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.

## I. INTRODUCTION

The Mori–Zwanzig (MZ) formulation is a technique originally developed in statistical mechanics^{31,57} 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 Hald^{19} 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 Nier^{21}, and Helffer and Nier^{20} 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.

## II. THE MORI–ZWANZIG FORMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

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

where $F:M\u21a6Rd$ and $\sigma :M\u2192Rd\xd7m$ are smooth functions, ** ξ**(

*t*) is

*m*-dimensional Gaussian white noise with independent components, and

*x*_{0}is a random initial state characterized in terms of a probability density function

*ρ*

_{0}(

**). The solution (1) is a**

*x**d*-dimensional stochastic (Brownian) flow on the manifold $M$.

^{26}As is well known, if $F:M\u21a6Rd$ and $\sigma :M\u2192Rd\xd7m$ 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)

By evaluating ** u**(

**) along the stochastic flow generated by the SDE (1) and averaging over the Gaussian white noise, we obtain**

*x*The evolution operator $F(t,0)$ is a Markovian semigroup^{36} generated by the following (backward) Kolmogorov operator:^{25,38,42}

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

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 $Q=I\u2212P$. By following the formal procedure outlined in Refs. 11, 54, and 55, we obtain

Applying the projection operator $P$ to (6) yields

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}

### A. EMZ equation with Mori’s projection operator

Let us consider the weighted Hilbert space $H=L2(M,\rho )$, where *ρ* is a positive weight function in $M$. For instance, *ρ* can be the probability density function of the random initial state *x*_{0}. Let

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

where $Gij=\u27e8ui(0),uj(0)\u27e9\rho $ and *u*_{i}(0) = *u*_{i}(** 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

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

In Eqs. (12a)–(12d), we have *u*_{j}(0) = *q*_{j}(0) = *u*_{j}(*x*_{0}) (*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}

### B. An example: EMZ formulation of the Ornstein–Uhlenbeck SDE

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

where *σ* and *θ* are positive parameters and *ξ*(*t*) is Gaussian white noise with the correlation function ⟨*ξ*(*t*), *ξ*(*s*)⟩ = *δ*(*t* − *s*). As is well-known, the Ornstein–Uhlenbeck process is ergodic and it admits a stationary (equilibrium) Gaussian distribution $\rho eq=N(0,\sigma 2/2\theta )$. Let *x*(0) be a random initial state with probability density function *ρ*_{0} = *ρ*_{eq}. The conditional mean and conditional covariance function of the process *x*(*t*) are given by

Averaging over the random initial state yields

At this point, we define the projection operators

The Kolmogorov backward operator associated with (13) is

By using the identity

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

Here, $M(t)=E\xi (t)[x(t)|x(0)]$ is the conditional mean of *x*(*t*), while $C(t)=Ex(0)[E\xi (t)[x(t)x(0)|x(0)]]$ 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.

## III. ANALYSIS OF THE EFFECTIVE MORI–ZWANZIG EQUATION

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 Nier^{20} 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 $e\u2212tK$ and $e\u2212tQKQ$. Clearly, if $K$ and $QKQ$ are dissipative, then $\u2212K$ and $\u2212QKQ$ are accretive. Unless otherwise stated, throughout this section, we consider scalar quantities of interest, i.e., we set *M* = 1 in Eq. (2).

### A. Analysis of the Kolmogorov operator

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

where $Xi(x)$ (0 ≤ *i* ≤ *m*) denotes a first-order partial differential operator in the variable *x*_{i} 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

*N*be a real number. Define

**is a multi-index of arbitrary order. Note that $Pol0N$ is the set of infinitely differentiable functions growing at most polynomially as ‖**

*α***‖ →**

*x**∞*. Similarly, we define the space of

*k*th order differential operators with coefficients growing at most polynomially with

**as**

*x*It is straightforward to verify that if $X\u2208PolkN$ and $Y\u2208PollM$, then the operator commutator $[X,Y]=XY\u2212YX$ is in $Polk+l\u22121N+M$.

