Path integral derivation and numerical computation of large deviation prefactors for non-equilibrium dynamics through matrix Riccati equations

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large deviation prefactors for non-equilibrium dynamics through matrix Riccati equations

Freddy Bouchet, Julien Reygner

To cite this version:

Freddy Bouchet, Julien Reygner. Path integral derivation and numerical computation of large devia- tion prefactors for non-equilibrium dynamics through matrix Riccati equations. 2021. �hal-03319835�

Freddy Bouchet · Julien Reygner

Path integral derivation and numerical

computation of large deviation prefactors for non-equilibrium dynamics through matrix Riccati equations

Abstract For many non-equilibrium dynamics driven by small noise, in physics, chemistry, biology, or economy, rare events do matter. Large deviation theory then explains that the leading order term of the main statistical quantities have an ex- ponential behavior. The exponential rate is often obtained as the infimum of an action, which is minimized along an instanton. In this paper, we consider the computation of the next order sub-exponential prefactors, which are crucial for a large number of applications. Following a path integral approach, we derive the dynamics of the Gaussian fluctuations around the instanton and compute from it the sub-exponential prefactors. As might be expected, the formalism leads to the computation of functional determinants and matrix Riccati equations. By contrast with the cases of equilibrium dynamics with detailed balance or generalized de- tailed balance, we stress the specific non locality of the solutions of the Riccati equation: the prefactors depend on fluctuations all along the instanton and not just at its starting and ending points. We explain how to numerically compute the pref- actors. The case of statistically stationary quantities requires considerations of non trivial initial conditions for the matrix Riccati equation.

Keywords Non-equilibrium statistical physics·Rare events·Large deviation theory·Arrhenius law ·Sub-exponential prefactors·Stochastic differential equations

F. Bouchet

Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, Lyon, France E-mail: [email protected]

J. Reygner

CERMICS, Ecole des Ponts, Marne-la-Vall´ee, France E-mail: [email protected]

1 Introduction

1.1 Rare events, instantons and sub-exponential prefactors

Many systems in physics, chemistry, economics or biology can be described by stochastic differential equations with small noise. In such cases many statisti- cal quantities, for instance the invariant distribution, first exit times, mean first passage times or transition probabilities have asymptotic exponential behavior C^εexp(−I/ε), whereI is the exponential rate,ε is the noise amplitude, andC^ε is a sub-exponential prefactor (see below for a more precise definition).

This mathematical remark has profound consequences in physics. The most classical examples are the definitions of thermodynamic potentials which are rates in this sense, or the Arrhenius law. Besides static properties, from a dynamical perspective, when conditioned on the occurence of a rare event, path probabilities often concentrate close to a predictable path, called instanton. This is a key and fascinating property for the dynamics of rare events and of their impact, which was first observed in statistical physics, for the nucleation of a classical supersatu- rated vapor [27]. Soon after, a similar concentration of path probabilities has been studied in gauge field theories [10, 40], for instance for the Yang–Mills theory. In- stantons continue to have number of applications in modern statistical physics, for instance to describe excitation chains at the glass transition [26], reaction paths in chemistry [24], escape of Brownian particles in soft matter [39].

The computation of the rate I is the subject of classical techniques using Laplace asymptotics, for instance in classical or path integrals [10, 40, 22]. At the mathematical level, this is the subject of large deviation theory, see for instance the Freidlin–Wentzell theory for small noise large deviations [17]. However, for most genuine applications, computing the rateIis not sufficient and a proper estimation of the sub-exponential prefactorC^ε, or of its asymptotic behavior in the limit of small noiseε↓0, is required. From a field theory perspective, such computations require the estimation of the path integrals at next to leading order. Such com- putations are very classical in the field theory context and involve the estimation of expectations over Gaussian processes, for which solutions of Riccati equations are needed. Several classical tools and approaches have been devised in quantum field theory and for equilibrium problems, often on a case by case basis (see for instance [8, 40] as examples among many others).

Recently, rare events, instantons, transitions rates, have been studied in far from equilibrium systems and non-equilibrium steady states, were one starts from dynamics without detailed balance. The statistical mechanics approaches have then be extended to scientific fields so far unexpected. For instance rare events, instantons, and related concepts have been used in turbulence [18, 28, 19, 6, 11], atmosphere dynamics [6, 36], climate dynamics [33], astronomy [38, 1], among many other examples. Moreover, a large effort has been pursued to develop dedi- cated numerical approches to compute the related instantons [21].

For all of these cases, it is essential to go beyond the computation of the expo- nential rates and to compute the sub-exponential prefactors. Then, although for- mally classical ideas still apply, several simplifications related to equilibrium dy- namics no more occur. Often, the extent of possible analytic simplifications is limited, and one has to rely more on numerical simulations for actual computa-

tions. The aim of this paper is to develop the theoretical analysis at a level were it will be useful for devising numerical algorithms. We will thus consider stochastic differential equations with small noise, and develop the formalism and show the potential for numerical computations. The numerical computation will be illus- trated on a simple example.

At a technical level, we will derive the needed matrix Riccati equations, dis- cuss their initial conditions which are not trivial, for instance in the case of sta- tistically stationary quantities, and explain the relation between the matrix Riccati equations and the quantities of interest.

1.2 Setting

In this article, we consider systems described by the stochastic differential equa- tion

dX_s^ε=b(X_s^ε)ds+√

2εdWs, s≥0, (1)

inR^d, whereb:R^d→R^dis a smooth vector field,Wis ad-dimensional Brownian motion, andε>0 can be interpreted as a temperature parameter. In general, the diffusion processX^εdefined by (1) is not reversible, so that it serves as a model for physical systems driven out of equilibrium. In particular, when the process pos- sesses a stationary distributionP^ε — called thenon-equilibrium steady state, the latter may not be explicit. Still, the Freidlin–Wentzell theory [17] asserts that in theε↓0 limit and under suitable assumptions,P^ε satisfies a large deviation prin- ciple, with a rate functionV called thequasipotential. We shall formally denote by

limε↓0−εlogP^ε(x) =V(x) (2) this large deviation principle, and we refer to [12] for standard material on large deviation theory. The quasipotential is known to play the role of an entropy func- tion for non-equilibrium models [17, 22, 3].

