Stochastic action for tubes: Connecting path probabilities to measurement

The trajectories of diffusion processes are continuous but non-differentiable, and each occurs with vanishing probability. This introduces a gap between theory, where path probabilities are used in many contexts, and experiment, where only events with non-zero probability are measurable. Here we bridge this gap by considering the probability of diffusive trajectories to remain within a tube of small but finite radius around a smooth path. This probability can be measured in experiment, via the rate at which trajectories exit the tube for the first time, thereby establishing a link between path probabilities and physical observables. Considering $N$-dimensional overdamped Langevin dynamics, we show that the tube probability can be obtained theoretically from the solution of the Fokker-Planck equation. Expressing the resulting exit rate as a functional of the path and ordering it as a power series in the tube radius, we identify the zeroth-order term as the Onsager-Machlup stochastic action, thereby elevating it from a mathematical construct to a physical observable. The higher-order terms reveal, for the first time, the form of the finite-radius contributions which account for fluctuations around the path. To demonstrate the experimental relevance of this action functional for tubes, we numerically sample trajectories of Brownian motion in a double-well potential, compute their exit rate, and show an excellent agreement with our analytical results. Our work shows that smooth tubes are surrogates for non-differentiable diffusive trajectories, and provide a direct way of comparing theoretical results on single trajectories, such as path-wise definitions of irreversibility, to measurement.

A fundamental concern in stochastic dynamics is to meaningfully quantify the probability of a given trajectory. These probabilities fully characterize a given stochastic dynamics and are indispensable in applications. For example, path-wise definitions of irreversibility as ratios of probabilities of forward-and time-reversed trajectories, are central to the field of stochastic thermodynamics [18,19]. As a second example, reaction pathways between states, obtained from the most probable path connecting them, are essential to the study of rare events such as chemical reactions or conformational changes in biomolecules [20][21][22].
For any diffusive dynamics, as for example the overdamped Langevin equation [2][3][4], which is the most widely used model for stochastic dynamics, the probability of any single trajectory is zero. Consequently, over the last decades, much work has been going into quantifying relative probabilities of Langevin paths [23][24][25][26][27][28][29][30][31][32][33][34][35]. However, because it is not possible to directly access experimentally the ratio of two vanishingly small quantities, * jk762@cam.ac.uk hitherto these theoretical results could not be put to the experimental test. More generally, the fact that a given individual stochastic trajectory occurs with probability zero is the reason that no theoretical result pertaining to individual stochastic trajectories can be checked directly in experiment. Figure 1. For a one-dimensional system, a smooth path ϕ is shown as black solid line, around which a tube of radius R is indicated as grey shaded area. Initial and final position of ϕ are shown as horizontal dotted lines. While the blue trajectory is a realization of the Langevin Eq. (1) which stays inside the tube at all times, the orange trajectory leaves the tube before the final time t f , and therefore contributes to the exit rate from the tube.
We here overcome this limitation, by shifting the focus from individual stochastic trajectories to the finite-radius tubular ensemble, comprised of all stochastic trajectories that remain within a small but finite threshold distance R from a smooth reference path ϕ(t), see Fig. 1 for an illustration. The name tubular ensemble is motivated by the fact that this neighborhood around the reference path is a tube in spacetime. The probability to observe any of those stochastic trajectories, which is called the sojourn probability, is nonzero, and can be measured directly in experiment or simulation, simply by counting which ratio of observed trajectories remains within the threshold distance from the reference path until the final observation time. Thus, considering this ensemble yields a systematic approach to regularizing and connecting to experiment the theoretical discussion of path probabilities, which are recovered as tubes shrink to zero radius. Importantly, our work elevates stochastic actions, a widely used theoretical concept to quantify ratios of path probabilities, to physical observables. This allows, for the first time, the testing of theoretical results involving path probabilities directly in experiment.
The relevance of the tubular ensemble, however, goes beyond serving as a bridge between theory and experiment. In physical applications, one is typically not interested in a single path, but rather in a pathway, that is a family of trajectories that remain within a threshold distance of a reference path. This is precisely the family of trajectories that the tubular ensemble describes.
Our work establishes the tubular ensemble as a generalization of the very concept of an individual stochastic trajectory, which allows to connect to experiment or simulation any question related to individual paths in systems subject to stochastic dynamics. For the overdamped Langevin equation, we provide a conceptually simple derivation of the sojourn probability. In the limit R → 0 we recover the Onsager-Machlup (OM) stochastic action Lagrangian, which is known to characterize relative path likelihoods [23][24][25][26][27][28][29][30][31][32][33][34][35]; in particular, we show explicitly that this Lagrangian appears as a contribution to the exit rate with which stochastic trajectories first leave the tubular neighborhood around ϕ. By calculating the first radius-dependent corrections to the OM Lagrangian, we go beyond single-trajectory asymptotics.
The remainder of this paper is organized as follows. In Sect. II we discuss our general theory for N -dimensional Langevin dynamics. In Sect. III we illustrate our general results by considering explicitly the special case of barrier crossing in a one-dimensional system, N = 1. We in particular show how our theoretical predictions can be compared directly to observables from simulated Langevin time series. We close in Sect. IV by summarizing our results, and discussing their further implications.

II. THEORY
We consider the overdamped Langevin equation, which for an N -dimensional coordinate X t ≡ X(t) ≡ (X 1 (t), ..., X N (t)), is given bẏ where D = k B T /γ is the diffusivity, β −1 = k B T is the inverse thermal energy with k B the Boltzmann constant and T the temperature, γ is the friction coefficient, F (x, t) is a deterministic, possibly time-dependent, force, and ξ is Gaussian white noise with vanishing mean and unit covariance matrix. We assume that D is position-independent, extension of our results to position-dependent diffusivity is discussed in the conclusions.
A. The tubular ensemble One approach to relative path likelihoods of overdamped Langevin dynamics is to derive a formal pathintegral representation of the propagator associated with Eq. (1), and then to use the resulting symbolic expression as a basis for relative path probabilities [23-28, 36, 37]. However, this approach suffers from ambiguities arising from the time-discretization of the short-time propagator [38,39]. In essence, the formal expression one obtains depends on which of infinitely many time-discretization schemes one uses [38]; while for most purposes these discretizations are equivalent, the theoretically derived most probable path, which is sometimes thought of representing the typical behavior of the dynamics, depends on the choice of discretization [39].
A different route towards quantifying relative path probabilities is to consider the tubular ensemble, which consists of those realizations X t of the Langevin Eq. (1) that stay inside a ball of radius R with center a smooth reference path ϕ(t), t ∈ [t i , t f ] [29][30][31][32][33][34][35]40], up to time t ≤ t f , where ||X|| ≡ X 2 1 + ... + X 2 N ; see Fig. 1 for an illustration of X ϕ R . We use the name tubular ensemble for X ϕ R because a ball with time-dependent center is a tube in spacetime (x, t), c.f. Fig. 1. The corresponding sojourn probability is the probability for a stochastic trajectory X to remain closer than a distance R to ϕ until time t; for finite R this probability of course depends on the distribution of initial positions X ti ∼ P i inside the tube. Because the probability of any individual trajectory is zero for Langevin dynamics, the sojourn probability vanishes as R → 0. The relative probability for two reference paths ϕ, φ can still be quantified by [29][30][31][32][33][34][35] where the stochastic action S[ϕ], which is a functional of the smooth path ϕ, is found to be with the Onsager-Machlup (OM) Lagrangian The literature concerned with deriving Eq. (6) via the ensemble Eq. (2) is rather technical [29][30][31][32][33][34], and is focused on the tubular ensemble in the singular single-trajectory limit R → 0.
The key difference between the previous literature and our derivation, is that, instead of working directly with the Langevin Eq. (1), we consider the equivalent description of the stochastic process inside the tube via the Fokker-Planck equation (FPE) [3,4] with a time-dependent spatial domain given at time t by as illustrated by the grey shaded area in Fig. 1, and subject to absorbing boundary conditions at the tube boundary, so that P ϕ R (x, t) describes the distribution of those particles that have never left the tube until time t. Once Eq. (7) is solved for given initial condition X ti ∼ P i , the sojourn probability up to time t is simply the survival probability where here and in the following we suppress the dependence on the initial condition P i unless it is relevant for the discussion. From Eq. (10) in turn we obtain the instantaneous exit rate α ϕ R (t) at which stochastic trajectories leave the tube for the first time, defined by As we show in the following subsections, this yields is the free-diffusion steady-state exit rate out of a ball of radius R, withλ (0) 1 the negative of the eigenvalue with the smallest absolute value of the Laplace operator on the unit ball B 1 with absorbing boundary conditions, L OM is the OM Lagrangian defined in Eq. (6), and L (2) is a quadratic correction to the exit rate, which we calculate in this work. According to Eq. (12), for small radius R the exit rate is dominated by free diffusion. The OM Lagrangian is the first correction to freely diffusive exit from the tube, and with L (2) we include finite-radius effects beyond OM theory. Our derivation directly relates L OM to an experimentally measurable exit rate from a fictitious tube around a smooth reference path ϕ; despite the appearance of the term α free in the mathematical literature on the subject [33,34], this connection between stochastic action and a physical exit rate has not been made explicit before.
In the following subsections we discuss our general theory, outlined just above, for N -dimensional Langevin dynamics. In Sect. II B we derive a perturbative expression for the propagator of the FPE, Eq. (7), with absorbing boundary conditions. Based on this propagator, we in Sect. II C calculate the instantaneous exit rate, defined in Eq. (11), as a power series in the tube radius R, which finally leads to Eq. (12).

