Abstract
We study the trajectories of a solution \(X_t\) to an Itô stochastic differential equation in \({\mathbb { R}}^d\), as the process passes between two disjoint open sets, \(A\) and \(B\). These segments of the trajectory are called transition paths or reactive trajectories, and they are of interest in the study of chemical reactions and thermally activated processes. In that context, the sets \(A\) and \(B\) represent reactant and product states. Our main results describe the probability law of these transition paths in terms of a transition path process \(Y_t\), which is a strong solution to an auxiliary SDE having a singular drift term. We also show that statistics of the transition path process may be recovered by empirical sampling of the original process \(X_t\). As an application of these ideas, we prove various representation formulas for statistics of the transition paths. We also identify the density and current of transition paths. Our results fit into the framework of the transition path theory by Weinan and Vanden-Eijnden.
Similar content being viewed by others
1 Introduction
In this article we study solutions \(X_t \in {\mathbb { R}}^d\) of the Itô stochastic differential equation
where \((W_t, {\mathcal {F}}^W_t)\) is a standard Brownian motion in \({\mathbb { R}}^d\), defined on a probability space \((\Omega , {\mathcal {F}}, {\mathbb { P}})\). This diffusion process in \({\mathbb { R}}^d\) has generator
where \(a:= \sigma \sigma ^{\mathrm {T}}\) is a symmetric matrix. We suppose that \(\sigma (x)\) is smooth and that \(a(x)\) is uniformly positive definite and bounded:
holds for some \(\Lambda > \lambda > 0\). Although the vector field \(b\) may not be bounded, we suppose that \(b\) is smooth and satisfies conditions that guarantee the ergodicity of the Markov process \(X_t\) and the existence of a unique invariant probability distribution \(\rho (x) > 0\) satisfying the adjoint equation
We also assume that for some \(\alpha > 1\),
for all \(R > 0\), where \(\tau _1\) is the first hitting time of \(X_t\) to the unit ball \(\{ z \in {\mathbb { R}}^d \;|\; |z| \le 1\}\). For example, it follows from Theorems 2 and 3 of [34] that these assumptions will hold if
for some \(r > 1 + (d/2)\).
Suppose that \(A, B \subset {\mathbb { R}}^d\) are two bounded open sets with smooth boundary and such that \(\bar{A}\) and \(\bar{B}\) are disjoint. Because the process is ergodic, \(X_t\) will visit both \(A\) and \(B\) infinitely often. Inspired by the transition path theory developed by Weinan and Vanden-Eijnden [15, 24] (see also the review article [16]), our main interest is in those segments of the trajectory \(t \mapsto X_t\) which pass from \(A\) to \(B\). These transition paths and are defined precisely as follows. First, for \(k \ge 0\), define the hitting times \(\tau _{A,k}^+\) and \(\tau _{B,k}^+\) inductively by
and for \( k \ge 0\),
We will call these the entrance times. Then define the exit times
These times are all finite with probability one, and \(\tau _{A,k}^+ \le \tau _{A,k}^- < \tau _{B,k}^+ \le \tau _{B,k}^- < \tau _{A,k+1}^+\) for all \(k \ge 0\) (see Fig. 1). If \(t \in [\tau _{A,k}^-, \tau _{B,k}^+]\) for some \(k\), we say that the path \(X_t\) is \(A \rightarrow B\) reactive. Let \(\Theta = (\bar{A \cup B})^{c}\), and hence \(\partial \, \Theta = \partial A \cup \partial B\). For \(k \in {\mathbb {N}}\), the continuous process \(Y^k:[0,\infty ) \rightarrow \bar{\Theta }\) defined by
is the \(k{\text {th}}\) \(A \rightarrow B\) reactive trajectory or transition path. Observe that \(Y_0^k = X_{\tau _{A,k}^-} \in \partial A\), that \(Y^k_t = X_{\tau _{B,k}^+} \in \partial B\) for all \(t \ge \tau _{B,k}^+ - \tau _{A,k}^-\), and that \(Y^k_t \in \Theta \) for all \(t \in (0,\tau _{B,k}^+ - \tau _{A,k}^-)\). Unlike the entrance times, the exit times \(\tau _{A, k}^-\) and \(\tau _{B, k}^-\) are not stopping times with respect to the natural filtration. So, one cannot apply the strong Markov property to \(X_t\) at times \(\tau _{A,k}^-\) and \(\tau _{B,k}^-\). Indeed, the law of the process \(Y_t^k\) is very different from that of the process \(X_t\) starting at a point in \(\partial A\).
Illustration of a trajectory with entrance and exit times. The transition path from \(A\) to \(B\) is marked in red (color figure online)
Our main results describe the probability law of these transition paths in terms of a transition path process, which is a strong solution to an auxiliary stochastic differential equation. In particular, empirical samples of the reactive portions of \(X_t\) may be regarded as sampling from the transition path process. The motivation comes from the study of chemical reactions and thermally activated processes where understanding these reactive trajectories are crucial [6, 12]. In these applications, the domains \(A\) and \(B\) are usually chosen as regions in configurational space corresponding to reactant and product states. Mathematically, our results fit into the framework of the transition path theory [15, 16, 24].
Having identified the transition path process, we can compute statistics of the transition paths by sampling directly from the transition path SDE, rather than using acceptance/rejection methods or very long-time integration on the original SDE. Our theoretical results might be used to analyze numerical methods of sampling reactive trajectories.
We will now describe our main results and their relation to other works. Proofs are deferred to later sections.
1.1 The transition path process
Our definition of the transition path process is motivated by the Doob \(h\)-transform as follows. Let \(\tau _A\) and \(\tau _B\) denote the first hitting time of \(X_t\) to the sets \(\bar{A}\) and \(\bar{B}\), respectively:
Let \(q(x)\ge 0\) be the forward committor function:
which satisfies \(L q(x) = 0\) for \(x \in \Theta = (\bar{A \cup B})^{c}\) and
By the maximum principle, \(q(x) > 0\) for all \(x \in \Theta \). By the Hopf lemma we also have
where \(\widehat{n}(x)\) will denote the unit normal exterior to \(\Theta \) (pointing into \(A\) and \(B\)). For \(x \in \Theta \), consider the stopped process \(X_{t \wedge \tau _A \wedge \tau _B}\) with \(X_0 = x\), and let \({\mathcal {P}}_x\) denote the corresponding measure on \({\mathcal {X}} = C([0,\infty ), \bar{\Theta })\):
where \({\mathcal {B}}\) is the Borel \(\sigma \)-algebra on \({\mathcal {X}}\). If \(\Lambda _{AB}\) denotes the event that \(\tau _A > \tau _B\), the measure \({\mathcal {Q}}_x^q\) on \(({\mathcal {X}},{\mathcal {B}})\) defined by
is absolutely continuous with respect to \({\mathcal {P}}_x\), if \(x \in \Theta \). By the Doob \(h\)-transform (see e.g. [28, Theorem 7.2.2]), we know that \({\mathcal {Q}}_x^q\) defines a diffusion process \(Y_t\) on \(C([0,\infty ), \bar{\Theta })\) with generator:
So, the effect of conditioning on the event \(\tau _B < \tau _A\) is to introduce an additional drift term. For \(x \in \Theta \), the transition probability for \(Y_t\) is
where \(p(t,x,dy)\) is the transition probability for \(X_t\) killed at \(\partial B\) [28, Theorem 4.1.1].