*non-degenerate*if there are two constants

*N*and

*C*such that

It was recently shown by Eckmann and Hairer^{12,13} that $K$ is hypoelliptic if the Lie algebra generated by the operators ${X0,\u2026,Xm}$ in (22) is non-degenerate. The main result can be summarized as follows:

*(Ref.* *13 **). Let* ${X0,\u2026,Xm}$ *and* *f* *in* *(22)* *satisfy the following conditions:*

$Xj\u2208Pol1N$

*for all**j*= 0, …,*m**and*$f\u2208Pol0N$*;**There exists a finite integer**M**such 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 rank**M**is non-degenerate;*

*Then, the operator* $K$ *defined in* *(22)* *and* $\u2202t+K$ *are 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 $\u2202t+K*$ 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

where $S\u2032(Rn)$ is the space of tempered distributions in $Rn$. The operator $\Lambda \u0304\beta $ is the product operator defined as $\Lambda \u0304\beta \u2254(1+\Vert x\Vert 2)\beta /2$, while Λ^{α} is a pseudo-differential operator (see Refs. 13, 14, and 21) that reduces to

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

which induces the Sobolev norm ‖·‖_{α,β}. Throughout this paper, ‖·‖ denotes the standard *L*^{2} norm. With the above definitions, it is possible to prove the following important estimate on spectrum of the Kolmogorov operator $K$:

*(Ref.*

*13*

*). Let*$K\u2208Pol2N$

*be an operator of the form*

*(22)*

*satisfying conditions 1 and 2 in Proposition 1. Suppose that the closure of*$K$

*is a maximal-accretive operator in*$L2(Rn)$

*and that for every*

*ϵ*> 0,

*there are two constants*

*δ*> 0

*and*

*C*> 0

*such that*

*for all*$u\u2208S(Rn)$

*. If, in addition, there exist two constants*

*δ*> 0

*and*

*D*> 0

*such that*

*then*$K$

*has compact resolvent when considered as an operator acting on*$L2(Rn)$

*, whose spectrum*$\sigma (K)$

*is contained in the following cusp-shaped region*$SK$

*of the complex plane (see*Fig. 1

*):*

*for some positive constant*

*C*

_{1}

*and*$M\u2208N$

*.*

We remark that in Ref. 13, the cusp $SK$ is defined as $SK={z\u2208C:Rez\u22650,|Imz|<(8C1)M/2(1+Rez)M}$. 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

In a series of papers, Hèrau, Nier, and Helffer^{20,21} proved that the Kolmogorov operator $K$ corresponding to classical Langevin dynamics generates a semigroup $e\u2212tK$ 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).

*Suppose that*$K$

*satisfies all conditions in Theorem 1. If the spectrum*$\sigma (K)$

*of*$K$

*in*$L2(Rn)$

*is such that*

*then for any*$0<\alpha <min(Re\sigma (K)/{0})$

*, there exists a positive constant*

*C*=

*C*(

*α*)

*such that the estimate*

*holds for all*$u0\u2208L2(Rn)$

*and for all*

*t*> 0

*, where*

*π*

_{0}

*is the spectral projection onto the kernel of*$K$

*.*

*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 integral

^{37}representation of the semigroup $e\u2212tK$ given by

*z*≥ 0, we have |

*z*+ 1|

^{2/M}≥ (8

*C*

_{1})(1 + Re

*z*)

^{2}. A substitution of this inequality into (29) yields, for all $u\u2208L2(Rn)$,

*C*

_{α}> 0 such that $\Vert (K\u2212\alpha )u\Vert \u2265C\alpha \Vert u\Vert $. On the other hand, $K$ is also a real operator, which implies that for all complex numbers $z=(\alpha +iy)\u2209\sigma (K)$, we have