Let us define the large deviationprefactor C^ε(x)to the non-equilibrium steady stateP^εby the identity

P^ε(x) =C^ε(x)exp

−V(x) ε

. (3)

An equivalent formulation of (2) is the assertion that limε↓0εlogC^ε(x) =0, so that no precise information on the prefactor can be obtained merely from large deviation theory. However, combining notions from this theory with a WKB ap- proximation, we derived an asymptotic equivalent of the prefactor in [5], recalled in Equation (17) below.

The purpose of the present article is to continue our study of the prefactor, by proposing an alternative method to obtain (17), based on the path integral for- mulation, and presenting a numerical method to effectively compute the terms appearing in this formula. Both tasks rely on the study of matrix-valued Riccati equations satisfied by quantities related to the fluctuations of the diffusion process X^εaround given deterministic paths.

1.3 Organization of the paper

In Section 2, we state the asymptotic equivalent of the prefactor obtained in [5] and recall the notions of large deviation theory involved in this formula. In Section 3, we describe an alternative method leading to the same formula, which is based on the path integral formulation of the non-equilibrium steady state and establishes a connection between the prefactor and the fluctuations of the processX^ε around its most probable path. The relation with sharp asymptotics for mean exit times is discussed in Section 4. Section 5 is dedicated to the numerical computation of the asymptotic equivalent of the prefactor, which we apply on an illustrative example in Section 6.

1.4 Notations

The Euclidian norm and inner product ofR^dare respectively denoted byk · kand h·,·i. We writeIdfor the identity matrix of sized×d. Given a smooth functionf : R^d→R(typically,Vorbk,ℓk), we denote by∇f and∇²fthe gradient and Hessian matrix of f. Similarly, given a smooth vector fieldF:R^d→R^d(typicallyborℓ),

∇F(x)refers to the matrix with coefficient∂_jFi(x)at thei-th row and j-th column, and∇²F(x)(y,y)refers to the vector whosei-th coordinate ishy,∇²Fi(x)yi. The symbols∆and div refer to the usual Laplacian and divergence operators of scalar functions. The closure of an open setD⊂R^dis denoted by ¯D.

2 Prefactor for the stationary distribution

In this section, we first precise the assumptions under which we shall work. We then provide a very brief summary of the main notions from the Freidlin–Wentzell theory on which we shall rely. Finally, we provide an asymptotic equivalent for the prefactorC^ε(x), obtained in [5] through a WKB approximation.

2.1 Assumptions on the vector fieldb

Throughout this section, we make the following assumptions:

(A1) the deterministic system ˙x=b(x)possesses a unique equilibrium point ¯x∈R^d, which attracts all the trajectories,

(A2) for allε>0, the SDE (1) possesses a unique stationary measureP^ε.

Remark 1 Assumption (A1) allows to streamline the exposition of our results. In the more general case where the deterministic system ˙x=b(x)possesses several (isolated) equilibrium points, our arguments can be adapted and produce state- ments that are valid in the neighborhood of the equilibrium points.

2.2 Action functional and quasipotential

We recall a few notions from the Freidlin–Wentzell theory [17, Chapter 4] that will be useful in the paper.

Theaction functionalfor the stochastic differential equation (1) is defined for a trajectoryφ= (φ_s)t1≤s≤t2on the time interval[t1,t2]by

A_t

1,t₂[φ] = (_Rt

s=t21L(φ_s,φ˙_s)ds ifφ is absolutely continuous,

+∞ otherwise, (4)

where the LagrangianL writes

∀(x,v)∈R^d×R^d, L(x,v) =1

4kv−b(x)k². (5) It describes the large deviations of the trajectory(X_s^ε)_s∈[t1,t2]whenε↓0 [17, The- orem 1.1, p. 86].

Thequasipotentialwith respect to ¯xis defined by the variational formula V(x) =inf{A_t

1,t2[φ],φ_t1=x,¯ φ_t2=x,t1<t2}, (6) where the infimum runs over all finite time intervals[t1,t2]and trajectoriesφ= (φs)t₁≤s≤t₂. The quasipotential is nonnegative. We shall furthermore assume that it solves the Hamilton–Jacobi equation

∀x∈R^d, h∇V(x),∇V(x)i+hb(x),∇V(x)i=0, (7) see the discussion in [17, pp. 100–101]. Equivalently, the vector fieldℓdefined by

b(x) =−∇V(x) +ℓ(x) (8)

satisfies

∀x∈R^d, h∇V(x), ℓ(x)i=0. (9) Then, the quasipotential satisfies the identity

V(x) = Z 0

s=−∞

L(ϕ_s^x,ϕ˙_s^x)ds=A_−∞,0[ϕ^x], (10) where ϕ^x = (ϕ_s^x)s≤0 is the minimum action path joining ¯x tox, defined as the unique solution to the backward Cauchy problem

(ϕ˙_s^x=∇V(ϕ_s^x) +ℓ(ϕ_s^x), s≤0, ϕ₀^x=x, lim

s→−∞ϕ_s^x=x.¯ (11)

In the sequel of the paper, we shall callϕ^xthefluctuation path, as it describes the most probable path followed by a fluctuation of the diffusion processX^εjoining ¯x tox.

Remark 2 Assuming that the quasipotentialV is smooth enough and differentiat- ing the equalityh∇V, ℓi=0 twice, we get the identity

d k=1

∑

(∂_k∇²V)ℓ_k+∇²V∇ℓ+∇ℓ^⊤∇²V+

d k=1

∑

∂_kV∇²ℓ_k=0, (12) which shall be useful in the course of the paper.