B. Perturbative solution of FPE in tube interior
FPE in dimensionless streaming coordinates. To eliminate the time-dependence of the spatial domain Eq. (8), we introduce the dimensionless streaming variables where τ D ≡ L 2 /D is the time scale on which a particle diffuses over the typical length scale L of the external force F . The domain forx is then independent of time and given by the unit ball, We furthermore define a dimensionless probability den-sityP , dimensionless forceF , and a dimensionless path ϕ asP where (x, t) and (x,t) are related as defined in Eq. (14).
Here and below, dimensionless quantities are always indicated by a tilde. In dimensionless form the FPE, Eq. (7), becomes with the dimensionless tube radius and the dimensionless apparent Fokker-Planck (FP) op-eratorF app , given bỹ where∇ denotes the gradient with respect tox with components∇ j ≡ ∂/∂x j , and whereφ ≡ ∂tφ. A dot over a function in dimensionless (dimensionful) form always signifies a derivative with respect to dimensionless (dimensionful) time. For example,φ = L/τ Dφ . Dots are used interchangeably with the symbols ∂ t , ∂t. As can be seen directly from Eq. (21), with respect to the coordinate system (x,t), the velocity of the path ϕ acts as a fictitious spatially constant force inside the tube, so that we obtain an apparent total forcẽ which is why we callF app the apparent dimensionless FP operator. In dimensionless streaming coordinates, the time-depedendent absorbing boundary condition, Eq. (9), becomes which is independent of time. This is the principal advantage of transforming to streaming coordinates. FPE in terms of the instantaneous eigenbasis. We expand the probability distributionP ϕ in Eq. (19) in terms of the instantaneous FP eigenstatesρ n (x,t) as At timet the eigenvalues −λ n (t) and eigenfunctions ρ n (x,t) of the apparent dimensionless FP operator F app (t) fulfill the eigenvalue equatioñ and the absorbing boundary conditionsρ n (x,t) = 0 for ||x|| = 1. We assume the eigenvalues to be ordered, i.e.λ n ≤λ m for n < m, and due to the absorbing boundary condition we haveλ 1 > 0. We assume that at any timet there exists a steady-state solutionρ ss (x,t) of Eq. (19) with reflecting boundary conditions atB; we do not requireρ ss to be normalized. Using this instantaneous steady-state we introduce the instantaneous inner product With respect to this inner product, the FP operatorF app is self-adjoint so that the absorbing-boundary eigenfunctionsρ n can be chosen orthogonal at each timet [4]. If at any timet the force F (x, t) inside the domain B ϕ R (t) originates from a potential U (x, t), such that F = −∇U , then the instantaneous steady-state solution is given bỹ whereŨ (x,t) ≡ βU (x, t), and the dot indicates the standard Euclidean inner product on R N . We emphasize that Eq. (27) does not require a global potential for F , but only a local potential inside the ball B ϕ R (t). If such a local potential does not exist, the instantaneous nonequilibrium steady stateρ ss has to be determined by other means [41].
Expanding the probability distributionP ϕ in Eq. (19) in terms of the instantaneous FP eigenstates as given by Eq. (24), and projecting the equation ontoρ n using the inner product Eq. (26), yields where n ∈ N and a dot here denotes a derivative with respect tot. Because the apparent FP operator is timedependent, both the eigenvaluesλ n and the inner products ρ n ,ρ m , ρ n ,ρ n , are functions oft. The FPE, Eq. (7), with absorbing boundary conditions is equivalent to Eq. (28); once the latter is solved, the dimensionless probability density inside the tube is obtained from Eq. (24), which can be recast in physical units using Eq. (16). SinceF app depends on , so do the quantitiesλ n , ρ n ,ρ m , ρ n ,ρ n , which appear in Eq. (28). From Eq. (21) it is apparent that the ratio of the drift to the diffusion is of order and, therefore, to lowest order the spectrum is that of a free diffusion inside a unit ball. The eigenvalues, eigenfunctions, and steady-state distributions are independent oft at this order, and therefore, any time-dependence of the eigenfunctions must be at least of order . This implies that the ratio of the offdiagonal to diagonal terms in Eq. (28) is at least of order 3 . Thus, mode-coupling effects are sub-dominant and the uncoupled dynamics provides a good first approximation for small . In the context of time-dependent perturbation theory in quantum mechanics, this is known as the adiabatic approximation [42]. Perturbative calculation of the instantaneous FP spectrum. In App. A, we discuss in detail the calculation of both the instantaneous eigenvalues and eigenfunctions as perturbation series in , and calculate explicit expressions for the eigenvaluesλ n to order 3 , and for the eigenfunctionsρ n to order . For n = 1, and if the force F inside the tube is given by a potential also for n > 1, we furthermore calculate explicitly the contributionρ (2) n at order 2 .
Perturbative solution of the FPE. In App. B we in detail derive a solution to Eq. (28), given bỹ where n > 1 in Eq. (31), for a one-dimensional system N = 1 we have k = 6 and for N ≥ 2 we have k = 5, and where we defineΛ n ≡λ n + 2 ρ n ,ρ n ρ n ,ρ n , The solution Eqs. (30), (31), is valid after an initial transient time, i.e. fort and neglects terms that are exponentially small as compared to Eqs. (30), (31).
The form of Eqs. (30), (31) allows for an intuitive interpretation. Initially all eigenmodes are excited, with their respective amplitudeã n (t i ) determined by the initial condition. The dynamics of each mode is dominated by the adiabatic exponential decay, and after an initial relaxation time the mode n = 1 (which decays slowest) dominates the probability distribution Eq. (24); this is represented by the first term in the bracket in Eq. (30). The leading-order effect of the mode coupling is twofold. First, during their initial decay the modes n > 1 can transfer some of their initial amplitudeã n (t i ) to the n = 1 mode, as described by the second term in the bracket in Eq. (30). Second, after their initial decay the n > 1 modes can be excited instantaneously by the lowest mode n = 1, as described by Eq. (31).
For a particle initially localized atx i , we have a deltapeak initial condition,P ϕ (x,t i ) ≡P i (x) = δ(x −x i ), so that the initial amplitude of the n-th mode is given bỹ Substituting the resulting coefficients Eq. (30), (31), into the eigenmode expansion Eq. (24) of the propagator then yields where k = 6 for a one-dimensional system, N = 1, and k = 5 for N ≥ 2. Equation (36)  , and a part that only depends on (x i ,t i ); Thus, while the total probability to have remained inside the tube until timet is affected by the initial condition, after the initial relaxation timeτ rel the spatial probability distribution inside the tube is independent of the initial condition.
Using Eq. (36), we can express the solution for an ar-bitrary initial distributionP i inside the tube as from which the survival probability, Eq. (10), follows in dimensionless form as Complementary to the survival probability is the normalized probability densityP n,ϕ inside the tube at any timẽ t, defined as which describes the distribution inside the tube of those particles that have stayed until the current timet. Using Eqs. (36), (37), the distribution Eq. (39) can be shown to be independent ofP i .
C. Exit rate from tube For a particle starting at time t i according to a distribution X ti ∼ P i inside the tube, the instantaneous exit rate is given by where P ϕ is the survival probability defined in Eq. (10). Using Eq. (14), (16), the dimensionless instantaneous exit rate Eq. (40) is defined asα where the dot denotes a derivative with respect tot, and Using the steady-state FP solution Eqs. (36)(37)(38), the exit rate Eq. (41) is evaluated to yield α ϕ (t) =λ withĨ and where we used that ρ m ,ρ 1 Ĩ m is of order 2 , c.f. App. A 3. Equation (42), which is valid after the initial transient decay timeτ rel defined in Eq. (34), is independent of the initial distributionP i ; this is because in Eq. (36) the initial condition only contributes an overall prefactor independent of (x,t), which does not affect the relative change of particles inside the tube quantified by Eq. (41). With Eq. (42) the instantaneous exit rate is expressed solely in terms of the instantaneous FP spectrum inside the tube. Expanding the quantities that appear in Eq. (42) in powers of , and using the symmetry properties of these quantities, c.f. App. A, a power series expansion of the exit rate is obtained as whereα free =λ where at Eq. (46) we use the perturbative result forλ (2) n , c.f. App. A, the definition of the OM Lagrangian L OM is given in Eq. (6), and wherẽ Note that we suppress the dependence on ϕ in the notation of theα (k) , and thatα free is independent of ϕ. Using Eq. (41) the exit rate in physical units can be obtained from Eqs. (44)(45)(46)(47); note that according to Eq. (41), a scaling k inα ϕ (dimensionless form) translates to a scaling R k in α ϕ R (physical units). The order-2 term in Eq. (12) is thus given by L (2) =α (2) /(τ D L 2 ); according to Eq. (21) the instantaneous FP spectrum depends on (ϕ,φ); because of the additional time derivative in Eq. (47), the term L (2) additionally depends onφ.
Equations (44)(45)(46)(47), which express the exit rateα ϕ fully in terms of the perturbative spectrum of the FP operator inside the tube, are one of the main results of this paper. The equations show that for small tube radius 1, the exit from the tube is dominated by the steady-state freediffusion exit rate given by Eq. (45); this is consistent with the fact that the Langevin Eq. (1) is on short times dominated by the noise term ξ (as opposed to the deterministic force F ). From Eq. (44) we see that the freediffusion exit rate in fact diverges as 1/ 2 , which gives a physical picture as to why the probability for observing the single path ϕ is zero.
According to Eqs. (44), (46), the first correction to the free-diffusion exit rate, which occurs at order 0 , is given by the OM Lagrangian L OM ; this establishes a direct link between L OM and the physical observable α ϕ R . The next correction Eq. (47), which is quadratic in the tube radius, is still in the adiabatic limit, meaning that only the n = 1 eigenvalue and eigenfunction appear in Eq. (47).