*C*

_{α}along the line

*γ*

_{int}, which leads to

*γ*

_{ext}is defined by all complex numbers

*z*=

*x*+

*iy*such that $|z+1|=(8C1)M/2(1+Rez)M$. In addition, if $z\u2209SK$, then the norm of the resolvent is bounded by $\Vert (K\u2212z)\u22121\Vert \u22648C1|z+1|\u22121/M=(x+1)\u22121$. Combining these two inequalities yields

*π*

_{0}is a projection operator into the kernel of $K$. This allows us to write $\Vert e\u2212tKu0\u2212\pi 0u0\Vert =\Vert e\u2212tK(u0\u2212\pi 0u0)\Vert \u2264\Vert u0\u2212\pi 0u0\Vert $. By combining this inequality with (32), (34), (36), and (37), we see that there exists a constant

*C*=

*C*(

*α*) such that

In the following corollary, we derive an upper bound for the norm of the derivatives of the semigroup $e\u2212tK$:

*Suppose that*$K$

*satisfies all conditions listed in Theorem 2. Then, for any*

*t*> 0

*, the*

*n*

*th order time derivative of the semigroup*$e\u2212tK$

*satisfies*

*where*

*C*

*is a positive constant,*

*α*

*and*

*π*

_{0}

*are defined in Theorem 2, and*

*M*

*is the constant defining the cusp*

*(28)*

*.*

*t*> 0,

*z*=

*x*+

*iy*is in

*γ*

_{int}, then we have that |

*z*| is bounded by constant. By using the uniform boundedness of the resolvent (35), we obtain

*z*=

*x*+

*iy*is in

*γ*

_{ext}, then $|z|<|z+1|=(8C1)M/2(1+x)M$ and $\Vert (z\u2212K)\u22121\Vert \u22648C1|z+1|\u22121/M=(1+x)\u22121$. A substitution of these estimates into the second integral at the right-hand side of (42) yields, for all

*t*> 0,

*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

*π*

_{0}is a finite rank operator that admits the canonical form (in

*L*

^{2})

*L*

^{2}-adjoint of $K$, we have

*t*> 0. Combining these results with the triangle inequality, we have that for any fixed

*t*> 0 and any $n\u2208N$,

Inequality (39) suggests that the flow defined by the semigroup $e\u2212tK$ has bounded derivatives in time. We emphasize that estimate (39) is not sufficient to prove the convergence of the formal power series expansion of $e\u2212tK$ since

### B. Analysis of the projected Kolmogorov operator

In this section, we analyze the semigroup $e\u2212tQKQ$ generated by the operator $QKQ$, where $K$ is the Kolmogorov operator (4) and $P$ and $Q=I\u2212P$ 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.

*Let*$P:L2(Rn)\u2192L2(Rn)$

*be a finite-rank, symmetric projection operator. If*$K$

*satisfies all conditions listed in Theorem 1, then the operator*$QKQ$

*is also maximal accretive and has a compact resolvent. Moreover, the spectrum of*$QKQ$

*lies within the cusp*

*for some positive constants*$CQ$

*and integer*$MQ$

*.*

^{15}the adjoint of a maximal-accretive operator is accretive, and therefore,

^{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 $e\u2212tQKQ$ in $L2(Rn)$. Next, we show that if $K$ satisfies the hypoelliptic estimate $\Vert u\Vert \delta ,\delta \u2264C(\Vert u\Vert +\Vert Ku\Vert )$, then so does $QKQ$, i.e.,

*L*

^{2}into the weighted Sobolev space $S\delta ,\delta $ defined in (24). At this point, we recall that

*S*

^{δ,δ}is compactly embedded into

*L*

^{2}(Lemma 3.2, Ref. 13). Hence, $(QKQ+I)\u22121$ is compact from

*L*

^{2}into

*L*

^{2}, 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 $K\u2208Pol2N$. Then, for

*δ*= max{2,

*N*}, we have the bound

*δ*> 0, we can find an integer $MQ>0$ and a constant

*C*such that

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 $e\u2212tQKQ$ and the regularity of the EMZ orthogonal dynamics induced by $e\u2212tQKQ$.