2.3 WKB derivation of the formula for the prefactor

Injecting both the decomposition (8) of the vector fieldband the ansatz (3) in the stationary Fokker–Planck equation

ε∆P^ε−div(bP^ε) =0 (13)

shows thatP^εis equal to the Gibbs measure P_Gibbs^ε (x) = 1

Z^εexp

−V(x) ε

, Z^ε=

Z

x∈R^dexp

−V(x) ε

dx, (14) if and only if the condition

∀x∈R^d, divℓ(x) =0 (15)

holds. In this case, the prefactorC^εdoes not depend onxand the Laplace approx- imation forZ^ε provides the asymptotic equivalence

C^ε_ε≃

↓0

sdet∇²V(x)¯

(2πε)^d (16)

as soon as∇²V(x)¯ is positive-definite.

If the condition (15) does not hold, an expansion in powers ofε can be per- formed in (13), following the usual WKB approximation method. This leads to the prefactor equivalent

C^ε(x)_ε≃

↓0

s

det∇²V(x)¯ (2πε)^d exp

− Z0

s=−∞divℓ(ϕ_s^x)ds

, (17)

which was derived in [5, Section 3], see also previous results in [9, 31, 32, 35]. In accordance with the discussion above, the exponential term somehow describes the accumulation of ‘non-Gibbsianness’ of the stationary measure along the fluc- tuation pathϕ^x. We shall call this term thenon-Gibbsianness correction.

3 Derivation of the formula from the path integral formulation

In this section, we still work under Assumptions (A1) and (A2), and describe an alternative method to the WKB approximation in order to derive the formula (17) for the prefactor to the stationary distribution P^ε. The method is based on the path integral formulation ofP^ε, in which a Laplace approximation is performed, leading to the computation of an infinite-dimensional Gaussian integral. The lat- ter involves the fluctuations of the processX^ε around the pathϕ^x, and thanks to the Feynman–Kac formula, it is computed by solving a backward matrix Riccati equation.

Although it is certainly less straightforward than the direct WKB approxima- tion sketched in the previous section, the method presented here has the conceptual advantage to establish a connection between the prefactor, written under the form of a functional determinant, and the process of fluctuations ofX^εaround the path ϕ^x. The latter process was recently studied in the mathematical literature [30, 34].

3.1 Laplace approximation in the path integral formulation

When X₀^ε =x0∈R^d, the probability forX_t^ε to be close to x∈R^d writes in the formal path integral formulation

Px₀[X_t^ε≃x] = Z φt=x

φ0=x0

D[φ]exp

−A_0,t[φ]

ε

, (18)

where we used Ito’s convention to approximate the path integral. As a conse- quence, the stationary distributionP^ε writes

P^ε(x) = lim

t→+∞

Z φt=x φ0=x0

D[φ]exp

−A_0,t[φ] ε

= lim

t→−∞ Z _φ0=x

φt=x0

D[φ]exp

−A_t,0[φ] ε

,

(19)

where at the second line we considered trajectories defined on[t,0],t<0 rather than on[0,t],t>0. Givent<0 andx∈R^d, we therefore introduce the notation

p^ε_t(x) = Z φ₀=x

φt=x0

D[φ]exp

−A_t,0[φ] ε

. (20)

Notice that thet→ −∞limit of this quantity should not depend onx0, hence from now on we fixx0=x.¯

The second-order expansion of the action functional on[t,0]around the fluc- tuation path defined by (11) writes

A_t,0[φ]≃A_t,0[ϕ^x] +δA_t,0

δ φ [ϕ^x](φ−ϕ^x) +1 2

δ²A_t,0

δ φ² [ϕ^x](φ−ϕ^x,φ−ϕ^x). (21) In thet→ −∞limit, the first-order term vanishes because of the optimality con- dition onϕ^x. The second-order term rewrites

1 2

δ²A_t,0

δ φ² [ϕ^x](δ φ,δ φ)

=1 4

Z0

s=t kδφ˙s−∇b(ϕ_s^x)δ φsk²− hϕ˙_s^x−b(ϕ_s^x),∇²b(ϕ_s^x)(δ φs,δ φs)i ds

=1 4

Z0

s=t kδφ˙_s+Q^x_sδ φ_sk²+2hδ φ_s,R^x_sδ φ_si ds,

(22)

where the matricesQ^x_sandR^x_sare defined, fors≤0, by Q^x_s=−∇b(ϕ_s^x), R^x_s=−

d k=1

∑

∂kV(ϕ_s^x)∇²bk(ϕ_s^x). (23) At this stage, let us anticipate on the numerical discussion of Section 5, and point out that rewriting

∇V(ϕ_s^x) = 1

2(ϕ˙_s^x−b(ϕ_s^x)), (24)

thanks to (11), shows that the matricesQ^x_sandR^x_scan be computed from the mere knowledge of the vector fieldbwith its space derivatives, and the fluctuation path ϕ^x together with its time derivative. In particular, it is not necessary to compute neither the quasipotential nor its space derivatives along the fluctuation path.