III. ONE-DIMENSIONAL SYSTEMS AND NUMERICAL EXPERIMENTS
In the present section we consider the special case of a one-dimensional system, N = 1, for which it is straightforward to calculate explicit expressions for the results derived in Sect. II. To illustrate and verify our perturbative analytical results, we compare to numerical simulations throughout; in Sect. III A we introduce the corresponding example system, a double-well system with a barrier-crossing transition path ϕ. While in Sect. III B we discuss the normalized probability density inside the tube, we in Sect. III C consider the exit rate.

A. Model
For a length scale L and a time scale T , we consider the Langevin Eq. (1) with a diffusion coefficient D = L 2 /T , so that τ D = T . We consider a force F that is given as the gradient of a potential, F (x) = −(∂ x U )(x), and for the potential U (x) use a quartic double well, with βU 0 = 2, as illustrated on the right-hand side of Fig. 2. For the smooth reference path ϕ we choose a barrier crossing path, parametrized as where for κ, which controls the maximal barrier crossing speed, we use κ = 10; we furthermore choose t i = 0,   In the plot on the left-hand side, the potential is shown as colormap in the background, with the two minima of the potential represented by horizontal dashed lines. The reference path ϕ, defined in Eq. (50), is shown as solid black line. Around the reference path, a tube of radius = R/L = 0.5 is depicted by a shaded grey region. The vertical dashed lines denote the timest = 0.1, 0.5, 0.6, which are considered in Fig. 3.

B. Perturbative solution of FPE in tube interior
As we show in detail in App. C 1, for N = 1 the eigenvalue/eigenfunction Eq. (25) can be solved recursively for increasing k, and the solution at order k is of the form n,c (x,t) are polynomials inx of order ≤ k, and depend ont viaẼ l (t), 1 ≤ l ≤ k, which are given bỹ In App. C 1, we give explicit expressions forλ n,c , up to k = 5. Using the perturbative spectrum Eq. (51), the propagator Eq. (36) can be calculated as a power series in . From the propagator, in turn, the perturbation series for the normalized probability densityP n,ϕ inside the tube is obtained using Eq. (39). It is found thatP n,ϕ is of the form where the coefficientsÑ c (x,t), are polynomials inx of order ≤ k, and depend ont viaẼ l (t), 1 ≤ l ≤ k, as defined in Eq. (52). The explicit expressions forÑ For all times displayed, we observe that while for = 0.1, 0.5, numerical and perturbative results agree very well with each other, for the largest radius considered, = 0.7, clear deviations between the two are discernible. At the timet = 0.1 considered in Fig. 3 (a), the path ϕ is close to the minimum at x = −L and has a small velocity, c.f. Fig. 2. While for the smallest radius = 0.1 the probability density is almost symmetric aroundx = 0, indicating that the dynamics inside the tube is dominated by free diffusion, for = 0.5, 0.7 the influence of the potential leads to a slight shift of the most probable position towards small negative values ofx. The perturbative probability density for = 0.7 takes on negative values close tox = −1, which is clearly unphysical and signifies a breakdown of the perturbative results of order 5 . In Fig. 3 (b) we show probability densities at timet = 0.5, which according to Fig. 2 is when the path ϕ traverses the barrier top. Despite the fact that at the maximum the potential U is a symmetric function ofx, all probability densities shown in Fig. 3 (b) are tilted towards negative values ofx. This is because the velocity of the path ϕ leads to a ficticious force, as seen explicitly in Eq. (22); due to this fictitious force the symmetry of the potential U is broken at the barrier top, which leads to the tilted probability densities observed in the figure. This effect is less pronounced at small , where the dynamics inside the tube is dominated by free diffusion, as compared to the apparent deterministic force due to U andφ. In Fig. 3 (c) we consider the timet = 0.6, at which the path ϕ descents from the barrier top towards the minimum at x = L, c.f. Fig. 2. Here we observe that even though the force resulting from the potential U pushes towards the positivex-direction, due to the velocity of the path ϕ the apparent force Eq. (22) leads to a probability density that is still slightly tilted towards the negativex-direction, i.e. uphill in the potential energy landscape.
In the Supplementary Material (SM) [43] we provide videos that show the full time evolution of the normalized probability density for radius = 0.1, 0.3, 0.5, 0.7. For = 0.1, 0.3, numerical and analytical results show perfect agreement throughout. Consistent with Fig. 3, for = 0.5 small deviations between numerical and analytical density are observed, and become most pronounced as the path ϕ ascends the barrier (t ≈ 0.45); however, given the size of the tube the agreement between numerical and analytical probability density is remarkably good overall. For = 0.7 the breakdown of our perturbative results can be observed; the analytical probability takes on negative values and at times deviates considerably from the numerical data.
Overall, from Fig. 3 (a), (b), (c), and also the supplementary videos, we conclude that for small to intermediate tube radius, our analytic result Eq. (53) very well approximates the actual FP dynamics inside the tube.

