zg(p,q)eβH(p,q)/2dpdq.
(80)

Lemma 5.
Suppose that the particle interaction potentialV(q) in(68)satisfies Hypothesis 1. Then, for any set of observables{wj}j=1msatisfyingwj,viρeq=0andKwj=vj, we have that the kernel ofQ̃K̃Q̃is given by
Ker(Q̃K̃Q̃)=Ker(K̃)Ran(P̃)Span{wjeβH/2}j=1m,
(81)
whereK̃andP̃are defined in(74)(74) and (79), respectively. In particular, ifPis defined asP()=,pjρeqpj, wherepjis the momentum ofjth particle, then we have
σ(Q̃K̃Q̃)iR{0}.
(82)

Proof.
We first prove (81). To this end, let us first define the finite-dimensional space
W=Ker(K̃)Ran(P̃)Span{wjeβH/2}j=1m.
(83)
If uKer(Q̃K̃Q̃), then Q̃K̃Q̃u=0. This implies K̃Q̃uP̃K̃Q̃u=0. Equivalently,
K̃u=K̃P̃u+P̃K̃Q̃u=j=1mu,vieq/2K̃vieβH/2+j=1mK̃Q̃u,vieq/2vieβH/2Span{vjeβH/2}j=1mSpan{K̃vjeβH/2}j=1m.
(84)
Since Kwj=UK̃U1wj=vj, we have K̃wjeβH/2=vjeβH/2. This implies that uW and Ker(Q̃K̃Q̃)W. Let f be an arbitrary element in W. Then,
f=αeβH/2+j=1mρjvjeβH/2+j=1mθjwjeβH/2.
{α, ρ1, …, ρm, θ1, …, θm} are the coordinates of f in the finite-dimensional space W. By using the definition of P and the fact that viρeq=vi,wjρeq=0 and vj2ρeq=1, we obtain
P̃f=j=1mρjvjeβH/2Q̃f=αeβH/2+j=1mθjwjeβH/2.
Therefore,
K̃Q̃f=P̃K̃Q̃f=j=1mθjvjeβH/2Q̃K̃Q̃f=0.
This proves that WKer(Q̃K̃Q̃), and therefore, (81) holds. In fact, the kernel of Q̃K̃Q̃ can be constructed by taking the union of three sets defined by the following conditions:
  1. Q̃u=0, which implies P̃u=0, i.e., uRan(P̃);

  2. Q̃u0,K̃Q̃u=0, which implies Q̃uKer(K̃). This is possible only if uRan(Q̃)Ker(K̃) since in this case we have Q̃u=u; and

  3. Q̃u0,K̃Q̃u0,Q̃K̃Q̃u=0, which implies K̃Q̃u=P̃K̃Q̃u0. This is possible only if Q̃u=u, K̃u0, and uSpan{wieβH/2}i=1m, provided that the set of observables {wj}j=1m satisfies wj,viρeq=0 and Kwj=vj.

Combining these three cases and using the fact that L2(R2d)=Ran(P̃)Ran(Q̃), we have
Ker(Q̃K̃Q̃)=Ran(P̃)Ker(K̃)Ran(Q̃)Ker(K̃)Ran(Q̃)Span{wjeβH/2}j=1m,=Ran(P̃)Ker(K̃)Span{wjeβH/2}j=1m.
Next, we prove condition (82) for P=,piρeqpj. Such a condition states that the only eigenvalue of Q̃K̃Q̃ on the imaginary axis iR is the origin. Equivalently, this means that for all uL2(R2d) such that Q̃K̃Q̃u=iλu (λR), we have that λ = 0. To see this, we first notice that Re(Q̃K̃Q̃)u=0. Since Q̃ is a symmetric operator, we have that Re(Q̃K̃Q̃)u=[Q̃(K̃+K̃*)Q̃]u/2=Q̃S̃Q̃u=0, where S̃=j=1dXi*Xi. This means that uKer(Q̃S̃Q̃). As before, Ker(Q̃S̃Q̃) can be constructed by taking the union of three different sets defined by the following conditions:
  1. Q̃u=0, which implies u = ρpj;