This observation suggests that the \(A\rightarrow B\) reactive trajectories should have the same law as a solution to the SDE
originating at a point \(Y_0 = y_0 \in \partial A\) and terminating at a point in \(\partial B\). While the SDE (1.12) admits strong solutions for \(y_0 \in \Theta \) since \(q(x) > 0\) in \(\Theta \), the drift term becomes singular at the boundary of \(A\), where \(q\) vanishes. Our first result is the following theorem which shows that there is still a unique strong solution to this SDE even for initial condition lying in \(\partial A\). For convenience, let us define the vector field
Theorem 1.1
Let \((\widehat{W},{\mathcal {F}}^{\widehat{W}}_t)\) be a standard Brownian motion in \({\mathbb { R}}^d\), defined on a probability space \((\widehat{\Omega }, \widehat{\mathcal {F}},{\mathbb {Q}})\). Let \(\xi :\widehat{\Omega } \rightarrow \bar{\Theta }\) be a random variable defined on the same probability space and independent of \(\widehat{W}\). There is a unique, continuous process \(Y_t:[0,\infty ) \rightarrow \bar{\Theta }\) which is adapted to the augmented filtration \(\widehat{\mathcal {F}}_t\) and satisfying the following, \({\mathbb {Q}}\)-almost surely:
where
Moreover, \(Y_t \not \in \bar{A}\) for all \(t > 0\).
The augmented filtration is defined in the usual way, \(\widehat{\mathcal {F}}_t\) being the \(\sigma \)-algebra generated by \({\mathcal {F}}_t^{\widehat{W}}\), \(Y_0\), and the appropriate collection of null sets so that \(\widehat{\mathcal {F}}_t\) is both left- and right- continuous. We will use \(\widehat{{\mathbb {E}}}\) to denote expectation with respect to the probability measure \({\mathbb {Q}}\).
Observe that if \(d = 1\), \(\sigma = 1/\sqrt{2}\) is constant, and \(b \equiv 0\), then \(q(x)\) is a linear function, and (1.12) corresponds to a Bessel process of dimension 3. For example, if \(A = (-\infty ,0)\), \(B = (1,\infty )\), we have
and the function \(Z_t = (Y_t)^2\) satisfies the degenerate diffusion equation
In this simple case, existence and uniqueness of a strong solution starting at \(Y_0 = 0\) can be shown using arguments involving Brownian local time (see [23, 30]). However, those arguments are not applicable to the more general setting we consider here. The work most closely related to Theorem 1.1 in a higher dimensional setting may be that of DeBlaissie [14] who proved pathwise uniqueness for certain SDEs having diffusion coefficients that degenerate like \(\sqrt{d(Z_t)}\) where \(d(z)\) is the distance to the domain boundary (as in (1.15)). In an earlier work, Athreya et al. [1] proved uniqueness for the martingale problem associated with a similarly degenerate diffusion in a positive orthant in \({\mathbb { R}}^d\). Nevertheless, those analyses do not apply to the case (1.12) considered here.
The next theorem shows that the law of the reactive trajectories is that of the process \(Y_t\) with appropriate initial condition. For this reason, we will call the process \(Y_t\) the transition path process.
Theorem 1.2
Let \(X_t\) satisfy the SDE (1.1). Let \(Y^k\) denote the \(k{\text {th}}\) \(A \rightarrow B\) reactive trajectory defined by (1.5). Let \(Y\) be defined as in Theorem 1.1. Then for any bounded and continuous functional \(F:C([0,\infty )) \rightarrow {\mathbb { R}}\), we have
The processes \(X_t\) and \(Y^k_t\) may be defined on a probability space that is different from the one on which \(Y_t\) is defined. The notation \(Y_0 \sim X_{\tau _{A,k}^-}\) used in Theorem 1.2 means that \(Y_0\) has the same law as \(X_{\tau _{A,k}^-}\), meaning \({\mathbb {Q}}(Y_0 \in U) = {\mathbb { P}}(X_{\tau _{A,k}^-} \in U)\) for any Borel set \(U \subset {\mathbb { R}}^d\).
1.2 Reactive exit and entrance distributions
The distribution of the random points \(X_{\tau _{A,k}^-}\) will depend in the initial condition \(X_0\). From the point of view of sampling the transition paths, however, there is a very natural distribution to consider for \(Y_0\), which is related to the “equilibrium measure” in the potential theory for diffusion processes [7, 8, 32]. To motivate this distribution formally, let \(h > 0\) and consider the regularized hitting times
where \(X_t\) satisfies (1.1). Then define
This is the probability that at some time \(s \in [0,h]\), the path \(X_t\) starting from \(x \in \partial A\) becomes a transition path, not returning to \(\bar{A}\) before hitting \(\bar{B}\). With this in mind, the quantity
may be interpreted as a rate at which transition paths exit \(A\), when the system is in equilibrium. Therefore, a natural choice for an initial distribution for \(Y_0 \in \partial A\) is:
By the Markov property, we have
where \(\rho (t,x,\cdot )\) is the density for \(X_t\), given \(X_0 = x\). Therefore, for any \(x \in \partial A\) we have
in the sense of distributions, although \(q\) is not \(C^2\) on \(\partial \Theta = \partial A \cup \partial B\). Hence \(\eta _{A,h}(x) \rightarrow \eta _A(x) = \rho (x)Lq(x)\) for \(x \in \partial A\). The distribution \(Lq\) is supported on \(\partial \Theta \). If \(\phi \) is a smooth test function supported on a set \(B_r(x)\), a small neighborhood of \(x \in \partial A\), then we have
where \(\widehat{n}(x)\) is the unit normal vector exterior to \(\Theta \), and \(\,\mathrm{d}\sigma _A\) is the surface measure on \(\partial A\). Since \(q = 0\) on \(\partial A\) and \(Lq = 0\) on \(\Theta \), this implies,
That is (after a similar calculation for points on \(\partial B\)),
in the sense of distributions. Restricting on \(\partial A\), we get
By switching the role of \(A\) and \(B\) in the above discussion, it is also natural to define a measure on \(\partial B\) as
Note that \(1 - q\) gives the forward committor function for the transition from \(B\) to \(A\) and that \(L q(x) = \eta _A(\mathrm{d}x) - \eta _B(\mathrm{d}x)\). Although the distributions \(\eta _A\) and \(\eta _B\) are positive (by (1.9)), they need not be probability distributions. Nevertheless, the mass of the two measures is the same.
Lemma 1.3
The measures \(\eta _A\) and \(\eta _B\) satisfy \(\eta _A(\partial A) = \eta _B(\partial B)\). That is,
This computation motivates us to define
We call these distributions the reactive exit distribution on \(\partial A\) and on \(\partial B\), respectively. The constant \(\nu \) is a normalizing constant so that \(\eta _A^-\) and \(\eta _B^-\) define probability measures on \(\partial A\) and \(\partial B\). By Lemma 1.3, the normalizing constant is the same for both measures. Our next result relates the reactive exit distribution on \(\partial A\) to the empirical reactive exit distribution on \(\partial A\), defined by
Proposition 1.4
Let \(\mu _{A,N}^-\) be the empirical reactive exit distribution on \(\partial A\) defined by (1.25). Then \(\mu _{A,N}^-\) converges weakly to \(\eta _A^-\) as \(N \rightarrow \infty \). That is, for any continuous and bounded \(f:\partial A \rightarrow {\mathbb { R}}\)
holds \({\mathbb { P}}\)-almost surely.