*Assume that*$K$

*satisfies all conditions listed in Theorem 1. Let*$P:L2(Rn)\u2192L2(Rn)$

*be a symmetric finite-rank projection operator. If the spectrum of*$QKQ$

*in*$L2(Rn)$

*satisfies*

*then for any*$0<\alpha Q<min(Re\sigma (QKQ)/{0})$,

*there exists a positive constant*$C=C(\alpha Q)$

*such that*

*for all*$u0\u2208L2(Rn)$

*and for all*

*t*> 0

*, where*$\pi 0Q$

*is the spectral projection onto the kernel of*$QKQ$

*.*

*Suppose that*$P$

*and*$K$

*satisfy all conditions listed in Theorem 4. Then, for any*

*t*> 0,

*the*

*n*

*th order derivative of the semigroup*$e\u2212tQKQ$

*satisfies*

*where the function*$BQ(t)$

*has the same form as*

*(40)*

*, with*

*α*

*replaced by*$\alpha Q$

*and*

*M*

*replaced by*$MQ$

*.*

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:

*Consider a scalar observable*

*u*(

*t*) =

*u*(

**(**

*x**t*))

*with the initial condition*

*u*(0) =

*u*

_{0}

*, and let*$P(\u22c5)=\u27e8(\u22c5),u0\u27e9u0$

*be a one-dimensional Mori’s projection*

*(9)*

*. Then, the EMZ memory kernel*

*(12c)*

*converges exponentially fast to the equilibrium state*$\u27e8QK*u0,\pi 0QKu0\u27e9$

*, with rate*$\alpha Q$

*. In other words, there exists a positive constant*

*C*

*such that*

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

where $\Vert \u22c5\Vert M$ denotes any matrix norm and ** G** is the Gram matrix (12a). In addition, the matrix $CQ$ has entries $CijQ=\u27e8QK*ui(0),\pi 0QKuj(0)\u27e9$, while $DijQ=\Vert QKui(0)\Vert \Vert Kuj(0)\Vert $. The proof of (62) follows immediately from the following inequality:

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

Let us now introduce the tensor product space $V=\u2297i=1mL2(Rn)$ and the following norm:

where ‖·‖ is the standard $L2(Rn)$ norm and $\Vert \u22c5\Vert M$ is any matrix norm. Then, from (64), it follows that

## IV. AN APPLICATION TO LANGEVIN DYNAMICS

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 gases^{27,38} 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 $\pi 0Q$ 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 $R2d$:

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*= (2

*γ*/

*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

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

where

is the Hamiltonian and *Z* is the partition function. At this point, we introduce the unitary transformation $U:L2(R2d)\u2192L2(R2d,\rho eq)$ defined by

where $L2(R2d;\rho eq)$ is a weighted Hilbert space endowed with the inner product

The linear transformation (71) is an isometric isomorphism between the spaces $L2(R2d)$ and $L2(R2d;\rho eq)$. In fact, for any $u\u0303\u2208L2(R2d)$, there exists a unique $u\u2208L2(R2d;\rho eq)$ such that $u\u0303=(e\u2212\beta H/2/Z)u$ and

By applying (71) to (68), we construct the transformed Kolmogorov operator $K\u0303=U\u22121KU$, which has the explicit expression

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

provided we set

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 $K\u0303$ and its formal adjoint $K\u0303*$ 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 $K\u0303=U\u22121KU$, we can transform the MZ projection operators $P$ and $Q$ into operators in the “flat” Hilbert space $L2(R2d)$ as $P\u0303=U\u22121PU$ and $Q\u0303=U\u22121QU$. The relationship between $L2(R2d)$, $L2(R2d;\rho eq)$, and the operators defined between such spaces can be summarized by the following commutative diagram:

The properties of all operators in $L2(R2d)$ and $L2(R2d;\rho eq)$ are essentially the same since $U$ is a bijective isometry. For instance, if $P$ is compact and symmetric, then $P\u0303$ 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 $K\u0303$ 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*(