  2. Q̃u0,S̃Q̃u=0, which imply uKer(S), i.e., u=αΦ(q)eβ4μp2, where Φ(q) is an arbitrary function of the coordinates q; and

  3. Q̃u0,S̃Q̃u0,Q̃S̃Q̃u=0, which imply P̃S̃Q̃u=S̃Q̃u.

The first condition implies that Q̃K̃Q̃u=0=iλu, i.e., λ = 0. Upon definition of g=Q̃u, the third condition implies that S̃g,pjeq/2pjeβH/2=S̃g. This is a linear ordinary differential equation (ODE) for g that has the unique solution g = θpjeβH/2 for some constant θ ≠ 0. However, it is easy to show that there is no u such that Q̃u=g=θpjeβH/2. In fact, if such u exists, then P̃Q̃u=P̃g=θpjeβH/20, which contradicts the operator identity P̃Q̃=0. Finally, the second conditions implies that if u=Φ(q)eβ4μp2, then P̃u=0 and Q̃u=u. Now consider Im(Q̃K̃Q̃)u=Q̃X0Q̃u=iλu. By using the conditions above, we obtain
Q̃X0Q̃u=X0Q̃uP̃X0Q̃u=X0uX0Q̃u,pjeq/2pjeβH/2=i=1dβ2μpiqiV(q)Φ(q)eβ4μp2piμqiΦ(q)eβ4μp2X0Q̃u,pjeq/2pjeβH/2=i=1dpi(fi(q))eβ4μp2=iλΦ(q)eβ4μp2.
(85)
The last equality holds if and only if fi(q) = 0 and λ = 0. This proves that Q̃K̃Q̃ has no purely imaginary eigenvalues.□

Remark.

Proving the existence and uniqueness of a set of observables {w1, …, wm} such that wj,viρeq=0 and Kwj=vj is not straightforward as it involves the analysis of a system of m hypo-elliptic equations Kwj=vj. Fortunately, this can avoided in some cases, e.g., when the observable vj coincides with the time derivative of wj. A typical example is the momentum pj of the jth particle. We also emphasize that in Lemma 5 we proved that QKQ has no purely imaginary eigenvalues if the projection operator P is chosen as P=,pjρeqpj. This result may not be true for other projections, i.e., QKQ can, in general, have purely imaginary eigenvalues.

Lemma 5 allows us to prove the following exponential convergence result for the semigroup etQKQ:

Proposition 3.
Suppose that the particle interaction potentialV(q) in(68)satisfies Hypothesis 1. LetPbe the projection operator(79). For any set of observables {w1, …, wm} satisfyingwj,viρeq=0,Kwj=K*wj=vj, andσ(QKQ)iR{0}, there exist two positive constantsCandαQsuch that
etQKQu0π0Qu0Leq2CeαQtu0Leq2
(86)
for allu0L2(R2d;ρeq)andt > 0. In(86),π0Qis the orthogonal projection onto the linear spaceKer(QKQ)=Ker(K)Ran(P)Span{wj}j=1m.

Proof.
Rewrite (86) as an L2(R2d) estimation problem
etQ̃K̃Q̃ũ0π̃0Q̃ũ0L2CeαQtũ0L2,
(87)
where π̃0Q̃=U1π0QU. The transformed Kolmogorov operator K̃ is of the form (22) with compact resolvent and a spectrum enclosed in cusp-shaped region of the complex plane shown in Fig. 1 (see Proposition 2). Then, by Theorem 3, the operator Q̃K̃Q̃ has exactly the same properties, provided P̃ is a symmetric, finite-rank projection. To derive estimate (86), we simply use the conclusions of Theorem 4. To this end, we need to make sure that the following two conditions are satisfied:
  • Condition 1. Ran(π0Q)=Ker(Q̃K̃Q̃)=Ker(K̃)Ran(P̃)Span{wjeβH/2}j=1m. Moreover, the L2-orthogonal space Ker(Q̃K̃Q̃) is an invariant subspace of operator Q̃K̃Q̃.

  • Condition 2. π̃0Q̃ is an orthogonal projection in L2(R2d).