As a consequence of the second-order expansion of the action functional, the Laplace approximation yields, forε>0 small but fixed,

p^ε_t(x)_t→−≃_∞exp

−A_t,0[ϕ^x] ε

× Z δ φ₀=0

δ φt=x¯−ϕ_t^x

D[δ φ]exp

− 1 4ε

Z 0

s=t kδφ˙s+Q^x_sδ φsk²+2hδ φs,R^x_sδ φsi ds

. (25) By (10), in the t → −∞ limit, A_t,0[ϕ^x] converges to the quasipotentialV(x), while (11) shows that ¯x−ϕ_t^xvanishes. Therefore the prefactorC^ε(x)defined by (3) is equivalent, whenε↓0, to thet→ −∞limit of the path integral

Z δ φ₀=0 δ φt=0

D[δ φ]exp

− 1 4ε

Z 0

s=t kδφ˙s+Q^x_sδ φsk²+2hδ φs,R^x_sδ φsi ds

= 1 ε^d/2

Zδ φ0=0 δ φt=0

D[δ φ]exp

−1 4

Z 0

s=t kδφ˙_s+Q^x_sδ φ_sk²+2hδ φ_s,R^x_sδ φ_si ds

(26) after rescalingδ φ by the factor√ε. Since the term appearing in the time inte- gral above is quadratic inδ φ, the path integral is an infinite-dimensional Gaussian integral, the computation of which usually reduces to the computation of a func- tional determinant. In the next paragraph, we adopt a different strategy to compute its value, based on the use of the Feynman–Kac formula.

3.2 Feynman–Kac formula and backward matrix Riccati equation Let us define, for allt≤0 andx,y∈R^d,

u(x;t,y) = Z δ φ₀=0

δ φt=y

D[δ φ]exp

−1 4

Z 0

s=t kδφ˙s+Q^x_sδ φsk²+2hδ φs,R^x_sδ φsi ds

, (27) so that

C^ε(x)≃

ε↓0

1 ε^d/2 lim

t→−∞u(x;t,0). (28)

The path integral in (27) writes as the expectation u(x;t,y) =Et,y

exp

−1 2

Z 0

s=thY_s^x,R^x_sY_s^xids

δ₀(Y₀^ε)

, (29)

where(Y_s^x)s≤0is the diffusion process defined by dY_s^x=−Q^x_sY_s^xds+√

2dW_s, s≤0, (30)

andEt,y[·]refers to the expectation under whichY_t^x=ywhileδ0denotes the Dirac distribution in 0. The linear process(Y_s^x)s≤0describes the scaled Gaussian fluctu- ations of the original process around(ϕ_s^x)s≤0.

By the Feynman–Kac formula, the functionu(x;t,y)is the solution to the back- ward parabolic problem







−∂tu(x;t,y) =∆yu(x;t,y)− hQ^x_ty,∇yu(x;t,y)i −1

2hy,R^x_tyiu(x;t,y), t<0, u(x; 0,y) =δ0(y).

(31) Since the diffusion process (Y_s^x)s≤0 describes Gaussian fluctuations around the fluctuation path, we shall look for a solution to (31) with the Gaussian ansatz

u(x;t,y) = 1

p(4π)^dη_t^xexp

−hy,K_t^xyi 4

, (32)

where, for allt <0, η_t^x>0 and the matrixK_t^x is symmetric. Furthermore, the conditionu(x; 0,y) =δ₀(y)implies the limit behavior

limt↑0(K_t^x)⁻¹=0, lim

t↑0η_t^x=0, lim

t↑0η_t^xdetK_t^x=1. (33) Injecting the ansatz (32) into the problem (31), we get the system of ordinary differential equations

∀t<0,

(η˙_t^x=−η_t^xtrK_t^x,

K˙_t^x= (K_t^x)²+Q_t^x⊤K_t^x+K_t^xQ_t^x−2R_t^x. (34) The second equation is a backward matrix Riccati equation, it is solved in Ap- pendix A.2. We obtain the explicit result

∀t<0, K_t^x=−2∇²V(ϕ_t^x) + (Z_t^x)⁻¹, (35) with

Z_t^x= Z0

s=t

exp

_Zt

r=sA^x_rdr exp _Zt

r=sA^x_rdr _⊤

ds, A^x_r=∇²V(ϕ_r^x) +∇ℓ(ϕ_r^x),

(36) and⌊exp(·)⌋denotes the time ordered exponential of matrices, the definition and a few properties of which are recalled in Appendix A.1. From (35) we deduce in Appendix A.3 that

t→−lim∞η_t^x= 1

2^ddet(∇²V(x))¯ exp

2 Z 0

s=−∞divℓ(ϕ_s^x)ds

. (37)

Combining this result with the identities (28) and (32) allows to recover the for- mula (17). As a conclusive statement, we write the asymptotic equivalence for the non-equilibrium steady state

P^ε(x)_ε≃

↓0

s

det∇²V(x)¯ (2πε)^d exp

−V(x) ε −

Z 0

s=−∞divℓ(ϕ_s^x)ds

. (38)

4 Application to mean exit times

LetD⊂R^d be a domain containing an equilibrium point ¯xof the deterministic system ˙x=b(x). Let us define theexit timefromDby

τ_D^ε =inf{s≥0 :X_s^ε6∈D}. (39) Under suitable assumptions onD, the Freidlin–Wentzell theory asserts that

limε↓0εlogEx¯[τ_D^ε] = inf

y∈∂DV(y), (40)

where the subscript ¯xindicates thatX₀^ε=x, and¯ V still refers to the quasipotential with respect to ¯xdefined by (6).

When the deterministic system ˙x=b(x)possesses several stable equilibrium points and D denotes the basin of attraction of one of these points ¯x, then the diffusion process(X_s^ε)s≥0is calledmetastable, and the logarithmic equivalent (40) is called theArrhenius law.

In any case, the associatedprefactor L^ε_Dto the mean exit time fromDis defined by

Ex¯[τ_D^ε] =L^ε_Dexp 1

ε inf

y∈∂DV(y)

. (41)

In this section, we express this prefactor in terms of the functionC^ε(x)computed in (17).

4.1 Domain with a noncharacteristic boundary

In this subsection, we assume that Dis an open, smooth and connected subset ofR^d satisfying the following conditions, where we denote byn(y)the exterior normal vector aty∈∂D.