C. Exit rate from tube
Using the explicit expressions for the spectrum given in App. C, the expansion of the exit rate Eq. (44) in powers of is given bỹ where theẼ l ≡Ẽ l (t) are defined in Eq. (52), and a dot denotes a derivative with respect tot. We again consider barrier crossing in the double-well system, as defined in Eqs. (49), (50), and illustrated in Fig. 2. In Fig. 3 (d), (e), (f), we compare numerically calculated exit rates to perturbative results obtained from Eqs. (44), (54-56). In the plots the exit rate is shifted and rescaled according to so that i) the sign of a curve indicates whether the exit rate is enhanced or diminished as compared to the freediffusion limitα free , and ii) the magnitude yields the relative importance of the terms Eqs. (55), (56) as compared toα free . Numerical data is shown as solid colored lines, perturbative analytical results are given as broken colored lines. To gauge the importance of the quadratic correction Eq. (56) relative to the OM Lagrangian Eq. (55), we furthermore include plots of the latter as solid black lines in Fig. 3 (d), (e), (f). As detailed in App. C 5, we use as initial distributionP i for our simulations the instantaneous steady state of the FPE, so that there is no transient initial decay in our numerical data; a brief discussion of the transient effects of the initial condition on the exit rate is given in App. C 4. In Fig. 3 (d) we consider the radius = 0.1. As can be seen, the numerical and analytical results agree perfectly with each other, and also with the OM Lagrangian Eq. (55). This means that the quadratic correction Eq. (56) is not yet relevant at this radius. Relative deviations from the free-diffusion exit rateα free are less than 10% throughout, so that the exit rate is dominated by free diffusion. Figure 3 (e) shows data for the intermediate radius = 0.5. Numerical and perturbative analytical results agree very well with each other, with minor deviations att ≈ 0.5. However, clear deviations between numerical data and the OM Lagrangian Eq. (55) are visible, meaning that the quadratic correction Eq. (56) to the exit rate is now relevant. The deviations between our perturbative/numerical results and OM theory are twofold. First, when the path is close to the minima, the OM action underestimates the true (numerical) exit rate. During these times, the numerical exit rate is rather insensitive to the exact position of the tube center within the well, because the rate limiting step to exit the tube is to climb the potential barrier, which is expected to be rather insensitive to the exact position of the tube center in the well. The second difference between our perturbative/numerical results and OM theory is that during barrier crossing, the numerical exit rate is delayed as compared to the OM Lagrangian. From the magnitude of the rescaled exit rate Eq. (57), we conclude that for = 0.5, the free-diffusion exit rate is of the same order as the corrections Eq. (55), (56). Figure  3 (f) shows data for the largest radius = 0.7. Overall the perturbative result Eq. (44) still shows reasonable agreement with the numerical exit rate, which is surprising since the corresponding probability density at times deviates strongly from the numerical results, c.f. Fig. 3 (a), and the supplementary videos. However, clear deviations between numerical and analytical exit rate can be discerned, most prominently during barrier crossing att ≈ 0.5. Numerical exit rate and OM Lagrangian Eq. (55) disagree considerably, showing the importance of the quadratic correction Eq. (56). During barrier crossing, the contributions to the exit rate from Eqs. (55), (56) are about 5 times larger than the free-diffusion exit rateα free .
In summary, Fig. 3 (d), (e), (f) shows that our perturbative results Eqs. (44), (54-56) describe the exit rate quantitatively up to a tube radius well comparable to the typical length scale of the potential U , and in particular beyond the regime where the OM Lagrangian is applicable.
To close this section, we illustrate how finite-radius exit rates obtained directly from measured trajectories compare to our perturbative analytical results. For this, we consider a tube radius = R/L = 0.5, as also discussed in Fig. 3 (e). Figure 4 depicts the exit rate obtained directly from a large number of independent simulated time series. As Fig. 4 shows, the exit rate obtained directly from Langevin time series agrees well with our perturbative result Eq.  to measure the exit rate for a finite-radius tube directly from time series, without fitting any model to the data. Note that since the FPE, Eq. (7), with absorbing boundary conditions is equivalent to the Langevin Eq. (1), with trajectories being discarded once they first cross the absorbing boundary, it is expected that Fig. 3 (e) and Fig. 4 lead to the same conclusions; indeed, the agreement between numerical FP solution and results obtained from Langevin simulations is an important consistency check for our numerics. Apart from illustrating how our results directly connect to measured time series, the analysis based on Langevin trajectories also highlights two features that appear when extracting the exit rate from recorded data. First, since all Langevin simulations are initiated at x = −L, which can be thought of as a definite experimental initial condition, the exit rate shows a short transient relaxation period for timest 0.05, see App. C 4 for further discussion. Second, the number of trajectories inside the tube decreases over time, so that the statistics for calculating the exit rate become successively worse; this explains why the exit rate measured from Langevin trajectories starts to become noisy around t ≈ 0.5.