Proof of Condition 1.
In Lemma 5, we have shown that Ker(Q̃K̃Q̃)=Ker(K̃)Ran(P̃)Span{wjeβH/2}j=1m. Hence, we just need to prove that Ker(Q̃K̃Q̃) is an invariant subspace of the operator Q̃K̃Q̃. To this end, we recall that the projection operator P̃ is a symmetric operator; therefore, Ran(P̃)=Ran(P̃*). In Ref. 20, Helffer and Nier proved that Ker(K̃)=Ker(K̃*)=eβH/2. By following the same mathematical steps that lead us to Eq. (81), we obtain
Ker(Q̃K̃Q̃)=Ker(Q̃K̃*Q̃)=Ker(K̃)Ran(P̃)Span{wjeβH/2}j=1m.
(88)
We now verify that Q̃K̃Q̃ maps the linear subspace Ker(Q̃K̃Q̃) into itself, i.e., that for any uKer(Q̃K̃Q̃), we have that Q̃K̃Q̃uKer(Q̃K̃Q). To this end, we notice that if wKer(Q̃K̃Q̃), then
Q̃K̃Q̃u,w=u,Q̃K̃*Q̃w=0.
This follows directly from Ker(Q̃K̃Q̃)=Ker(Q̃K̃*Q̃). On the other hand, if uKer(Q̃K̃Q̃), then Q̃K̃Q̃u0, and therefore, we must have Q̃K̃Q̃uKer(Q̃K̃Q̃). Next, consider the following orthogonal decomposition of the Hilbert space L2(R2d):
L2(R2d)=Ker(Q̃K̃Q̃)Ker(Q̃K̃Q̃).
If we define a projection operator π0Q̃ with range Ran(π0Q̃)=Ker(Q̃K̃Q̃), then for any ũ0L2(R2d), we have the orthogonal decomposition
ũ0=π0Q̃ũ0+(ũ0π0Q̃ũ0),whereπ0Q̃ũ0Ker(Q̃K̃Q̃),ũ0π0Qũ0Ker(Q̃K̃Q̃).
Since Ker(Q̃K̃Q̃) is an invariant subspace of Q̃K̃Q̃ and therefore of etQ̃K̃Q̃, we have that etQ̃K̃Q̃(ũ0π0Qũ0)Ker(Q̃K̃Q̃) for all t > 0. On the other hand, since U is an unitary transformation, we have σ(Q̃K̃Q̃)iR={0}. These facts allow us to deform the domain of the Dunford integral representing etQ̃K̃Q̃ũ0π̃0Q̃ũ0 from [−i∞, +i∞] to the cusp SQ̃K̃Q̃, as we did in Theorem 2. This yields
etQ̃K̃Q̃ũ0π̃0Q̃ũ0=etQ̃K̃Q̃ũ0π̃0Q̃ũ0=12πiSQ̃K̃Q̃etzzQ̃K̃Q̃1ũ0dz.
At this point, we can follow the exact same procedure in the Proof of Theorems 2 and 3 to show that the semigroup estimate (86) holds true.

Proof of Condition 2.
We first recall that Ran(π̃0Q̃)=Ker(Q̃K̃Q̃). This implies that for all uL2 and all wKer(Q̃K̃Q̃), we have
π̃0Q̃u,w=u,[π̃0Q̃]*w=0.
Hence, [π̃0Q̃]*w=0 for all wKer([π̃0Q̃]*), which implies that Ker(Q̃K̃Q̃)Ker([π̃0Q̃]*). On the other hand, for all uL2 and all wKer([π̃0Q̃]*), we have
u,[π̃0Q̃]*w=π̃0Q̃u,w=0.
From these equations, it follows that Ker([π̃0Q̃]*)=Ker(Q̃K̃Q̃). Next, we decompose L2(R2d) as
L2(R2d)=Kerπ̃0Q̃Ranπ̃0Q̃,L2(R2d)=Kerπ̃0Q̃*Ranπ̃0Q̃*.
It follows from the above result that Ran(π̃0Q̃)=Ran([π̃0Q̃]*)=Ker(Q̃K̃Q̃) and Ker(π̃0Q̃)=Ker([π̃0Q̃]*)=Ker(Q̃K̃Q̃). For all u,wL2(R2d), we have that wπ̃0Q̃wKer(π̃0Q̃), which can be written as
π̃0Q̃u,wπ̃0Q̃w=u,[π̃0Q̃]*(wπ̃0Q̃w)=0.
Therefore, the operator π̃0Q̃ is an orthogonal projection. This completes the proof. In addition, since π̃0Q̃ has the range Ker(K̃)Ran(P̃)Span{wjeβH/2}j=1m, it can be shown that for the special case vieβHρeq=wieβHρeq=0 and vi,wjρeq=0, we have that π̃0Q̃ admits the explicit representation
π̃0Q̃=π̃0+P̃+i=1m,wiρeqwieβH/2.
(89)
The projection π̃0Q̃ can be transformed back to π0Q by using the mapping U defined in (71).□

Remark.
In general, the orthogonal projection onto Ker(K̃)Ran(P̃)Span{wjeβH/2}j=1m can be written as
π̃0Q̃=i=12m+1,ẽiẽi,
(90)
where {ẽi}i=12m+1 is an orthonormal basis of Ker(Q̃K̃Q̃) in L2(R2d).