A similar statement holds for the reactive exit distribution on \(\partial B\) and the empirical distribution of the points \(X_{\tau _{B,k}^-}\). The reactive exit distribution \(\eta _A^-(\mathrm{d}x)\) is related to the equilibrium measure \(e_{A, B}(\mathrm{d}x)\) in the potential theory for diffusion processes [7, 8], [32, Section 2.3]. In fact, the committor function \(q\) is known as the equilibrium potential in those works, and the equilibrium measure \(e_{A, B}(\mathrm{d}x)\) is given by \(Lq\) restricted on \(\partial A\) (see equation (2.11) of [7]). Specifically, we have
The reactive exit distribution was also used in the milestoning algorithm as in [35]. To the best of our knowledge, Proposition 1.4 for the first time characterizes the equilibrium measure from a dynamic perspective. In the case that the drift \(b(x) = - \nabla V(x)\) is a gradient field and \(\sigma = \sqrt{\epsilon } I\) is a multiple of the identity matrix, the constant \(\nu \) is related to the capacity of the sets \(A\) and \(B\):
(See definition (2.13) of [7] for \(\text {cap}_{A}(B)\).) The results we present here do not require that \(b(x)\) is a gradient field; nevertheless, the constant \(\nu \) still admits the integral representation given below in Proposition 1.8.
We also identify the limit of the empirical reactive entrance distribution on \(\partial B\), defined as
To describe its limit as \(N \rightarrow \infty \), let us denote by \(\widetilde{L}\) the adjoint of \(L\) in \(L^2({\mathbb { R}}^d, \rho (x) \mathrm{d}x)\), given by
This corresponds to the generator of the time-reversed process \(t \mapsto X_{T - t}\) [19]. Note that \(\widetilde{L} = L\) if the SDE (1.1) is reversible, i.e. \(L\) is self-adjoint in \(L^2({\mathbb { R}}^d, \rho (x) \,\mathrm{d}x)\). In addition to the forward committor function \(q(x)\) (recall (1.7)), we also define the backward committor function \(\widetilde{q}(x)\) to be the unique solution of
with boundary condition
In terms of \(\widetilde{q}\), we define the reactive entrance distribution on \(\partial B\) as
and analogously the reactive entrance distribution on \(\partial A\)
Again, \(\nu \) is a normalizing constant so that these are probability measures; \(\nu \) is the same as the constant in (1.23). The following proposition justifies the definition of the reactive entrance distribution.
Proposition 1.5
Let \(\mu _{B,N}^+\) be the empirical reactive entrance distribution on \(\partial B\) defined by (1.27). Then \(\mu _{B,N}^+\) converges weakly to \(\eta _B^+\) as \(N \rightarrow \infty \). That is, for any continuous and bounded \(f:\partial B \rightarrow {\mathbb { R}}\)
holds \({\mathbb { P}}\)-almost surely.
A similar statement holds for the reactive entrance distribution on \(\partial A\) and the empirical distribution of the points \(X_{\tau _{A, k}^+}\).
Remark 1.6
If the SDE (1.1) is reversible, we have \(\widetilde{q} = 1 - q\), and hence \(\eta _A^+(\mathrm{d}x) = \eta _A^-(\mathrm{d}x)\) and \(\eta _B^+(\mathrm{d}x) = \eta _B^-(\mathrm{d}x)\).
In view of Proposition 1.4, \(\eta _A^-\) is a natural choice for the distribution of \(Y_0\). With this choice, the transition path process \(Y_t\) characterizes the empirical distribution of \(A \rightarrow B\) reactive trajectories, as the next theorem shows:
Theorem 1.7
Let \(X_t\) satisfy the SDE (1.1). Let \(Y^k\) denote the \(k{\text {th}}\) \(A \rightarrow B\) reactive trajectory defined by (1.5). Let \(Y\) be the unique process defined by Theorem 1.1 with initial distribution \(Y_0 \sim \eta _A^-(\mathrm{d}x)\) on \(\partial A\) defined by (1.23), and let \({\mathcal {Q}}_{\eta _A^-}\) denote the law of this process on \({\mathcal {X}} = C([0,\infty ))\). Then for any \(F \in L^1({\mathcal {X}},{\mathcal {B}},{\mathcal {Q}}_{\eta _A^-})\), the limit
holds \({\mathbb { P}}\)-almost surely.
In particular, the limit \( \widehat{{\mathbb {E}}}[F(Y)]\) is independent of \(X_0\). Using Theorem 1.7, several interesting statistics of the transition paths can be expressed in terms of the quantities we have defined. Actually, Proposition 1.4 is an immediate corollary of Theorem 1.7, by choosing \(F(Y^k) = f(Y^k_0)\), so we will not give a separate proof of Proposition 1.4.
1.3 Reaction rate
Let \(N_T\) be the number of \(A \rightarrow B\) reactive trajectories up to time \(T\):
The reaction rate \(\nu _R\) is defined by the limit
and it is the rate of the transition from \(A\) to \(B\). Also, the limits
and
are the expected reaction times from \(A \rightarrow B\) and \(B \rightarrow A\), respectively. The reaction rate from \(A \rightarrow B\) and \(B\rightarrow A\) are then given by \(k_{AB} = T_{AB}^{-1}\) and \(k_{BA} = T_{BA}^{-1}\). Another interesting quantity is the expected crossover time from \(A \rightarrow B\)
which is the typical duration of the \(A \rightarrow B\) reactive intervals. Observe that \(C_{AB} < T_{AB}\). Similarly, we define
The next result identifies these limits in terms of the committor functions and the reactive exit and entrance distributions.
Proposition 1.8
The limits (1.31), (1.32), (1.33), (1.34), and (1.35) hold \({\mathbb { P}}\)-almost surely, and
Here \(u_B(x) = {\mathbb {E}}[\tau ^X_{B} \;|\; X_0 = x ]\) is the mean first hitting time of \(X_t\) to \(\bar{B}\), and \(v_B(x) = \widehat{{\mathbb {E}}}[ \tau ^Y_B \;|\; Y_0 = x]\) is the mean first hitting time of \(Y_t\) to \(\bar{B}\). Similarly, if \(q\) is replaced by \((1 - q)\) in the definition of \(Y\), then \(v_{A}(x) = \widehat{{\mathbb {E}}}[ \tau ^Y_A \;|\; Y_0 = x]\). Recall that \(\nu \) is the normalizing factor for the reactive exit and entrance distributions.
The formulas for \(\nu _R\), \(T_{AB}\), and \(T_{BA}\) were obtained in [15]. We believe the formulas for \(C_{AB}\) and \(C_{BA}\) are new. We also note that the crossover time for the transition path process in one dimension was recently studied in [4, 10] by other methods.