IV. SUMMARY AND CONCLUSIONS
In this work we establish the finite-radius tubular ensemble, which consists of all stochastic trajectories that stay close to a smooth reference path ϕ, as a physically and mathematically useful concept to regularize and extend the path probabilities of individual stochastic trajectories. We in particular derive explicit expressions for the probability to observe any path of the tubular ensemble, thus generalizing the Onsager-Machlup (OM) stochastic action. Our results have several important consequences.
From a mathematical perspective, we evaluate and study the probability P (X ϕ R ), i.e. the probability that a stochastic trajectory stays close to a given smooth reference path, for finite radius R. We therefore focus on a measure, which is in contrast to previous work, which aimed to define probability densities on the space of all continuous paths, and therefore always involved the singular limit R → 0 [29]. Compared to the approach to path probabilities via path integrals [23][24][25][26][27][28], an advantage of our approach is that at no point we need to discretize time. Therefore, none of the technical/conceptual difficulties arising from different timediscretization schemes discussed in the literature arise [28,38,39]. Furthermore, in our theory smooth and nondifferentiable stochastic trajectories are cleanly disentangled. The former are used to parametrize a set (a moving ball with finite radius), the latter are confined to this set.
In a sense, the approach used in the present paper is opposite to Freidlin-Wentzell theory [44]. While Freidlin and Wentzell also consider the tubular ensemble Eq. (2), they investigate the double limit of vanishing radius R → 0 and temperature 1/β = k B T → 0. Practically speaking, in their analysis the deterministic force in the Langevin Eq. (1) is assumed to be the dominant term. In our perturbative calculation at constant temperature, on the other hand, we perturb around the freediffusion solution of the FPE, which means that in our analysis the random force term in the Langevin Eq. (1) is assumed to be the dominant term on short length scales. That random noise dominates over deterministic forces at short length-and time scales is a basic feature of the Langevin equation and is in fact the reason why a typical realization of Eq. (1) is nowhere differentiable.
Our theory for the finite-radius tubular ensemble Eq. (2) is also highly relevant from a physical perspective. By establishing a direct relation between exit rate and stochastic action Lagrangian, we put the latter within reach of experiments. Indeed, substituting Eqs. (5), (11), into Eq. (4), it follows that While directly measuring the probability of an individual given path is experimentally unfeasible, simply because that probability is zero, directly measuring experimentally the probability for a trajectory to stay inside a moving ball with finite radius is possible with present-day technology [9,45,46]. This means the right-hand side of Eq. (58) can be measured for finite R, as was done in Fig. 4, and then extrapolated to the limit R → 0; this can be done without fitting a stochastic model to the time series. Equation (58) thus allows to compare model-free measurements of exit rates (right-hand side of the equation) to theoretical predictions for the stochastic action Lagrangian (left-hand side of the equation). This will allow to experimentally validate theoretical predictions for the stochastic action Lagrangian as a measure for relative path likelihoods. Equation (58) can furthermore serve as an operational and experimentally relevant definition for the action Lagrangian for other models of stochastic dynamics, for example those used to describe active particles [47]. While irrelevant in the limit R → 0, for finite tube radius it will be important to understand in more depth how transient effects due to the initial distribution X ti ∼ P i affect the sojourn probability. A basis for investigating such boundary effects is given by the full perturbative solution considered in App. B.
While we assume a smooth path ϕ, our derivation in fact only uses that it is twice differentiable. The first derivativeφ emerges from applying the coordinate transformation Eq. (14) to the FPE, c.f. Eq. (21). The second derivative enters because the FPE in terms of the instantaneous eigenbasis, Eq. (28), contains the time-derivative of FP eigenfunctions; since these eigenfunctions depend onφ, their derivative depends onφ. It will be interesting to extend our theory to reference paths ϕ that are continuous, but not differentiable, such as realizations of the Langevin Eq. (1). A starting point for this would be to investigate how the FPE transforms under a nondifferentiable coordinate transformation [48].
Another possible extension of our theory is to include position-dependent diffusivity, i.e. to replace the constant diffusion coefficient D by a function D(x). Assuming that the diffusivity varies slowly along the tube, a first approximation is to simply replace D by D(x) in our results. In view of the exit rate Eq. (12), the sojourn probability is then given by where the diffusivity in the OM Lagrangian Eq. (6) is now evaluated at D(ϕ(t)). Equation (59) shows that for position-dependent diffusivity, for small tube radius R the leading order difference in sojourn probabilities along two paths ϕ, φ, is the mean free-diffusion exit rate along the paths, and the OM action is now a subleading-order correction. Thus, in the limit R → 0, instead of Eq. (6) one would rather want to consider an action to quantify physically observed relative path probabilities. Intuitively, a particle is more likely to diffuse away from a given reference path in a region with large diffusivity, as compared to a region with low diffusivity. In the mathematical literature the leading-order effect due to free diffusion, given by Eq. (60), is usually scaled away, essentially by introducing a position-dependent tube radius R(x) such that D(x)/R(x) 2 is constant as a function of x [29]. Thus, before applying the OM theory in systems with position-dependent diffusivity, one should decide whether one wants to quantify relative path probabilities using a spatially constant threshold R, in which case one would want to use Eq. (60) as action, or using a varying threshold R(x) ∼ D(x), in which case the OM action is the leading order difference in sojourn probabilities [29,37]. The present work on the tubular ensemble Eq. (2) offers an intuitive picture on (relative) path probabilities for the Langevin Eq. (1), providing a physical approach to this hitherto rather technical subject. Since any question that can be posed for individual stochastic trajectories is straightforwardly extended to the tubular ensemble, and through that is made accessible to simulation or experiment, the theory presented here is expected to find many applications in the future. The results will be particularly useful for the field of stochastic thermodynamics, where the concept of individual trajectories, and ratios of their probabilities, is employed extensively [18,19,47]. In the present appendix, we perturbatively solve the eigenvalue Eq. (25) up to order 2 . For this, we first expand the right-hand side of the equation as a power series in .