(B1) The deterministic system ˙x=b(x)possesses a unique equilibrium point ¯xin D, which attracts all the trajectories started fromD, andhb(y),n(y)i<0 for ally∈∂D.

(B2) The functionV isC¹inD, andh∇V(y),n(y)i>0 for ally∈∂D.

Under these assumptions, we proceed as in Subsection 2.2 and define the vector fieldℓ: ¯D→R^dby the identityb=−∇V+ℓ. Thenℓstill satisfies the orthogonal- ity relation (9).

(B3) The minimum ofV over∂Dis reached at a single pointy^∗, at which

µ^∗=h∇V(y^∗) +ℓ(y^∗),n(y^∗)i>0, (42) and the quadratic formh^∗:ξ7→ hξ,∇²V(y^∗)ξihas positive eigenvalues on the hyperplanen(y^∗)^⊥={ξ∈R^d:hξ,n(y^∗)i=0}.

By Assumption (B1), [17, Theorem 4.1, p. 106] applies and yields (40). In [5, Eq. (4.25)], the integral formula

λ_D^ε = Z

y∈∂Dh∇V(y) +ℓ(y),n(y)iC^ε(y)exp

−V(y) ε

dy (43)

was derived for theexit rateλ_D^ε=Ex¯[τ_D^ε]⁻¹. Using the second-order expansion of V in the neighborhood ofy^∗in this formula, we obtain the equivalent

L^ε_Dε≃

↓0

1 C^ε(y^∗)µ^∗

s deth^∗ (2πε)^d−1

ε≃↓0

1 µ^∗

s2πεdeth^∗ det∇²V(x)¯ exp

Z0

s=−∞divℓ(ϕ_s^y∗)ds

(44)

for the prefactor to the mean exit time fromD.

4.2 The Eyring–Kramers formula for metastable states

In this subsection, we assume that the deterministic system ˙x=b(x)possesses two stable equilibrium points ¯x1and ¯x2, whose basins of attractions are separated by a smooth hypersurfaceS.

We call D the basin of attraction of ¯x1 and formulate the following set of assumptions.

(C1) All the trajectories of the deterministic system ˙x=b(x)started onSremain inSand converge to a single equilibrium point x^∗∈S; besides, the matrix

∇b(x^∗)possessesd−1 eigenvalues with negative real part and a single positive eigenvalueλ^∗.

(C2) Denoting byV the quasipotential with respect to ¯x1still defined by (6), there exists a unique (up to time shift) trajectoryρ = (ρ_t)t∈R⊂Dsuch that

t→−lim∞ρt=x¯1, lim

t→+∞ρt=x^∗, and V(x^∗) =A_−∞,+∞[ρ]. (45) (C3) V is smooth in the neighborhood of(ρt)t∈R, and the vector fieldℓdefined by

the identityb=−∇V+ℓsatisfies the orthogonality relation (9).

In this context,V reaches its minimum onSat the pointx^∗, and we refer to [2] for a proof of the Arrhenius law (40) based on the Freidlin–Wentzell theory. Further- more, the pathρ is called theinstantonand it satisfies

∀t∈R, ρ˙t=∇V(ρt) +ℓ(ρt). (46) As a consequence, for anyt∈R, the fluctuation path(ϕ_s^x)s≤0joining ¯xtox=ρ_t coincides with the instanton, in the sense that

∀s≤0, ϕ_s^x=ρs+t. (47) In order to describe the prefactorL^ε_Din this case, we formulate the following supplementary assumption.

(C4) The matrixH^∗=limt→+∞∇²V(ρt)exists and hasd−1 positive eigenvalues and 1 negative eigenvalue.

Under Assumptions (C1–4), a formula was obtained in [5, Eq. (1.10)] to de- scribe the sharp asymptotics of the expected time taken by the process to reach the neighborhood of ¯x2, which is the contents of the so-calledEyring–Kramers for- mulain the context of reversible diffusion processes [7]. Large deviation theory shows that with overwhelming probability, the path taken by the process to reach

¯

x2passes close tox^∗. At this point, sinceb(x^∗) =0, the process has a probability close to 1/2 to turn back intoDand a probability close to 1/2 to reach the neigh- bourood of ¯x2. Therefore, the expected time taken by the process to exitDis half the time described by the Eyring–Kramers formula. As a consequence, dividing the right-hand side of [5, Eq. (1.10)] by 2, we get the estimate

L^ε_Dε≃

↓0

π λ^∗

s |detH^∗| det∇²V(x¯1)exp

_{Z +∞}

t=−∞divℓ(ρ_t)dt

(48) for the prefactor to the mean exit time fromD.

Remark 3 Notice that in the case addressed in Subsection 4.1,L^ε_Dis proportional to√ε, while in the present case,L^ε_Ddoes not depend onε.

4.3 Comments on the non-Gibbsianness correction

In the formula obtained forL^ε_D in the cases of both Subsections 4.1 and 4.2, the non-Gibbsianness correctiondiscussed at the end of Section 2 appears. In partic- ular for the Eyring–Kramers formula, the identity (47) allows to relate the integral term of (48) to fluctuation paths by the remark that

Z _+∞

t=−∞divℓ(ρ_t)dt= lim

t→+∞

Z ₀

s=−∞divℓ(ϕ_s^ρt)ds. (49)

As a consequence, in the sequel of this paper, we focus on the numerical evaluation of this term.

Recent rigorous derivations of the Eyring–Kramers formula for nonreversible diffusion processes have been obtained in [25, 29], but to our knowledge, they are restricted to the case of processes whose invariant distribution is given by the Gibbs measure (14) and therefore do not include the non-Gibbsianness correction.

5 Effective computation of the prefactor

In Section 4, we showed that sharp asymptotics for mean exit times essentially follow from an accurate computation of the prefactorC^ε to the stationary distri- bution, which was the object of Sections 2 and 3. In this section, we therefore come back to the framework of Assumptions (A1–2) and focus on the numerical evaluation of this prefactor, whose equivalent is given by (17).