**) satisfies the following weak ellipticity hypothesis:**

*q**The particle interaction potential* *V*(** q**)

*is of class*$C\u221e(Rd)$

*, and for all*$q\u2208Rd$,

*it satisfies the following conditions:*

$\u2200\alpha \u2208Nd$

*such that*|| = 1*α**,*$|\u2202q\alpha V(q)|\u2264C\alpha 1+\Vert \u2207V(q)\Vert 2$*for some positive constant**C*_{α}.*There exists*$M\u2208N$*, and**C*≥ 1*, such that*$C\u22121(1+\Vert q\Vert 2)1/(2M)\u22641+\Vert \u2207V(q)\Vert 2\u2264C(1+\Vert q\Vert 2)M/2$*.*

Hypothesis 1 holds for any particle interaction potential that grows at most polynomially at infinity, i.e., *V*(** q**) ≃ ‖

**‖**

*q*^{M}as $q\u2192\u221e$. With this hypothesis, it is possible to prove the following proposition:

*(Ref.*

*20*

*). Consider the Langevin equation*

*(67)*

*with particle interaction potential*

*V*(

**)**

*q**satisfying Hypothesis 1. Then, the operator*$K\u0303$

*defined in*

*(74)*

*has a compact resolvent and a discrete spectrum bounded by the cusp*$SK$

*. Moreover, there exists a positive constant*

*C*

*such that the estimate*

*holds for all*$u\u03030\u2208L2(R2d)$

*and for all*

*t*> 0

*, where*$\pi \u03030$

*is the orthogonal projection onto the kernel of*$K\u0303$

*in*$L2(R2d)$

*.*

By using the isomorphism (71), we can rewrite Proposition 2 in $L2(R2d;\rho eq)$ as

where $\pi 0=U\pi \u03030U\u22121$ is the orthogonal projection $\pi 0(\u22c5)=E[(\u22c5)]$. 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 $Q\u0303K\u0303Q\u0303$ and for the semigroup $e\u2212tQ\u0303K\u0303Q\u0303$ generated by the closure of $Q\u0303K\u0303Q\u0303$. According to Theorem 3, the spectrum of $Q\u0303K\u0303Q\u0303$ is bounded by the cusp $SQ\u0303K\u0303Q\u0303$, provided that $P$ is an orthogonal finite-rank projection operator. On the other hand, Theorem 4 establishes exponential convergence of $e\u2212tQ\u0303K\u0303Q\u0303$ to equilibrium if $Q\u0303K\u0303Q\u0303$ satisfies condition (57). It is left to determine the exact form of the spectral projection $\pi \u03030Q\u0303$, i.e., the projection onto the kernel of $Q\u0303K\u0303Q\u0303$ (see Theorem 4), and verify condition (57). To this end, we consider a general Mori-type projection $P$ and its unitarily equivalent version $P\u0303=U\u22121PU$,

where ${vj}j=1m={vj(q,p)}j=1m$ are zero-mean, i.e., $\u27e8vi\u27e9\rho eq=0$, orthonormal basis functions. In (79), we used the shorthand notation

*Suppose that the particle interaction potential*

*V*(

**)**

*q**in*

*(68)*

*satisfies Hypothesis 1. Then, for any set of observables*${wj}j=1m$

*satisfying*$\u27e8wj,vi\u27e9\rho eq=0$

*and*$Kwj=vj$,

*we have that the kernel of*$Q\u0303K\u0303Q\u0303$

*is given by*

*where*$K\u0303$

*and*$P\u0303$

*are defined in*(74)