ACKNOWLEDGMENTS
Taylor expansion of the force. The multidimensional Taylor expansion of the force F around the tube center ϕ(t) is given by where (x−ϕ(t)) αi ≡ x αi −ϕ αi (t) is the α i -th component of the vector x − ϕ(t). Substituting Eq. (A1) into the definition of the dimensionless force Eq. (17) and using Eq. (14), we obtain that where we use the Einstein sum convention for the indices α ≡ (α 1 , ..., α k−1 ), abbreviatex α ≡x α1 ...x α k−1 , and the vector-valued (k − 1)-multilinear formẼ k is defined as where dimensionless quantities (as indicated by a tilde) and quantities with physical dimensions are related via Eqs. (14), (18). Note that if the derivatives of the force commute, e.g. if the force is a smooth function of position for a time t, thenẼ k is symmetric in the (α 1 , ..., α k−1 ). If the force is locally given by a potential U as F = −∇U , then the j-th vector component ofẼ k is given bỹ so that Eq. (A3) a multivariate Taylor expansion of the dimensionless potentialŨ = βU +x ·φ around the tube centerx = 0. Consequently, in that case the unnormalized instantaneous steady state distribution inside the tube is given bỹ Hierarchy of equations for the spectrum. Inserting the power series Eq. (A3) into the eigenvalue Eq. (25), we obtaiñ where the dot denotes a scalar product and∇ denotes the gradient operator with vector components∇ i ≡ ∂/∂x i .
Expanding both the instantaneous eigenvalues and eigenfunctions as power series in , as defined in Eq. (29), substituting these into Eq. (A7), and demanding that the resulting equation hold at each power k , we obtain a hierarchy of equations which for the n-th eigenvalue/eigenfunction pair at order k read where we use the convention that for k = 0, the sums on the right-hand side are zero. For the absorbing boundary conditions to be fulfilled independently of , they need to hold at each order separately, so that for all k ∈ {0, 1, 2, ...} we havẽ While any solution to Eqs. (A8), (A9) can be used in practice for the spectrum, the solution to these equations is not unique. To fix the solution uniquely, we introduce a normalization condition ρ n ,ρ n = 1. Inserting the power series expansion Eq. (29) for the eigenfunction into this normalization condition, and demanding that the condition hold at each power of , we obtain for k = 0 that where we use the convention that for k = 1 the sum on the right-hand side is zero and the expansion ofρ −1 ss in powers of is discussed in App. A 2. Note that for any k, there only appear perturbation termsρ (l) n with l < k on the right-hand side of Eq. (A11).
At order k = 0, the right-hand side of Eq. (A8) vanishes, so that the equation is reduced to the eigenvalue equation of the Laplace operator. Thus,λ Assuming the spectrum has been obtained up to order k − 1, the contribution at order k is calculated as follows. An equation forλ where we used the normalization condition Eq. (A10) forρ n has been obtained via Eq. (A12), the righthand side of Eq. (A8) is known, so that to obtainρ (k) n the inhomogeneous Helmholtz Eq. (A8) with boundary conditions Eq. (A9) has to be solved. While in general this can be done using the corresponding Green's function, we calculate the spectrum to order 2 directly using a simple ansatz below. Before that, however, we establish some general properties of the spectrum which follow from parity symmetry.
Parity properties of the spectrum. We introduce the parity operatorP, defined by its action on a function f as Consequently, for products of functions f , g, it holds thatP(f g) = (Pf )(Pg), and for the gradient we havẽ P∇ = −∇. Therefore the operatorP commutes with the Laplacian,P∇ 2 =∇ 2P , so that we can assume that the eigenfunctionsρ (0) n of the Laplacian diagonalize∇ 2 andP simultaneously, so that and furthermore that Thus,ρ (k) n has the same parity asρ (0) n if k is even, and the opposite parity asρ (0) n if k is odd. We now calculate the lowest order contributions to the N -dimensional FP spectrum; higher-order results for one-dimensional systems are given in App. C 1.
Order 1 contribution to the spectrum. For k = 1, Eq. (A15) yieldsλ (1) n = 0. Substituting this into Eq. (A8) for k = 1, we obtaiñ As can be verified by direct substitution, a solution to this inhomogeneous Helmholtz equation is given bỹ where the dot denotes the standard Euclidean inner product between the two N -dimensional vectorsẼ 1 (t),x. Becauseρ Order 2 contribution to the spectrum. For k = 2, Eq. (A12) becomes where we useλ (1) n = 0. Substituting Eq. (A18) into Eq. (A19), and performing the integrals, we obtain the second correction for the eigenvalue as where Substituting the definition ofẼ k,α , Eq. (A4), and using Eqs. (17), (18), it follows that which is the OM stochastic action in units of 1/τ D ≡ D/L 2 . To calculateρ (2) n , we insert Eqs. (A15), (A18), (A20) into the right-hand side of Eq. (A8) (with k = 2), resulting iñ This equation can be solved directly for the case where the force inside the tube is given as the gradient of an instantaneous potential, F = −∇U . According to Eq. (A5), in that case the 2-tensor (or vector-valued one form)Ẽ 2 is symmetric, i.e. we haveẼ j 2,i =Ẽ i 2,j , and direct substitution shows that Eq. (A23) is solved bỹ n is radially symmetric, as is the case for n = 1. In that caseρ n depends onx only via ||x||, and consequently there is a scalar function f such that∇ρ Using this, it is readily verified that Eq. (A24) is a solution to Eq. (A23).
Order 3 contribution to the eigenvalue. According to Eq. (A15), we haveλ

Parity properties of the reflecting-boundary steady state
In the present section we discuss the perturbative calculation and parity properties of both the steady statẽ ρ ss and its multiplicative inverseρ −1 ss ≡ 1/ρ ss . Perturbative calculation ofρ ss . According to Eq. (21), the instantaneous steady stateρ ss is the solution of the boundary value problem with boundary condition n ·j ss | ∂B = 0, wherej ss ≡ −∇ρ ss + F appρss , wheren is the outwardpointing unit normal vector onB, and whereF app = F −φ, as defined in Eq. (22). If the forceF originates from a potential,F = −∇Ũ , then the (unnormalized) instantaneous steady state is a Boltzmann distribution, c.f. Eqs. (27), (A6). Using the Taylor expansion of the exponential function, an expansion in powers of forρ ss is then obtained from Eq. (A6).
We now discuss how to perturbatively calculateρ ss for the general case, in which the forceF need not have an instantaneous potential inside the tube. Substituting into Eq. (A25) the power series expansion Eq. (A3) ofF , we obtaiñ where the dot denotes the standard Euclidean inner product. Expanding the instantaneous steady state as power series in ,ρ substituting this expansion into Eq. (A27), and demanding that the resulting equation hold at each power k , we obtain a hierarchy of equations which at order k reads where we use the convention that for k = 0, the sum on the right-hand side is zero. Inserting the power series expansions Eq. (A3), (A28), into the boundary condition Eq. (A26), and demanding that the resulting equation be fulfilled at each power k , we obtain where k ≥ 0 and we use the convention that for k = 0, the sum on the right-hand side is zero. While at order 0 , the (unnormalized) solution to Eqs. (A29), (A30) is simply given byρ ss . Similar to the parity properties of the FP spectrum, via induction in k it can be shown thatPρ where the parity operatorP is defined in Eq. (A13). Perturbative calculation and parity properties ofρ −1 ss . By definition of the inverse, it holds that Substituting the power series expansion Eq. (A28) ofρ ss and the expansioñ into Eq. (A34), and demanding that the equation hold at any order of , we obtain a recursive system of equations for the expansion ofρ −1 ss given by where in Eq. (A36) we use thatρ (0) ss = 1. Using Eqs. (A31), (A32), it follows from Eq. (A37) that Note that in Eq. (A39) only the symmetric part of the 2-tensorẼ 2 contributes. According to Eq. (A36), the parity of ρ −1 ss (0) is 1.
Using induction, and applying the parity operator to Eq. (A37), it furthermore follows that for all k.

Properties of power series expansions derived from parity
We now derive properties of some power series expansions used in the main text.
Integral over FP eigenfunction. We consider which we expand in a power series In particular, since the lowest eigenfunction of the Laplace operator (inside a unit ball and with absorbing boundary conditions) is even, we havẽ Inner product of FP eigenfunctions. We consider c.f. Eq. (26). The power series expansion of this inner product is given by where with the power series expansions Eq. (29), (A35). If the integrand has odd parity, the integral on the right-hand side vanishes; applying the parity operator to the integrand we calculatẽ where we use Eqs. (A16), (A40), and i + j + k = l. Thus, we have ρ n ,ρ m (l) = 0 if l odd and p n p m = 1, l even and p n p m = −1.
Inner product of FP eigenfunctions including time derivative. Since taking a time derivative does not change spatial parity we, similar to the previous case, have for the power series expansion .
(A54) Sinceρ 1 has even parity, p 1 = 1, we have according to Eqs. (A43), (A52), that the expansions in powers of of both ρ m ,ρ 1 ,Ĩ m , only have nonzero terms at even powers of if p m = 1, and at odd powers of if p m = −1; therefore, regardless of p m the product ρ m ,ρ 1 Ĩ m only contains even powers of , i.e.
The lowest order term of the expansion is therefore