1.4 Density of transition paths
We now consider the distribution \(\rho _R\) as defined in [15]:
where \(R\) is the random set of times at which \(X_t\) is reactive:
This distribution on \(\Theta \) can be viewed as the density of transition paths. By Proposition 1.8, and Theorem 1.7, we can describe \(\rho _R\) in terms of the transition density for \(Y_t\). Specifically, for any continuous and bounded function \(f:{\mathbb { R}}^d \rightarrow {\mathbb { R}}\), we have
Here \(Q_R(t,\eta _A^-,z)\) is the density of \(Y_t\), with \(Y_0 \sim \eta _A^-\), and killed at \(\partial B\)
and \(t_{B}\) is the first hitting time of \(Y_t\) to \(\bar{B}\). Hence, for \(z \in \Theta \),
Proposition 1.9
For all \(z \in \Theta \),
This formula for \(\rho _R\) was first derived in [15, 22].
1.5 Current of transition paths
The density \(Q_R(t,\eta _A^-,z)\) satisfies the adjoint equation
where \((L^q)^{*}\) is the adjoint of \(L^q\):
and \(K\) is defined by (1.13). Integrating from \(t = 0\) to \(t = \infty \) we see that \(\rho _R(z)\) satisfies
In divergence form, this equation is
where the vector field
is continuous over \(\bar{\Theta }\). The vector field \(J_R(z)\), identified in [15], may be regarded as the current of transition paths (see Remark 1.13). Observe that if the SDE (1.1) is reversible, we have \(\widetilde{q} = 1 - q\) and
and hence the current given by (1.41) simplifies to
This was observed already in [15]. The current was also discussed in potential theory in the context of reversible Markov chains, see e.g. [9].
On the boundary, the current (1.41) is related to the reactive exit and entrance distributions.
Proposition 1.10
We have
and hence,
As an immediate corollary, we have an additional formula for the reaction rate.
Corollary 1.11
Let \(S\) be a set with smooth boundary that contains \(A\) and separates \(A\) and \(B\), we have
where \(\widehat{n}\) is the unit normal vector exterior to \(S\).
The current \(J_R\) generates a (deterministic) flow in \(\bar{\Theta }\) stopped at \(\partial B\):
where \(t_B = t_B(z)\) is the time at which \(Z_t\) reaches \(\partial B\). As \(J_R\) is divergence free in \(\Theta \), \(J_R \cdot \widehat{n} < 0\) on \(\partial A\), and \(J_R \cdot \widehat{n} > 0\) on \(\partial B\), \(t_B(z)\) is finite for any \(z \in \bar{\Theta }\). The flow naturally defines a map \(\Phi _{J_R}: \partial A \rightarrow \partial B\): given any point \(z \in \partial A\), we define
Proposition 1.12
For any \(f \in C^1({\mathbb { R}}^d)\),
In particular,
where \(\Phi _{J_R, *}(\eta _A^-)\) is the pushforward of the measure \(\eta _A^-\) by the map \(\Phi _{J_R}\).
Hence, \(J_R\) characterizes “the flow of reactive trajectories” from \(A\) to \(B\).
Remark 1.13
Note that by Propositions 1.4 and 1.5, the left hand side of (1.45) is equal, \({\mathbb { P}}\)-almost surely, to the limit
If \(X_t\) was differentiable, we would have
Combining this with Proposition 1.12, we arrive at a formal characterization of \(J_R\)
This formal expression was used in [15] to define \(J_R\).
1.6 Related work
As we have mentioned, our work is closely related to the transition path theory developed by Weinan and Vanden-Eijnden [15, 16, 24], which is a framework for studying the transition paths. In particular, based on the committor function, formula for reaction rate, density and current of transition paths were obtained in [15]. Our main motivation is to understand the probability law of the transition paths. The main results Theorems 1.1, 1.2, and 1.7 identify an SDE which characterizes the law of the transition paths in \(C([0,\infty ))\). Therefore, as an application of these results, we are able to give rigorous proofs for the formula for reaction rate, density and current of transition paths in [15]. We note that in the discrete case, a generator analogous to (1.10) was also proposed very recently in [33] for Markov jumping processes.
Our results may be useful in the design of numerical path-sampling algorithms. Specifically, the results indicate that with knowledge of the committor function \(q(x)\) one can bias the sampling of \(X_t\) in order to directly sample the reactive trajectories, without an acceptance/rejection procedure. Of course, this assumes knowledge of the committor function, which is certainly non-trivial as it involves solving a high dimensional PDE; \(q(x)\) is explicit only in the simplest of cases (such as when \(d=1\)). We refer to [16, 27] and references therein for efforts in numerical approximations of committor functions. Nevertheless, our theoretical results might be used to analyze methods of sampling reactive trajectories. In particular, it would be important to know what sort of approximation of \(q\) could be used to efficiently sample the reactive trajectories. This issue is related to importance sampling algorithms for rare events (see e.g. [13, 36]). We plan to explore these issues more in future works.
The transition paths start at \(\partial A\) and terminate at \(\partial B\), and hence they can be viewed as paths of a bridge process between \(\bar{A}\) and \(\bar{B}\). In this perspective, our work is related to the conditional path sampling for SDEs studied in [20, 21, 29, 31]. In those works, stochastic partial differential equations were proposed to sample SDE paths with fixed end points. However, the paths considered were different from the transition paths as their time duration is fixed a priori. It would be interesting to explore SPDE-based sampling strategies for the transition path process identified in Theorem 1.1.
Let us also point out that in the work we present here we do not assume that the noise \(\sigma \) is small, as is the case in the asymptotic results of [7, 8, 10], which we have mentioned already, and also in some other works, such as the large deviation theory of Freidlin and Wentzell [17].
After this paper was submitted for publication, both Sznitman and one of the editors brought to our attention the relevant work of Meyer et al. [25]. If we define the non-decreasing processes
where \({\mathbb {Z}}^+\) is the set of non-negative integers, then the triple \((X_t,V_t^A,V_t^B)\) is a Markov process on \({\mathbb { R}}^d \times {\mathbb {Z}}^+ \times {\mathbb {Z}}^+\). Moreover, the exit times \(\tau _{A,k}^-\) defined above coincide with the random times \(L_k = \sup \{ t \ge 0 \;|\; (X_t,V_t^A,V_t^B) \in \bar{A} \times \{ k + 1\} \times \{k\} \}\). Although it is not a stopping time, \(L_k\) is a coterminal time, as defined in [25]. Theorem 5.1 of [25] applied to \((X_{t+ L_k},V_{t+L_k}^A,V_{t + L_k}^B)\) then implies that for \(t > 0\), \(Y_t^k\) is a strong Markov process with transition probability (1.11). In particular, this implies that for any \(t_0 > 0\) and any bounded and continuous functional \(F: C([t_0, \infty )) \rightarrow {\mathbb {R}}\), we have
This is similar to but weaker than the statement of Theorem 1.2, which also applies to \(t_0 = 0\). Moreover, the results of [25] do not identify the reactive exit distribution, which plays an important role in Theorem 1.7.
The rest of the paper is organized as follows. Theorems 1.1 and 1.2 are proved in Sect. 2. In Sect. 3 we prove Lemma 1.3, Proposition 1.5 and Theorem 1.7 related to the reactive entrance and exit distributions. As we have mentioned, Proposition 1.4 follows immediately from Theorem 1.7, so we do not give a separate proof of it. Propositions 1.8, 1.9, 1.10, Corollary 1.11, and Proposition 1.12 are proved in Sect. 4.