For a givenx∈R^d, the evaluation of the right-hand side of (17) requires the computation of the following quantities:

– the fluctuation path(ϕ_s^x)s≤0,

– the divergence ofℓalong the fluctuation path, – the determinant of∇²V(x).¯

The main difficulty to access these quantities is that in general, the transverse decomposition (8) of the vector fieldbis not explicit, and therefore neither∇V norℓare straightforward to obtain.

As a first step, one can remark that thanks to (8),

∀s≤0, divℓ(ϕ_s^x) =divb(ϕ_s^x) +∆V(ϕ_s^x), (50) so that computing the Hessian matrix∇²V(ϕ_s^x) of the quasipotential along the fluctuation path is sufficient to obtain divℓ(ϕ_s^x)and det∇²V(x).¯

Motivated by the large deviation principle (2), specific methods have been de- veloped in the computational physics community to evaluate the quasipotentialV. Thegeometric Minimum Action Method (gMAM) [37, 23] is a numerical proce- dure which computes the fluctuation pathϕ^xand returns the value ofV(x)given by (10). Once we are supplied withϕ^x, we could virtually iterate the method de- scribed above in order to compute the values of V in the neighborhood of the fluctuation path, and hence approximate∇²V(ϕ_s^x)for selected pointsson a time grid. However, computing a fluctuation path for each evaluation ofV is too costly and we shall look for an algorithm that avoids doing so. Our method relies on the fact that∇²V(ϕ_t^x)satisfies a forward matrix Riccati equation, which was already observed in [31, 32].

Proposition 1 Let x∈R^d. If V is smooth, then the family of matrices(H_t^x)_t≤0

defined by

H_t^x=∇²V(ϕ_t^x) (51) satisfies theforwardmatrix Riccati equation

H˙_t^x=−2(H_t^x)²+Q^x_t^⊤H_t^x+H_t^xQ^x_t+R^x_t, (52) complemented with the limit condition

t→−lim∞H_t^x=∇²V(x).¯ (53)

Proof By (11), the time derivative ofH_t^xwrites H˙_t^x=

d k=1

∑

(∂_kV(ϕ_t^x) +ℓk(ϕ_t^x))∂_k∇²V(ϕ_t^x). (54) On the one hand, using (8) and (23) yields

d k=1

∑

∂kV(ϕ_t^x)∂k∇²V(ϕ_t^x) =

d k=1

∑

∂kV(ϕ_t^x)∇²(−bk(ϕ_t^x) +ℓk(ϕ_t^x))

=R_t^x+

d k=1

∑

∂_kV(ϕ_t^x)∇²ℓ_k(ϕ_t^x),

(55)

while on the other hand, the identity (12) yields

d k=1

∑

ℓk(ϕ_t^x)∂_k∇²V(ϕ_t^x)

=−∇²V(ϕ_t^x)∇ℓ(ϕ_t^x)−∇ℓ(ϕ_t^x)^⊤∇²V(ϕ_t^x)−

d k=1

∑

∂_kV(ϕ_t^x)∇²ℓk(ϕ_t^x),

(56)

so that

H˙_t^x=−∇²V(ϕ_t^x)∇ℓ(ϕ_t^x)−∇ℓ(ϕ_t^x)^⊤∇²V(ϕ_t^x) +R^x_t. (57) We now compute

Q_t^x⊤H_t^x+H_t^xQ^x_t =2(H_t^x)²−∇²V(ϕ_t^x)∇ℓ(ϕ_t^x)−∇ℓ(ϕ_t^x)^⊤∇²V(ϕ_t^x), (58) which yields (52) and completes the proof.

Remark 4 The backward matrix Riccati equation (34) and the forward Riccati equation (52) are related by the fact that ifHtis a solution to (52), thenKt=−2Ht

is a solution to (34). This remark is employed in Appendix A.2 to solve (34) by quadrature.

As was already noted in Section 3.1, the coefficientsQ_t^x and R^x_t of the for- ward matrix Riccati equation (52) can be computed from the mere knowledge ofbandϕ^x. As a consequence,∇²V(ϕ_t^x)can be obtained by numerical integra- tion of (52). Therefore we are left with two tasks: computing the limit condition

∇²V(x)¯ for (52), and integrating this equation on(−∞,0]. These tasks are dis- cussed in the respective Sections 5.1 and 5.2. The whole method is then summa- rized in Section 5.3.

5.1 Determination of the limit condition Thet→ −∞limit condition forH_t^xis given by

H¯ :=∇²V(x),¯ (59) which we assume to be positive-definite. This matrix satisfies the stationary ver- sion of (52), which writes

2 ¯H²=Q¯^⊤H¯+H¯Q,¯ (60) with ¯Q:=−∇b(x). In the context of optimal control, this equation is referred to¯ as acontinuous time algebraic Riccati equation (CARE). Alternatively, since ¯xis a stable equilibrium point of the dynamical system ˙x=b(x), the eigenvalues of Q¯have nonnegative real parts. Let us assume that these eigenvalues have positive real parts. Then ¯H⁻¹solves the continuous Lyapunov equation

2I_d=H¯⁻¹Q¯^⊤+Q¯H¯⁻¹, (61) so that

H¯⁻¹=2 Z+∞

t=0 exp(−tQ)¯ exp(−tQ¯^⊤)dt. (62)

5.2 Reparametrization and integration of the system

The system (52-53) is defined on the time interval(−∞,0], with a limit condition int=−∞. To facilitate its numerical integration, we first introduce a reparametriza- tion of time by the length of the fluctuation path, in the spirit of gMAM [23, 37].