Appendix B: Perturbative solution of the FPE
Perturbative solution of the FPE in terms of the instantaneous spectrum. We now derive an approximate solution of Eq. (28), which incorporations the coupling between eigenmodes to order 4 (and in the case of a one-dimensional systen, N = 1, to order 5 ). The following derivation is similar to what in quantum mechanics is called time-dependent perturbation theory [42,49]. To render the following calculation easier to read, we rewrite Eq. (28) as where we introducẽ Λ n (t) ≡λ n (t) + 2 ρ n ,ρ n ρ n ,ρ n t , with δ n,m the Kronecker delta. From the spectrum calculated in App. A 1, it follows that for all n, m we have thatΛ n = O( 0 ),C nm = O( 0 ), so that the explicit powers of on the right-hand side of Eq. (B1) represent the leading order scaling of each of the terms. According to Eq. (B1), the dynamics of each mode is for small dominated by the adiabatic exponential decay described by the instantaneous decay rateΛ n (t)/ 2 . We now derive an approximate solution to Eq. (B1) which incorporates the leading order effects of the mode coupling described by the coupling matrixC nm .
To separate the adiabatic mode decay and the interaction between modes, we introduceb n viã a n (t) =b n (t)P ad n (t,t i ), (B4) with the adiabatic propagatorP ad n for mode n given bỹ where in the sum bounds we make explicit the fact that C nn = 0, and where we definẽ with ∆Λ mn (t) ≡Λ m (t) −Λ n (t).
Integrating Eq. (B6), we obtaiñ To eliminateb m (t ) in the second term on the right-hand side of Eq. (B9), we reinsert the expression Eq. (B9), similar to the construction of the Dyson series in quantum mechanics [49]. Iterating this procedure, by reinserting Eq. (B9) once more in the result, we obtaiñ where we define The first term in Eq. (B10) represents the adiabatic decay of the n-th eigenmode, for which according to Eq. (B4)b n is constant; the remaining terms describe the mode coupling. Intuitively, one might interpret M (1) nm as describing the direct interaction between two modes n, m, M nmk , and their higher-order equivalents, also depend on ; the naive scaling argument that one substitution of Eq. (B9) corresponds to increasing the order in by one therefore breaks down. As we discuss now, for the steady-state solution ofb n to order 4 (and 5 for a one-dimensional system, N = 1), it is in fact sufficient to discuss the mode-coupling effects mediated by M (1) nm , M (2) nmk . Direct interaction between two modes. To lowest order, the coupling between two modes n, m, is given by M Note that scenario 3 can only occur for dimensions N ≥ 2; in one dimension, N = 1, the absorbing-boundary spectrum of the Laplace operator inside a finite interval is non-degenerate.
Direct interactions between modes, scenario 1. Since the eigenvalues of the Laplace operator are ordered, for small we have ∆Λ mn (t) ≡Λ m (t) −Λ n (t) > 0, and since n / ∈ eig(m) it holds that ∆Λ mn (t) = O( 0 ). Therefore, for small the exponential in the definition ofP ad mn (t,t i ), Eq. (B7), decays on a timescaleτ mn defined by so that for small we havẽ SinceC nm = O( 0 ), the integral in Eq. (B11) is in scenario 1 thus dominated byt ≈t i . Assuming thatC nm does not vary too rapidly on the time scaleτ mn , we Taylor expand aroundt =t i , Furthermore assuming that ∆Λ mn does not vary too much on the decay time scaleτ mn , we approximatẽ Physically speaking, with approximations Eqs. (B15), (B17), we assume that the apparent force (and hence the FP spectrum) inside the tube varies slowly as compared to the relaxation times of the individual modes. Inserting approximations Eqs. (B15), (B17), into Eq. (B11), the integral is evaluated to Fort −t i τ mn , the result Eq. (B18) simplifies to Direct interactions between modes, scenario 2. We first note thatP ad mn (t ,t i ) =P ad mn (t,t i )P ad nm (t,t ). Substituting this into Eq. (B11), we obtain Similar to the discussion of scenario 1 the termP ad nm (t,t ) decays exponentially ast is decreased fromt, with a characteristic decay time scaleτ mn defined by which for small is given bỹ Thus, in scenario 2 the integral in Eq. (B11) is dominated byt ≈t. Assuming thatC nm does not vary too rapidly on the time scaleτ mn , we Taylor expand aroundt, where a dot here denotes a derivative w.r.tt. Furthermore assuming that ∆Λ mn does not vary too much on the time scaleτ mn , we approximatẽ Inserting approximations Eqs. (B23), (B25), into Eq. (B20), in scenario 2 the integral is evaluated to After an initial transient decay time, i.e. fort −t i τ nm withτ mn defined in Eq. (B22), the result Eq. (B26) simplifies to Direct interactions between modes, scenario 3. According to the perturbative FP spectrum calculated in Appendix A 1, for n ∈ eig(m) we have c.f. Eqs. (29), (A15), (A20), (B2), and note that ρ n ,ρ n = O( ). It follows that so that to leading order in Eq. (B11) becomes Direct interactions between modes: summary. According to Eqs. (B10), (B19), (B27), (B30), the leading-order contribution to the coupling between two modes n, m scales with as n < m and n / ∈ eig(m), 3P ad mn (t,t i ), n > m and n / ∈ eig(m), , n ∈ eig(m), whereP ad mn (t,t i ) grows exponentially with an exponent that scales as 1/ 2 . These scalings are valid after an initial transient time of the order of where we assume that the order of magnitude ofτ mn is independent of the time at which ∆Λ mn (t) is evaluated in Eq. (B32), so that we omit the time-dependence in ∆Λ mn . From the leading-order scalings Eq. (B31) we can infer the largest term in the sum which appears in Eq. (B10). Assuming that all theb m (t i ) are of comparable order of magnitude, which term dominates in Eq. (B33) depends on n.
• For n = 1 only scenario 1 is relevant (note that the lowest eigenvalue of the Laplace operator is nondegenerate [50]); the leading-order correction tõ b n (t i ) is thus at order 3 , and all modes m > 1 contribute to this correction, meaning that all terms in Eq. (B33) are relevant.
• For n > 1 the dominant correction is given by scenario 2, m = 1; this is because the corresponding P ad mn (t,t i ) grows fastest, as which follows for small from the fact that we perturb around the ordered eigenvalues of the Laplace operator. In particular, note that even though in scenario 3, where n ∈ eig(m), the coupling between modes has a lower-order prefactor (order ), the fact that in scenario 2 the factorP ad mn (t,t i ) grows exponentially (with an exponent that scales as 1/ 2 ) makes this the dominant contribution. This means that for n > 1 the sum Eq. (B33) is dominated by the term m = 1, i.e.
which is expected to hold after a timeτ 1n as defined in Eq. (B32).
Intuitively, these results tell us that i) the dominant correction to the adiabatic decay of the lowest mode n = 1 is due to its interaction with the modes m > 1 during the initial relaxation of the initial conditions (note that C nm , ∆Λ mn in Eq. (B19) are evaluated att i ), and ii) the dominant correction to the adiabatic decay of any mode n > 1 is due to instantaneous excitation by the lowest mode m = 1 (note thatC nm , ∆Λ mn in Eq. (B27) are evaluated att), which after an initial relaxation is expected to be the dominant mode.
Higher order coupling between modes. From Eq. (B31) and the subsequent discussion we see that after the initial relaxation of the system, the interaction between two modes n = m leads to corrections of order 3 if n / ∈ eig(m) (with an exponentially growing factor if n > m), and of order if n ∈ eig(m). To calculate the leading order corrections tob n up to order 4 in the steady-state limit, we therefore only need to take into account two scenarios for the three-mode coupling described by Eq. (B12), namely 1. k > 1, m ∈ eig(k), n = 1. In this scenario, a mode k > 1 couples to a mode m = k from the same Laplace eigenspace (→ interaction of order ), which then couples to the lowest mode n = 1 (→ interaction of order 3 ).
2. k = 1, m > 1, n ∈ eig(m). In this scenario, the lowest mode k = 1 excites a mode m > 1 (→ interaction of order 3 , with an exponentially growing prefactor), which then couples to a mode n = m from the same Laplace eigenspace (→ interaction of order ).
Note that these cases are only relevant for dimensions N ≥ 2; since for N = 1 the spectrum of the Laplace operator is not degenerate, higher-order couplings between modes always scale as 6 for a one-dimensional system. Higher order coupling between modes, scenario 1. Since 1 = n < m the factorP ad mn (t ,t i ) decays exponentially as a function oft , so that thet -integral in Eq. (B12) is dominated byt ≈t i . We therefore approx-imateP ad and furthermore Taylor expand where at the last equality sign we use that for m ∈ eig(k) which vanishes (to order 2 ) ast −t i τ m1 (recall that in the current scenario n = 1), withτ m1 defined in Eq. (B32).
Higher order coupling between modes, scenario 2. Exchanging the two integrals that are present in Eq. (B12) after substituting Eq. (B11), we obtain where we useP ad km (t ,t i ) =P ad km (t,t i )P ad mk (t,t ). Since 1 = k < m, the factorP ad mk (t,t ) decays exponentially ast is decreased fromt, so that thet -integral is dominated byt ≈t. Similar to scenario 1, we therefore approximatẽ where at the last equality sign we use that for m ∈ eig(n) we haveP ad mn (t ,t i ) = 1 + O( ), c.f. Eq. (B29). Substituting Eqs. (B41-B43) into Eq. (B40), thet -integral is then evaluated using integration by parts to yield Comparing this result to Eq. (B27), we see that after an initial transient time, i.e. fort−t i τ 21 = max m>1 {τ m1 } (recall that in the current scenario k = 1), withτ m1 defined in Eq. (B32), the contribution to the amplitudeb n from Eq. (B44) is exponentially smaller than the contribution from Eq. (B27); thus the contribution from Eq. (B44) can be neglected ast −t i τ 21 .
Final result for approximate FP solution. Substituting the results Eqs. (B19), (B27), (B30), (B39), (B44), into Eq. (B10), we find that theb n are to exponentially leading order given bỹ where n > 1 in Eq. (B46), and k = 6 for a onedimensional system, N = 1, and k = 5 for N ≥ 2. These approximate expressions are valid after an initial transient decay timẽ withτ m1 defined in Eq. (B32). Substituting these results forb n into Eq. (B4), the coefficientsã n of the eigenfunction-expansion of the solution of the FPE are finally given bỹ where n > 1, and for a one-dimensional system, N = 1, we have k = 6, while for N ≥ 2 we have k = 5; to obtain Eq. (B48-B50) we furthermore use thatã n (t i ) =b n (t i ), and at Eq. (B50) we use Eq. (B48). The expressions Eqs. (B48), (B49) hold after the initial transient decay timeτ rel defined in Eq. (B47), and neglect both terms of the order O( k ), as well as terms exponentially small as compared to the leading-order contributions.