2 The transition path process
Proof of Theorem 1.1
Without loss of generality, we prove the theorem in the case that \(\xi \equiv y_0\) is a single point in \(\bar{\Theta }\). The interesting aspect of the theorem is that \(y_0\) is allowed to be on \(\partial \Theta \), since the drift term is singular at \(\partial \Theta \). If we assume that \(y_0 \in \Theta \), then existence of a unique strong solution up to the time \(\tau _A \wedge \tau _B\) follows from standard arguments, since \(K(y)\) is Lipschitz continuous in the interior of \(\Theta \). That is, if \(y_0 \in \Theta \), there is a unique, continuous \(\widehat{\mathcal {F}}_t\)-adapted process \(Y_t\) which satisfies
Moreover, if \(y_0 \in \Theta \), then we must have \(\tau _A > \tau _B > 0\) almost surely. This follows from an argument similar to the proof of [23, Proposition 3.3.22, p. 161]. Specifically, we consider the process \(z_t = 1/q(Y_t) \in {\mathbb { R}}\), which satisfies
where \(\tau = \tau _B \wedge \tau _\epsilon \) with \(\tau _\epsilon = \inf \{ t > 0 \mid q(Y_t) = \epsilon \}\). Since \(\tau < \infty \) with probability one, we have
Hence \({\mathbb {Q}}( \tau _\epsilon < \tau _B) \le q(\epsilon )(z_0-1)\). So, \({\mathbb {Q}}(\tau _A < \tau _B) \le \lim _{\epsilon \rightarrow 0} {\mathbb {Q}}( \tau _\epsilon < \tau _B)=0\).
Now suppose \(y_0 \in \partial A\). In consideration of the comments above, it suffices to prove the desired result with \(\tau _B\) replaced by \(\tau _r\), the first hitting time to \(\partial B_r(y_0) \cap \Theta \), where \(B_r(y_0)\) is a ball of radius \(r > 0\) centered at \(y_0\). Thus, we want to prove existence and pathwise uniqueness of a continuous \(\widehat{\mathcal {F}}_t\)-adapted process \(Y_t:[0,\infty ) \rightarrow \bar{\Theta }\) satisfying
where
It will be very useful to define a new coordinate system in the set \(B_r^+(y_0) = B_r(y_0) \cap \Theta \) and to consider the problem in these new coordinates. For \(r > 0\) small enough we can define a \(C^3\) map \((h^{(1)}(y),\dots ,h^{(d-1)}(y),q(y)): \overline{B_r^+(y_0)} \rightarrow {\mathbb { R}}^{d-1} \times [0,\infty )\), such that the scalar functions \(h^{(i)}(y): \overline{B_r^+(y_0)} \rightarrow {\mathbb { R}}\) satisfy
Furthermore, the map may be constructed so that it is invertible on its range and that the inverse is \(C^3\). The existence of such a map follows from the regularity of \(\partial A\), the regularity of \(q\), and the fact that \(\langle \widehat{n}, a \nabla q \rangle \ne 0\) on \(\partial A\) by (1.9).
For two initial points \(x_1, x_2 \in \Theta \), let \(Y^{x_1}_t\) and \(Y^{x_2}_t\) denote the unique solutions to (2.1) with \(Y^{x_1}_0 = x_1\) and \(Y^{x_2}_0 = x_2\) respectively. That is,
where \(\tau _B^x\) is the first hitting time of \(Y^x_t\) to \(\partial B\). Changing to the coordinate system defined by \((h^{(1)}(y),\dots ,h^{(d-1)}(y),q(y))\), we denote
Let \(\tau _r^{1}\) and \(\tau _r^{2}\) denote the first hitting times of \(Y^{x_1}_t\) and \(Y^{x_2}_t\) to the set \(\partial B_r(y_0) \cap \Theta \). The processes \((h_{1,t},q_{1,t})\) and \((h_{2,t},q_{2,t})\) are well-defined up to the times \(\tau ^1_r\) and \(\tau ^2_r\), respectively.
We can control the difference between \((h_{1,t},q_{1,t})\) and \((h_{2,t},q_{2,t})\):
Lemma 2.1
There is a constant \(C\) such that for all \(x_1, x_2 \in B_{r/2}(y_0) \cap \Theta \)
and
where \(\tau = \tau ^{1}_r \wedge \tau ^{2}_r\).
The proof of Lemma 2.1 will be postponed. One immediate corollary is the following.
Corollary 2.2
There is a constant \(C\) such that for all \(x_1, x_2 \in B_{r/2}(y_0) \cap \Theta \)
where \(\tau = \tau ^{1}_r \wedge \tau ^{2}_r\).
Proof
On the closed set \(\{ z \in {\mathbb { R}}^{d} \mid z = (h(y),q(y)), \ y \in \overline{B_r^+(y_0)} \}\), the map \(y \mapsto (h(y),q(y))\) is invertible with a continuously differentiable inverse. Hence there is a constant \(C\), depending only on the map \(y \mapsto (h(y),q(y))\) such that
By combining this bound with Chebychev’s inequality and Lemma 2.1 we obtain (2.5). \(\square \)
Now suppose \(y_0 \in \partial A\). Let \(\{x_n\}_{n=1}^\infty \subset \Theta \) be a given sequence such that \(x_n \rightarrow y_0\) as \(n \rightarrow \infty \). For each \(n\), define \(Y^{x_n}_t\) by (2.4), and let \(\tau ^n_r\) denote the first hitting time of \(Y^{x_n}_t\) to \(\partial B_r(y_0) \cap \Theta \). We may choose the points \(x_n\) so that \(|x_n - y_0| \le 25^{-n}\). Define \(\widehat{\tau }^n = \tau ^{n+1}_r \wedge \tau _r^{n}\). Applying Corollary 2.2, we conclude
Therefore, by the Borel–Cantelli lemma, the series
with probability one. Let us define
We will prove that \(\tau _r\) is positive:
Lemma 2.3
For all \(r > 0\) sufficiently small, \({\mathbb {Q}}(\tau _r > 0 ) = 1\).
In view of (2.6) and Lemma 2.3, we conclude that there must be a continuous process \(Y_t\) such that, with probability one,
uniformly on compact subsets of \([0,\tau _r)\), as \(n \rightarrow \infty \). Let us define
Lemma 2.3
For all \(r > 0\) sufficiently small, \({\mathbb {Q}}(\bar{\tau }_{r/2} \in (0,\tau _r)) = 1\), and \(\bar{\tau }_{r/2}\) is stopping time with respect to \(\widehat{\mathcal {F}}_t\).