Remark.

In Proposition 3, we assumed that Kwj=K*wj=vj. If this condition is not satisfied, then the operator π0Q (or π̃0Q) is no longer an orthogonal projection, and Eq. (89) does not hold. It is rather difficult to obtain an explicit expression for π0Q in this case. We also remark that estimating the convergence constant αQ in (86) is a non-trivial task since such constant coincides with the real part of the smallest non-zero eigenvalue of QKQ.

Proposition 3 allows us to prove that the EMZ memory kernel (12c) and the fluctuation term (12d) of the particle system converge exponentially fast to an equilibrium state for any observable (2).

Corollary 5.1.
Under the same hypotheses of Proposition 3 and Corollary 4.2, the one-dimensional memory kernelK(t)=u(0),KetQKQQKu(0)ρeq/u(0)2ρeqconverges to an equilibrium state exponentially fast in time, i.e.,
K(t)Ku0ρeqQK*u0ρeq+QK*u0,wρeqKu0,wρeqCeαQt,
(91)
whereKw=K*w=u.

Proof.

The corollary follows immediately from (60) and (89) and the fact that PQ=0.□

We emphasize that if w is known, then the equilibrium state can be calculated explicitly. It is straightforward to extend (91) to matrix-valued memory kernels (12c). By following the same steps that lead us to (62), we obtain

K(t)G1CQMCG1DQMeαQt,
(92)

where M denotes any matrix norm and G is the Gram matrix (12a). The entries of the matrix DQ and CQ are given explicitly by

DijQ=QK*ui(0)Leq2Kuj(0)Leq2,CijQ=QK*ui(0)ρeqKuj(0)ρeq+k=1mKuj(0),wk(0)ρeqQK*ui(0),wk(0)ρeq.

The components of the EMZ fluctuation term (12d) decay to an equilibrium state as well, exponentially fast in time. In fact, if we choose the initial condition as u0=QKu0, then (58) yields the following L2(Rn;ρeq)-equivalent estimate:

fj(t)QKuj(0)ρeq+k=1mQKuj(0),wk(0)ρeqLeq2CeαQtQKuj(0)Leq2.
(93)

Inequality (93) can be written in a vector form as

f(t)QKu(0)ρeq+k=1mQKu(0),wk(0)ρeqwk(0)VeqCeαQt(QKu1(0)Leq2,,QKum(0)Leq2)M,
(94)

where Veq is a norm in the tensor product space Veq=i=1mL2(Rn;ρeq), defined similarly to (65).

We developed a thorough mathematical analysis of the effective Mori–Zwanzig equation governing the dynamics of noise-averaged observables in nonlinear dynamical systems driven by multiplicative Gaussian white noise. Building upon recent work of Eckmann, Hairer, Helffer, and Nier13,20 on the spectral properties of hypoelliptic operators, we proved that the EMZ memory kernel and fluctuation terms converge exponentially fast (in time) to a computable equilibrium state. This allows us to effectively study the asymptotic dynamics of any smooth quantity of interest depending on the stochastic flow generated by the SDE (1). We applied our theoretical results to a particle system widely used in statistical mechanics to model the mesoscale dynamics of liquids and gases and proved that for smooth polynomial-bounded particle interaction potentials, the EMZ memory and fluctuation terms decay exponentially fast in time to a unique equilibrium state. Such an equilibrium state depends on the kernel of the orthogonal dynamics generator QKQ and its adjoint QK*Q. We conclude by emphasizing that the Mori–Zwanzig framework we developed in this paper can be generalized to other stochastic dynamical systems, e.g., systems driven by fractional Brownian motion with anomalous long-time behavior,1,10,32 provided there exists a strongly continuous semigroup for such systems that characterizes the dynamics of noise-averaged observables.

This research was partially supported by the Air Force Office of Scientific Research (AFOSR) (Grant No. FA9550-16-586-1-0092) and the National Science Foundation (NSF) (Grant No. 2023495)—TRIPODS: Institute for Foundations of Data Science. The authors would like to thank Professor F. Hérau, Professor B. Helffer, and Professor F. Nier for helpful discussions on the spectral properties of the Kolmogorov operator.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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