*(*

*74*

*) and (*

*79*

*)*

*, respectively. In particular, if*$P$

*is defined as*$P(\u22c5)=\u27e8\u22c5,pj\u27e9\rho eqpj$

*, where*

*p*

_{j}

*is the momentum of*

*j*

*th particle, then we have*

*u*∈

*W*and $Ker(Q\u0303K\u0303Q\u0303)\u2286W$. Let

*f*be an arbitrary element in

*W*. Then,

*α*,

*ρ*

_{1}, …,

*ρ*

_{m},

*θ*

_{1}, …,

*θ*

_{m}} are the coordinates of

*f*in the finite-dimensional space

*W*. By using the definition of $P$ and the fact that $\u27e8vi\u27e9\rho eq=\u27e8vi,wj\u27e9\rho eq=0$ and $\u27e8vj2\u27e9\rho eq=1$, we obtain

$Q\u0303u=0$, which implies $P\u0303u=0$, i.e., $u\u2208Ran(P\u0303)$;

$Q\u0303u\u22600,K\u0303Q\u0303u=0$, which implies $Q\u0303u\u2208Ker(K\u0303)$. This is possible only if $u\u2208Ran(Q\u0303)\u2229Ker(K\u0303)$ since in this case we have $Q\u0303u=u$; and

$Q\u0303u\u22600,K\u0303Q\u0303u\u22600,Q\u0303K\u0303Q\u0303u=0$, which implies $K\u0303Q\u0303u=P\u0303K\u0303Q\u0303u\u22600$. This is possible only if $Q\u0303u=u$, $K\u0303u\u22600$, and $u\u2208Span{wie\u2212\beta H/2}i=1m$, provided that the set of observables ${wj}j=1m$ satisfies $\u27e8wj,vi\u27e9\rho eq=0$ and $Kwj=vj$.

*λ*= 0. To see this, we first notice that $Re(Q\u0303K\u0303Q\u0303)u=0$. Since $Q\u0303$ is a symmetric operator, we have that $Re(Q\u0303K\u0303Q\u0303)u=[Q\u0303(K\u0303+K\u0303*)Q\u0303]u/2=Q\u0303S\u0303Q\u0303u=0$, where $S\u0303=\u2211j=1dXi*Xi$. This means that $u\u2208Ker(Q\u0303S\u0303Q\u0303)$. As before, $Ker(Q\u0303S\u0303Q\u0303)$ can be constructed by taking the union of three different sets defined by the following conditions:

$Q\u0303u=0$, which implies

*u*=*ρp*_{j};$Q\u0303u\u22600,S\u0303Q\u0303u=0$, which imply $u\u2208Ker(S)$, i.e., $u=\alpha \Phi (q)e\u2212\beta 4\mu \Vert p\Vert 2$, where Φ(

) is an arbitrary function of the coordinates*q*; and*q*$Q\u0303u\u22600,S\u0303Q\u0303u\u22600,Q\u0303S\u0303Q\u0303u=0$, which imply $P\u0303S\u0303Q\u0303u=S\u0303Q\u0303u$.

*λ*= 0. Upon definition of $g=Q\u0303u$, the third condition implies that $\u27e8S\u0303g,pj\u27e9eq/2pje\u2212\beta H/2=S\u0303g$. This is a linear ordinary differential equation (ODE) for

*g*that has the unique solution

*g*=

*θp*

_{j}

*e*

^{−βH/2}for some constant

*θ*≠ 0. However, it is easy to show that there is no

*u*such that $Q\u0303u=g=\theta pje\u2212\beta H/2$. In fact, if such

*u*exists, then $P\u0303Q\u0303u=P\u0303g=\theta pje\u2212\beta H/2\u22600$, which contradicts the operator identity $P\u0303Q\u0303=0$. Finally, the second conditions implies that if $u=\Phi (q)e\u2212\beta 4\mu \Vert p\Vert 2$, then $P\u0303u=0$ and $Q\u0303u=u$. Now consider $Im(Q\u0303K\u0303Q\u0303)u=Q\u0303X0Q\u0303u=i\lambda u$. By using the conditions above, we obtain

*f*

_{i}(

**) = 0 and**

*q**λ*= 0. This proves that $Q\u0303K\u0303Q\u0303$ has no purely imaginary eigenvalues.□