We will postpone the proof of Lemmas 2.3 and 2.4. Since \(\bar{\tau }_{r/2} < \tau _r\), \(Y^{x_n}_t \rightarrow Y_t\) uniformly on \([0,\bar{\tau }_{r/2}]\). Let us now replace \(Y_t\) by the stopped process \(Y_{t \wedge \bar{\tau }_{r/2}}\). Since each \(Y^{x_n}_t\) is \(\widehat{\mathcal {F}}_t\)-adapted, so is the limit \(Y_t\). We claim that \(Y_t\) satisfies
Since \(Y^{x_n}_t \rightarrow Y_t\) uniformly on \([0,\bar{\tau }_{r/2}]\), we have \((q(Y^{x_n}_t), h(Y^{x_n}_t)) \rightarrow (q(Y_t),h(Y_t))\) uniformly on \([0,\bar{\tau }_{r/2}]\), and \((q_t,h_t) = (q(Y_t),h(Y_t))\) satisfies
and
for all \(t \in [0,\bar{\tau }_{r/2}]\), where \((q_t^{x_n},h_t^{x_n}) = (q(Y^{x_n}_t), h(Y^{x_n}_t))\). (Recall \(q_0 = 0\).) Since \(q_s^{x_n} > 0\), the last limit can be bounded below using Fatou’s lemma:
Recall that \(|g(q_s,h_2)|^2 \ge C_r > 0\). In particular, with probability one, the random set \(H = \{ s \in [0,\bar{\tau }_{r/2}] \mid q_s = 0 \}\) must have zero Lebesgue measure; if that were not the case, then we would have
for all \(t\) in a set of positive Lebesgue measure, an event which happens with zero probability. Therefore, by Fubini’s theorem,
which implies that \({\mathbb {Q}}( s < \bar{\tau }_{r/2}, \; q_s = 0 ) = 0\) for almost every \(s \ge 0\). Since \(\bar{\tau }_{r/2} > 0\) almost surely, this implies that we may choose a deterministic sequence of times \(t_n \in (0, 1/n]\) such that, almost surely, \(q_{t_n} > 0\) for \(n\) sufficiently large. By then applying the same argument as when \(y_0 \in \Theta \), we conclude that \(q_t > 0\) for all \(t > t_n\). Hence, \(q_t > 0\) for all \(t > 0\) must hold with probability one.
Since \(q_t\) is continuous, we now know that for any \(\epsilon > 0\),
holds with probability one. In particular,
so that
almost surely. Since \(q_t\) is continuous at \(t = 0\), we also know that
almost surely. Returning to (2.11) we now conclude that
holds with probability one. Equation (2.9) for \(Y_t\) now follows from (2.10) and (2.13) by changing coordinates.
Except for the proofs of Lemmas 2.1, 2.3, and 2.4, we have now established existence of a strong solution \(Y_t\) to (2.2) (with \(r\) replaced by \(r/2\)). The uniqueness of the solution follows by the same arguments. Suppose that \(Y^{1}_t\) and \(Y^{2}_t\) both solve (2.2) with the same Brownian motion and the same initial point \(Y^1_0 = Y^2_0 = y_0\). Then Corollary 2.2 implies that, \({\mathbb {Q}}\) almost surely, \(Y^{1}_t = Y^{2}_t\) for all \(t \in [0,\tau ^1_r \wedge \tau ^2_r]\) where \(\tau ^1_r\) and \(\tau ^2_r\) are the corresponding hitting times to \(\partial B_r(y_0) \cap \Theta \). In particular, \(\tau ^1_r = \tau ^2_r\). This proves pathwise uniqueness. \(\square \)
We now prove Lemmas 2.1, 2.3 and 2.4 to complete the proof of Theorem 1.1.
Proof of Lemma 2.1
By Itô’s formula the process \((h_1,q_1) = (h_{1,t},q_{1,t})\) satisfies
for \(0 \le t \le \tau ^{1}_r\), where the functions \(g = \sqrt{2} (\nabla q)^{\mathrm {T}} \sigma \in {\mathbb { R}}^{d}\), \(f = L h \in {\mathbb { R}}^{d-1}\), and \(m = \sqrt{2}(\nabla h)^{\mathrm {T}} \sigma \in {\mathbb { R}}^{(d-1) \times d}\), are all Lipschitz continuous in their arguments over \(\bar{B_r^+}\). Similarly, \((h_2,q_2) = (h_{2,t},q_{2,t})\) satisfies
for \(0 \le t \le \tau ^{2}_r\). Notice that the choice of coordinates satisfying (2.3) has eliminated a potentially singular drift term in the equations for \(h_{1,t}\) and \(h_{2,t}\). On the other hand, the drift term in the equations for \(q_1\) and \(q_2\) blows up near the boundary \(q = 0\). Indeed, if \(r > 0\) is small enough, by (1.9) there is a constant \(C_r > 0\) such that
Hence,
Letting \(\tau = \tau _r^1 \wedge \tau _r^2\) and using (2.14) and (2.16), we compute
for \(0 \le t \le \tau \). In particular,
holds for all \(t \ge 0\).
From (2.15) and (2.17) we also compute
for \(0 \le t \le \tau \), where we have used the notation \(g_1 = g(q_1,h_1)\) and \(g_2 = g(q_2,h_2)\). We claim that there is a constant \(C\), depending only on \(r\), such that
holds for all \(t \le \tau \), with probability one. Both sides of (2.22) are invariant when \((q_1,h_1)\) and \((q_2,h_2)\) are interchanged. So, we may assume \(q_1 \le q_2\) without loss of generality. We consider the following two possibilities. First, suppose that
Using this and \(q_1 \le q_2\) we have
The other possibility is
In this case, we have (also using \(q_1 \le q_2\))
Therefore, since \(|g_1 | \ge C_r > 0\) (by 2.19), we must have
where \(C > 0\) depends only on \(r\). This establishes (2.22).