The length of the fluctuation path is defined by L^x:=

Z 0

s=−∞kϕ˙_s^xkds= Z 0

s=−∞kb(ϕ_s^x)kds, (63)

where the second identity follows from the orthogonality relation (9). For allt≤0, we denote

σt= Zt

s=−∞kϕ˙_s^xkds= Z t

s=−∞kb(ϕ_s^x)kds∈(0,L^x]. (64) The reparametrized fluctuation path is the trajectory(ϕ˜_σ^x)_σ∈[0,L^x_] defined by the identity

∀t≤0, ϕ_t^x=ϕ˜_σ^x_t, (65) and the continuous extension ˜ϕ₀^x=x. It is easily observed that, for all¯ σ ∈(0,L^x],

d

dσϕ˜_σ^x =∇V(ϕ˜_σ^x) +ℓ(ϕ˜_σ^x)

kb(ϕ˜_σ^x)k . (66) We now define

H˜_σ^x =∇²V(ϕ˜_σ^x), (67) so that ˜H_σ^x_t=H_t^xfor allt≤0. Then the family(H˜_σ^x)_σ∈[0,L^x_]satisfies the problem





 d

dσH˜_σ^x=kb(ϕ˜_σ^x)k⁻¹

−2(H˜_σ^x)²+ (Q˜^x_σ)^⊤H˜_σ^x+H˜_σ^xQ˜^x_σ+R˜^x_σ

, σ ∈(0,L^x],

H˜₀^x=H,¯

(68) with

Q˜^x_σt=Q_t^x, R˜^x_σt=R^x_t. (69) Notice that whenσ↓0, bothkb(ϕ˜_σ^x)kand−2(H˜_σ^x)²+(Q˜^x_σ)^⊤H˜_σ^x+H˜_σ^xQ˜^x_σ+R˜^x_σ vanish. Therefore in order to integrate this system starting fromσ=0, we have to provide ana prioriestimate of theσ↓0 limit of

d dσH˜_σ^x =

d l=1

∑

d

dσϕ˜_l,σ^x ∂_l∇²V(ϕ˜_σ^x). (70) First, a first-order expansion in (66) yields

d dσϕ˜_σ^x_σ≃

↓0

∇²V(x) +¯ ∇ℓ(x)¯

(ϕ˜_σ^x−x)¯

k∇b(x) (¯ ϕ˜_σ^x−x)¯k =(2 ¯H+∇b(¯x)) (ϕ˜_σ^x−x)¯

k∇b(¯x) (ϕ˜_σ^x−x)¯ k (71) thanks to (8). Each individual term in the right-hand side is computable, so that one can evaluate the quantities_dσ^dϕ˜_l,σ^x

|σ=0for alll∈ {1, . . . ,d}.

It remains to compute the third derivatives ofV at ¯xin order to evaluate the matrices∂_l∇²V(x),¯ l∈ {1, . . . ,d}. To this aim, we take the derivative of (12) with respect to thel-th coordinate, and evaluate the result at ¯x. Using the fact thatℓ(¯x) and∇V(x)¯ vanish, we get the matrix identity

0=

d k=1

∑

∂k∇²V(x)¯

(∂lℓk(x)) +¯ ∂l∇²V(x)¯

(∇ℓ(x)) +¯ ∇²V(x)¯

(∂l∇ℓ(x))¯ + (∂_l∇ℓ(x))¯ ^⊤ ∇²V(x)¯

+ (∇ℓ(x))¯ ^⊤ ∂_l∇²V(x)¯ +

d k=1

∑

(∂_klV(x))¯ ∇²ℓ_k(x)¯ . (72) Substituting the derivatives ofℓwith those ofb+∇V, and introducing the nota- tions

vi jl=∂i jlV(x),¯ hi j=∂i jV(x),¯ β_i,j=∂jbi(x),¯ γ_i,jl=∂jlbi(x),¯ (73) we finally obtain that, for alli,j,l∈ {1, . . . ,d},

0=

d k=1

∑

vi jk(βk,l+hkl) +vikl(βk,j+hjk) +hik(γk,jl+vjkl) + (γ_k,il+v_ikl)h_jk+ (β_k,i+h_ik)v_jkl+h_kl(γ_{k,i j}+v_{i jk}).

(74)

Since the coefficientshi j,β_i,j andγ_i,jlare known, the system of equations above inducesd³linear relations between thed³unknown coefficientsv_{i jl}— more pre- cisely, since both the left-hand side of (74) and the value ofv_{i jl} are invariant by permutation of the indicesi, jandl, the number of independent linear relations and unknown coefficients is reduced tod(d+1)(d+2)/6. The resolution of this system allows to reconstruct the matrices∂_l∇²V(x)¯ and completes the computa- tion of (70). Notice that these matrices also possess an explicit formulation as an integral along the fluctuation path associated with the linearized stochastic differ- ential equation

d ˜X_s^ε=∇b(x)¯ X˜_s^ε−x¯ ds+√

2εdWs, s≥0, (75)

see [4, Section 3.3 and Equation (3.39)].

5.3 Conclusion

Givenx∈R^dandε>0, the numerical procedure sketched above to compute the right-hand side of (17) can be summarized in the following steps.

1. Compute the fluctuation pathϕ^x, for example using gMAM [23, 37]. From this step, the value ofV(x)along the fluctuation path can be deduced thanks to (10), which allows to evaluate the right-hand side of (2).

2. Solve the stationary matrix Riccati equation (60) to get ¯H=∇²V(x). This can¯ be done either:

– by computing the integral (62), which solves the Lyapunov equation (61), and inverting the result;

– or by using a numerical solver for the CARE (60) directly.

3. ComputeL^xand for a given number of time stepsN≫1, compute times 0= tN>tN−1>···>t1>t0=−∞such that

∀n∈ {1, . . . ,N−1},

Zt_n+1 s=tn

kb(ϕ_s^x)kds=θ, (76) withθ=L^x/N.