Proving the existence and uniqueness of a set of observables {*w*_{1}, …, *w*_{m}} such that $\u27e8wj,vi\u27e9\rho eq=0$ 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 *v*_{j} coincides with the time derivative of *w*_{j}. A typical example is the momentum *p*_{j} of the *j*th 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 $P=\u27e8\u22c5,pj\u27e9\rho eqpj$. 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 $e\u2212tQKQ$:

*Suppose that the particle interaction potential*

*V*(

**)**

*q**in*

*(68)*

*satisfies Hypothesis 1. Let*$P$

*be the projection operator*

*(79)*

*. For any set of observables*{

*w*

_{1}, …,

*w*

_{m}}

*satisfying*$\u27e8wj,vi\u27e9\rho eq=0$

*,*$Kwj=K*wj=vj$,

*and*$\sigma (QKQ)\u2229iR\u2286{0}$,

*there exist two positive constants*

*C*

*and*$\alpha Q$

*such that*

*for all*$u0\u2208L2(R2d;\rho eq)$

*and*

*t*> 0

*. In*

*(86)*

*,*$\pi 0Q$

*is the orthogonal projection onto the linear space*$Ker(QKQ)=Ker(K)\u222aRan(P)\u222aSpan{wj}j=1m$

*.*

Condition 1. $Ran(\pi 0Q)=Ker(Q\u0303K\u0303Q\u0303)=Ker(K\u0303)\u222aRan(P\u0303)\u222aSpan{wje\u2212\beta H/2}j=1m$. Moreover, the

*L*^{2}-orthogonal space $Ker(Q\u0303K\u0303Q\u0303)\u22a5$ is an invariant subspace of operator $Q\u0303K\u0303Q\u0303$.Condition 2. $\pi \u03030Q\u0303$ is an orthogonal projection in $L2(R2d)$.

*t*> 0. On the other hand, since $U$ is an unitary transformation, we have $\sigma (Q\u0303K\u0303Q\u0303)\u2229iR={0}$. These facts allow us to deform the domain of the Dunford integral representing $etQ\u0303K\u0303Q\u0303u\u03030\u2212\pi \u03030Q\u0303u\u03030$ from [−

*i∞*, +

*i∞*] to the cusp $SQ\u0303K\u0303Q\u0303\u2032$, as we did in Theorem 2. This yields

*u*∈

*L*

^{2}and all $w\u2208Ker(Q\u0303K\u0303Q\u0303)\u22a5$, we have

*u*∈

*L*

^{2}and all $w\u2208Ker([\pi \u03030Q\u0303]*)$, we have

In Proposition 3, we assumed that $Kwj=K*wj=vj$. If this condition is not satisfied, then the operator $\pi 0Q$ (or $\pi \u03030Q$) is no longer an orthogonal projection, and Eq. (89) does not hold. It is rather difficult to obtain an explicit expression for $\pi 0Q$ in this case. We also remark that estimating the convergence constant $\alpha Q$ in (86) is a non-trivial task since such constant coincides with the real part of the smallest non-zero eigenvalue of $QKQ$.

### A. EMZ memory and fluctuation terms

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).

*Under the same hypotheses of Proposition 3 and Corollary 4.2, the one-dimensional memory kernel*$K(t)=\u27e8u(0),KetQKQQKu(0)\u27e9\rho eq/\u27e8u(0)2\u27e9\rho eq$

*converges to an equilibrium state exponentially fast in time, i.e.,*

*where*$Kw=K*w=u$

*.*

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

where $\Vert \u22c5\Vert M$ denotes any matrix norm and ** G** is the Gram matrix (12a). The entries of the matrix $DQ$ and $CQ$ are given explicitly by

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 $L2(Rn;\rho eq)$-equivalent estimate:

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

where $\Vert \u22c5\Vert Veq$ is a norm in the tensor product space $Veq=\u2297i=1mL2(Rn;\rho eq)$, defined similarly to (65).

## V. SUMMARY

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 Nier^{13,20} 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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

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