Returning to (2.21) and controlling the first term on the right hand side of (2.21) with (2.22), we conclude that
By combining (2.20) and (2.27) and applying Gronwall’s inequality, we conclude that
Using (2.21) and (2.22) we also obtain
where \(V_t\) is the martingale
By the Burkholder–Davis–Gundy inequality (e.g. [30, Sec IV.4]) and (2.28), we have
This, together with (2.28) and (2.29), gives us
Similar arguments for \(h_1 - h_2\) lead to
\(\square \)
Proof of Lemma 2.3
Suppose \(\tau _r = 0\) holds with probability \(\epsilon > 0\). Because of (2.6) we may choose \(m\) sufficiently large so that
holds with probability at least \(1 - \epsilon /2\). Therefore, with probability at least \(\epsilon /2\) we have both \(\tau _r = 0\) and
Recall that \(|Y^{x_m}_0 - y_0| \le 25^{-m}\). Let \(m\) be larger, if necessary, so that \(25^{-m} \le r/4\). This and (2.30) imply that
holds with probability at least \(\epsilon /2\). However, this contradicts the fact that \(Y^{x_n}_{\tau _r^{n}} \in \partial B_r(y_0)\) for all \(n\). Hence, we must have \(\tau _r > 0\) with probability one.\(\square \)
Proof of Lemma 2.4
The fact that \(\bar{\tau }_{r/2} > 0\) with probability one follows from an argument very similar to the proof of Lemma 2.3. The fact that \(\bar{\tau }_{r/2} < \tau _r\) will follow by showing that
holds with probability one. First, suppose that \(\tau ^{n}_r < \tau _r\) and that
Then by (2.6) we have
where \(R(n)\) is the series remainder
which converges to zero, with probability one, as \(n \rightarrow \infty \). So, with probability one, if there is an increasing sequence of such times \(\tau ^{{n_j}}_r \nearrow \tau _r\) as \(j \rightarrow \infty \), we see that (2.31) must hold. On the other hand, suppose there is no such sequence. Then we must have \(\tau ^{n}_r \ge \tau _r\) for \(n\) sufficiently large. Hence \(Y^{x_n}_t\) must converge to \(Y_t\) uniformly on the closed interval \([0,\tau _r]\). Suppose \(\tau ^{n}_r \ge \tau _r\) and \(\tau ^{n}_r = \sup _{k \ge n} \tau ^{k}_r\). Then for all \(k \ge n\), we have
Therefore, since \(Y^{x_n}_t\) is continuous on \([0,\tau ^{n}_r]\) and since \(\tau _r = \liminf _{k \ge 0} \tau ^{k}_r\), we have
Since \(Y^{x_n}_{\tau _r} \rightarrow Y_{\tau _r}\) in this case and \(Y_t\) is continuous on \([0,\tau _r]\), then with probability one, this case also implies that (2.31) holds. Having established that \(0 < \bar{\tau }_{r/2} < \tau _r\) we conclude that \(Y^{x_n}_{t} \rightarrow Y_t\) uniformly on \([0,\bar{\tau }_{r/2}]\). Since each \(Y^{x_n}_t\) is \(\widehat{\mathcal {F}}_t\)-adapted, so is the limit \(Y_t\). In particular, \(\bar{\tau }_{r/2}\) is a stopping time. \(\square \)
Remark 2.5
Let us point out that if \(y_0 \in \partial A\) and \(T > 0\) is sufficiently small, the equation
has a unique solution satisfying \(\bar{Y}(t) \in \Theta \) for all \(t \in (0,T]\). Indeed, let \(z(t)\) solve the ODE
for \(t \in [0,T]\), with \(z(0) = y_0\). For sufficiently small \(T\), \(z(s) \in \Theta \) for \(t \in (0,T]\). Hence \(q(z(s)) > 0\) for \(t \in (0,T]\) and the function \(F(t) = \int _0^t q(z(s))\,ds\) is invertible. Now, it is easy to check that the function \(\bar{Y}(t) = z(F^{-1}(t))\) is continuous on \([0,T]\) and satisfies (2.32). Moreover, \(\bar{Y}(t) \in \Theta \) for all \(t \in (0,T]\). In fact,
for small \(t\).
We state and prove two properties of the transition path process, which will be used later.
Proposition 2.6
Let \(F\) be a bounded and continuous functional on \(C([0,\infty ))\). Define
where \(Y_t\) satisfies (1.14). Then \(g \in C(\bar{\Theta })\).
Proof
Suppose that \(\{x_n\}_{n=1}^\infty \subset \bar{\Theta }\) and that \(x_n \rightarrow x \in \bar{\Theta }\) as \(n \rightarrow \infty \). We claim that there must be a subsequence \(\{ x_{n_j} \}_{j=1}^\infty \) such that, \({\mathbb {Q}}\)-almost surely,
where \(Y^j_t\) satisfies (1.14) with \(Y^j_0 = x_{n_j}\), and \(Y_t\) satisfies (1.14) with \(Y_0 = x\). Since \(F\) is bounded and continuous on \(C([0,\infty ))\), the dominated convergence theorem then implies that
Since the limit is independent of the subsequence, this implies that \(g(x)\) is continuous.
To establish (2.33), we must show that \(Y^j_t \rightarrow Y_t\) uniformly on compact subsets of \([0,\infty )\). This follows from Corollary 2.2, as in the proof of Theorem 1.1. \(\square \)
Proposition 2.7
For any \(R > 0\), there is a function \(h_R:[0,+\infty ) \rightarrow [0,1]\) such that \(\int _0^\infty h_R(t) \,dt < +\infty \) and
holds for all \(t \ge 0\).
Proof
If \(x \in \Theta \), then by the Doob h-transform, we know that
where \(\tau _{AB}\) is the first hitting time of \(X\) to \(\bar{A} \cup \bar{B}\). Let \(\alpha > 1\) be as in assumption (1.4). Since \(\bar{A} \cup \bar{B}\) has non-empty interior and since \(\sigma \sigma ^T\) is uniformly positive definite, assumption (1.4) implies that for each \(R > 0\) there is \(C_R\) such that
From this and Chebychev’s inequality, it follows that
holds for all \(t > 0\). So, for any \(\epsilon > 0\),
holds for all \(t > 0\) and \(x \in \{ x \in \Theta \mid \; |x| \le R,\, q(x) \ge \epsilon \}\).
The bound (2.35) does not include points near \(\partial A\), where \(q(x) < \epsilon \). Fix \(\epsilon \in (0,1)\) and define the set \(S = \{ x \in \Theta \mid q(x) < \epsilon \} \cup \bar{A}\). If \(\epsilon \) is small enough, this set is bounded and we may assume \(|x | < R\) for all \(x \in S\). Suppose \(Y_0 = x\) with \(x \in S \cap \bar{\Theta }\). Let \(q_t = q(Y_t)\), which satisfies
where \(g(y) = \sqrt{2}(\nabla q(y))^{\mathrm {T}} \sigma (y)\). By (1.9) we know that if \(\epsilon > 0\) is small enough, there is a constant \(C_\epsilon > 0\) such that \(|g(y)|^2 \ge C_\epsilon \) for all \(y \in \bar{S} \cap \bar{\Theta }\). Therefore, if \(Y_t \in \bar{S} \cap \bar{\Theta }\) for all \(t \in [0,T]\), we must have \(q_t \le \epsilon \) for all \(t \in [0,T]\) and
for all \(t \in [0,T]\). This happens only if the martingale \(M_t = \int _0^t g(Y_s) \,\mathrm{d}\widehat{W}_s\) satisfies
To control the probability of this event, for any \(\gamma > 0\), \(\beta > 0\), \(T > 0\), Chebychev’s inequality implies
By choosing \(\beta = \gamma /||g ||_\infty ^2\) we have \({\mathbb {Q}}(M_T \le - \gamma T ) \le e^{- \gamma ^2 C_1 T}\). Hence there is a constant \(C_2 > 0\) such that
holds for all \(T > 1\) and \(x \in \bar{S} \cap \bar{\Theta }\).
Now we combine (2.35) and (2.36). Let \(\tau _S = \inf \{ t > 0 \mid Y_t \in \partial S \}\). By (2.36) we have \({\mathbb {Q}} \left( \tau _S > t/2\mid Y_0 = x \right) \le e^{- C_3 t}\) holds for all \(x \in \bar{S} \cap \bar{\Theta }\). Therefore, since \(\tau _S\) is a stopping time, we conclude that
for all \(x \in \bar{S} \cap \bar{\Theta }\). Since the last expression is an integrable function of \(t\), this completes the proof.\(\square \)
Proof of Theorem 1.2
Since \(\tau _{A,n}^+\) is a stopping time, it suffices to prove the result for \(n = 0\). Fix \(\epsilon > 0\) and let \(S \supset \bar{A}\) be the open set
For \(\epsilon > 0\) small, this is a bounded set that separates \(A\) and \(B\). The boundary \(\partial S\) is an isosurface for \(q\): \(q(x) = \epsilon \) for \(x \in \partial S\). As \(\epsilon \rightarrow 0\), \(S\) shrinks to \(A\), and the Hausdorff distance \(d_{\mathcal {H}}(\partial S, \partial A)\) is \({\mathcal {O}}(\epsilon )\) (because of (1.9)).