Appendix C: Explicit results for one-dimensional systems
In the present section, we consider our theory for a one-dimensional system, N = 1.

Spectrum of the FPE
We now derive explicit expressions for the perturbative spectrum of the FPE, following the strategy from App. A 1. In particular we show that at order k, the perturbative contribution to the eigenfunction is given byρ n,s (x,t) · sin n π 2 (x + 1) (C1) +Q (k) n,c (x,t) · cos n π 2 (x + 1) , n,c (x,t) are polynomials inx of order ≤ k.
For N = 1, the Taylor expansion of the force, Eqs. (A1), becomes withẼ where (x, t) and (x,t) are related via Eq. (14). With this, the equation for the n-th eigenvalue/eigenfunction pair at order k , Eq. (A8), becomes where we use the convention that for k = 0 the sums on the right-hand side are zero, and eachρ (k) n fulfills the boundary conditions c.f. Eq. (A9). The normalization condition at order k is given by Eqs. (A10), (A10), where we note that For N = 1 the equation for theλ (k) n , Eq. (A12), becomesλ We now show how Eqs. (C4), (C5), (C7), can be solved recursively with increasing k, and that at order k the solution forρ (k) n is of the form Eq. (C1).

Effect of initial distribution inside tube on exit rate
As described in App. C 5, in the numerical examples in the main text we eliminate transient relaxation effects at the initial timet i by using the instantaneous FP steady state att i as initial distributionP i .
To illustrate the effect of the initial distributionP i on the finite-radius exit rateα ϕ we here numerically consider the initial conditionP i (x) = δ(x), which corresponds to a particle starting out at timet i at the center of the tube.
In Fig. 5 we compare numerical exit rates resulting from this delta-peak initial condition (dashed colored lines) to numerical exit rate corresponding to the instantaneous steady-state initial condition (solid colored lines). As in Fig. 3 (d), (e), (f), we shift and rescale From all rates the free-diffusion exit rate is subtracted and the result is divided by the free-diffusion exit rate, as defined in Eq. (57). All data shown is obtained from numerical simulations of the FPE, Eq. (19), from which the exit rate is calculated using using Eq. (41). Colored solid lines are replots of the corresponding curves in Fig. 3 (d), (e), (f), and denote exit rates obtained using the instantaneous steady-state as initial condition for the simulations, as explained in App. C 5. Colored dashed lines show exit rates obtained using a delta peak at the tube center as initial condition for the simulations. Vertical dashed lines denote the initial relaxation timeτ rel given in Eq. (C38).
exit rates according to Eq. (57). Using the perturbative results from App. C 1, the initial relaxation timeτ rel , defined in Eq. (B47), is given as power series in as This perturbative expression forτ rel is plotted in Fig. 5 as vertical dashed lines. Figure 5 (a) shows data for tube radius = 0.1. While the data corresponding to the steady-state initial condition (colored solid line) is practically constant on the time scale depicted, the exit rate corresponding to the delta-peak initial condition (colored dashed line) shows relaxation behavior; the curve starts at 2 ∆α ϕ /α free (0) = −1, which according to Eq. (57) corresponds to a vanishing exit rateα ϕ (0) = 0, consistent with the intuition that a particle starting in the center of a finite-radius ball needs a finite time to diffusive out of the ball. This exit rate then relaxes to the steady-state exit rate on a time scale well-approximated by Eq. (C38); for times larger thant ≈ 2·τ rel all knowledge of the initial condition has decayed and the two exit rates are indistinguishable. The data shown for the larger tube radii = 0.5, 0.7 in Fig. 5 (b), (c) shows the exact same behavior. As expected from the leading-order scalingτ rel ∼ 2 , the relaxation time increases with tube radius .

Numerical algorithm for one-dimensional FPE
To simulate the dimensionless FPE, Eq. (19), (21), we discretize space by introducing the grid and discretize time using a timestep ∆t, where i ∈ {1, ..., N },F j app,i ≡F app (x i ,t j ), and in accordance with the absorbing boundary conditions we definẽ P j ,0 =P j ,N +1 = 0 for all j. To obtain an explicit formula for the distribution at time (j + 1) · ∆t in terms of the distribution at time j · ∆t, Eq. (C41) is then solved for P j+1 ,i (forward Euler integration scheme). All numerical results in this work are obtained using N = 100, ∆t = 10 −7 .
To eliminate boundary effects due to the transient decay of the initial condition, we pre-equilibrate the system for every . Starting from a distributionP i (x) = sin(π(x + 1)/2), we simulate the FPE, Eq. (C41), for a short time of the order of τ rel , while holding the parameters for position and velocity of the pathφ constant at the initial valuesφ(t i ),φ(t i ). At the end of this preequilibration, the system is in the instantaneous steady state decay corresponding toφ(t i ),φ(t i ). This instantaneous steady state is then normalized and used as initial condition for the simulation (in whichφ,φ then vary with time). A brief discussion on the dependence of the exit rate on the initial condition is given in App. C 4.