4. Using (71), compute d dσϕ˜_σ^x

|σ=0≃(2 ¯H+∇b(x))¯ ϕ_t^x₁−x¯ ∇b(x)¯ ϕ_t^x₁−x¯

, (77)

and solve (74) to get the matrices∂_l∇²V(x),¯ l∈ {1, . . . ,d}.

5. Compute the approximation ˜H^[n]of ˜H_nθ^x =Htnby integrating the matrix-valued differential equation

d

dσH˜_σ^x =−2(H˜_σ^x)²+ (Q˜^x_σ)^⊤H˜_σ^x+H˜_σ^xQ˜^x_σ+R˜^x_σ

kb(ϕ˜_σ^x)k (78) on the gridσ ∈ {θ,2θ, . . . ,Nθ}, with initial conditions







H˜^[0]=H,¯ H˜^[1]=H˜^[0]+θ

d l=1

∑

d dσϕ˜_l,σ^x

|σ=0∂_l∇²V(x).¯ (79) Many schemes can be employed for this numerical integration; in the example of Section 6, we use a first-order implicit Euler scheme, which is observed to have satisfying stability properties.

6. Deduce from (50) that Z0

s=−∞

divℓ(ϕ_s^x)ds≃

N−1 n=0

∑

(tn+1−tn)

divb(ϕ_t^x_n) +tr ˜H^[n]

, (80)

where we recall that the timest0, . . . ,tNare chosen so that the length of the instanton on each time interval(t_n,t_n+1)be equal toθ.

6 Numerical illustration

In this section, we apply the method devised in Section 5 to a two-dimensional process which exhibitsbistability, in the sense that the associated vector fieldb: R²→R²possesses two stable equilibrium points. We are therefore in the context of Subsection 4.2.

We shall fix one equilibrium point ¯xand first compute the instanton(ρ_t)_t∈R. By (47), the integral term involved in the computation of the prefactor to the sta- tionary distribution at the pointxwrites

Z0

s=−∞divℓ(ϕ_s^x)ds= Z t

s=−∞divℓ(ρs)ds=J(t). (81)

Hence, computing J(t)for anyt ∈R amounts to computing a whole family of prefactors, at pointsx=ρ_t. In thet→+∞limit, we shall also obtain the value of the prefactor to the mean exit time from the basin of attraction of ¯x1, as is described in Subsection 4.2.

6.1 Presentation of the example

Forα>0, we consider the potential function onR²defined by V(x1,x2) =v1(x1) +v2(x2), v1(x1) = x⁴₁

4 −x²₁

2, v2(x2) =αx²₂

2, (82) the critical points of which are(−1,0),(0,0)and(1,0). It is easily checked that the first and third points are stable equilibria of the dynamical system ˙x=−∇V(x), while the second point is stable in the direction ofx2but unstable in the direction ofx1.

For a smooth scalar field c:R²→R to be chosen below, let us define the vector fieldbby

b=−∇V+ℓ, ℓ(x1,x2) =c(x1,x2)

−v^′₂(x2) v^′₁(x1)

. (83)

For any choice ofc, the Hamilton–Jacobi equation (7) is satisfied, the vector field bvanishes at the same points as−∇V, and the points(−1,0)and(1,0)are stable equilibria of the dynamical system ˙x=b(x). Thesaddle-point(0,0)is stable in one direction and unstable in one direction, but these directions may differ from the canonical vectors ofR².

Let us denote ¯x= (−1,0)andx^∗= (0,0). Theinstanton(ρt)t∈Ris the hetero- clinic orbit of the dynamical system

˙

x=∇V(x) +ℓ(x) (84)

joining ¯xint=−∞tox^∗int= +∞. The instanton, the level lines ofV and the field lines ofbare plotted on Figure 1 for the choice

c(x1,x2) =βx1, β >0. (85) The numerical illustrations of this section are plotted forα=0.5 andβ =3.

6.2 Stationary matrix Riccati equation

With our definition of the vector fieldband the choice (85) for the scalar fieldc, the matrix ¯Q=−∇b(x)¯ appearing in (60) writes

Q¯=

v^′′₁(x¯1) c(x)v¯ ^′′₂(x¯2)

−c(x)v¯ ^′′₁(x¯1) v^′′₂(x¯2)

=

2 −β α

2β α

, (86)

where we have used the fact that∇V(x) =¯ 0 for the first identity. The numerical resolution of (60) yields the expected result

H¯= 2 0

0α

. (87)

0

−1 1

−1.4 −1.2 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8

0

−1 1

−0.8

−0.6

−0.4

−0.2 0.2 0.4 0.6 0.8

Fig. 1 The level lines ofV(pale blue), some field lines associated withb(red) and the instanton (thick blue) joining ¯x= (−1,0)tox^∗= (0,0).

6.3 Third derivatives ofV at ¯x

In order to determine the initial condition for the computation of ˜H^[n], we write the coefficients appearing in (74):

– the computation of ¯Hperformed above yields

h11=2, h12=0, h22=α; (88) – the computation of∇b(x)¯ yields

β_1,1=−2, β_1,2=β α, β_2,1=−β, β_2,2=−α; (89) – the computation of the second derivatives ofbyields

γ_1,11=6, γ_1,12=−β α, γ_1,22=0,

γ2,11=10β, γ2,12=0, γ2,22=0. (90) The system of linear equations (74) contains 4 equations, corresponding to the choices{i,j,l}={1,1,1},{1,1,2},{1,2,2},{2,2,2}, which write











0=36+6v111−3βv112,

0=6β α+β αv111+ (4+α)v112−2βv122, 0=2β αv₁₁₂+2(1+α)v₁₂₂−βv222, 0=3β αv122+3αv222.

(91)

The unique solution of this system is

v111=−6, v112=v122=v222=0, (92)