Recalling that \(\tau _{A,0}^+ = \inf \{ t \ge 0 \mid X_t \in \bar{A} \}\), we define
which is a stopping time with respect to \({\mathcal {F}}_t\). Then for \(k \ge 0\), we define inductively the stopping times (see Fig. 2)
Observe that \(r_{S,k} < r_{A,k} < r_{S,k+1}\), although it is possible that \(r_{B,k} = r_{B,k+1}\). Let \(r_{AB,k} = r_{A,k} \wedge r_{B,k}\), which is finite with probability one. We also define the random time
Left panel The set \(S\) and random times \(\tau _{S, j}\). Right panel Zoom-in of the boxed region together with stopping times \(r_{S, k}\) and \(r_{A, k}\)
Although \(\tau _{S,j}\) is not a stopping time with respect to \({\mathcal {F}}_t\), the relation
holds \({\mathbb { P}}\)-almost surely.
Now, let
and let \(h_0 = \tau _{S,0} - \tau _{A,0}^-\). Since \(F\) is bounded and continuous, and since \(h_0 \rightarrow 0\) (\({\mathbb { P}}\) almost surely) as \(\epsilon \rightarrow 0\), we have
We will show that
where \(g(x) = \widehat{{\mathbb {E}}}[F(Y_\cdot )\mid Y_0 = x]\).
Let \(M\) be the unique (random) integer such that
Equivalently, \(M = \min \{ k \ge 0 \mid r_{B,k} < r_{A,k} \}\). Since \(r_{B,k} > r_{A,k}\) for all \(k < M\), we have
Observe that the event \(\{k \le M\}\) coincides with the event that \(r_{B,j} > r_{A,j}\) for all \(j < k\), so the event \(\{k \le M\}\) is measurable with respect to \({\mathcal {F}}_{r_{S,k}}\). Therefore, we have
where
The last equality follows from the Doob \(h\)-transform (since \(x \in \partial S \subset \Theta \) here). Since \(q(x) = \epsilon \) for all \(x \in \partial S\), this means
where \(g(x) = \widehat{{\mathbb {E}}}[F(Y_{\cdot } )\mid Y_0 = x]\). Note that the random integer \(M\) depends on \(\epsilon \).
Let \(A_j\) denote the event \(\{j < M\}\), which occurs if and only if \(r_{A,k} < r_{B,k}\) for all \(k \in \{0,1,\dots ,j\}\). Since \(q(x) = \epsilon \) for all \(x \in \partial S\), the event \(A_j\) is independent of \(X_{r_{S,j}} \in \partial S\). Moreover, \(P(A_j) = (1 - \epsilon )^{j+1}\), since
Similarly, \({\mathbb { P}}( M = j) = \epsilon (1 - \epsilon )^j\). Now we evaluate (2.40):
Now let \(\epsilon \rightarrow 0\). Since \(g(x)\) is bounded and is continuous up to \(\partial A\) by Proposition 2.6, we have (by the dominated convergence theorem)
\(\square \)
3 Reactive exit and entrance distributions
Proof of Lemma 1.3
The equality (1.22) is equivalent to
Using (1.19), it is then equivalent to
which is obvious.\(\square \)
Before proving Proposition 1.5, we will need to establish some properties of the entrance and exit distributions and of the harmonic measure associated with the generator \(L\). These results will also be used later in the paper. First, using integration by parts, we have
Lemma 3.1
Let \(D \subset {\mathbb { R}}^d\) be open with smooth boundary. Let \(\phi , \psi \in C^2(D) \cap C^1(\bar{D})\) and bounded. Then
where \(\widehat{n}(x)\) is the exterior normal vector at \(x \in \partial D\).
Let us recall some tools from potential theory (see for example the books [28, 32] and also [7, 8] where potential theory was applied to analyze diffusion processes with metastability). The harmonic measure \(H_D(x, \mathrm{d}y)\) is given by the Poisson kernel corresponding to the boundary value problem
Therefore, for \(f \in C(\partial D)\),
is the unique solution to (3.2). Similarly, the harmonic measure \(\widetilde{H}_D(x, \mathrm{d}y)\) corresponds to the generator \(\widetilde{L}\) (recall (1.28)). For the boundary value problem
the solution is given by
The harmonic measures have a probabilistic interpretation: \(H_D(x, \mathrm{d}y)\) (resp. \(\widetilde{H}_D(x, \mathrm{d}y)\)) gives the probability that the process associated with the generator \(L\) (resp. \(\widetilde{L}\)) first strikes the boundary \(\partial D\) at \(\mathrm{d}y\) after starting at \(x\). In particular,
We also define the harmonic measures for the conditioned processes as
For \(x \in \Theta \) this is a measure on \(\partial B\). For \(x \in \partial A\) where \(q(x) = 0\), we may define \(H_{\Theta }^q(x,\mathrm{d}y)\) through a limit:
Recall that \(q(y) = 1\) for \(y \in \partial B\).
Recall the reactive exit and entrance measures \(\eta _A^-\), \(\eta _A^+\), \(\eta _B^-\) and \(\eta _B^+\). They are connected by harmonic measures as follows:
Proposition 3.2
Proof
We prove (3.8) first. If \(f \in C(\partial B)\), let \(u_f(x)\) solve \(Lu = 0\) in \(\Theta \) with
Hence \(u(x) \widetilde{q}(x) = 0\) on \(\partial \Theta \). By applying (3.1) with \(\phi (x) = \widetilde{q}(x)\) and \(\psi (x) = u_f(x)\), we obtain
From (3.7) and (1.23), we see that for all \(x \in \partial A\),
Hence for any \(f \in C(\partial B)\), we have
Combining this with (3.12), we conclude that
which proves (3.8).
To prove (3.9), let \(\psi \) solve \(L \psi = 0\) for \(x \in \bar{B}^{c}\) with \(\psi = f\) on \(\partial B\). Then by (3.1) with \(\phi = 1 - \widetilde{q}\), we have
Applying (3.1) with the function \(\phi \equiv 1\), we also find that
Therefore, since \(1 - \widetilde{q} = 1\) on \(\partial B\), we conclude that
We arrive at (3.9) noting that
We omit the proof of (3.10) which is analogous to that of (3.9) by switching the role of \(A\) and \(B\). \(\square \)
By combining (3.9) and (3.10) we immediately obtain the following:
Corollary 3.3
Let \(P_B(x,\mathrm{d}y)\) be the probability transition kernel
on \(\partial B\), and let \(P_A(x,\mathrm{d}y)\) be the probability transition kernel
on \(\partial A\). Then
and
That is, \(\eta _B^+\) and \(\eta _A^+\) are invariant under \(P_B\) and \(P_A\), respectively.
We are ready to return to the proof of Proposition 1.5.
Proof of Proposition 1.5
We first verify that \(\eta _B^+\) is a probability measure. Taking \(\psi = q\) and \(\phi = \widetilde{q}\) in (3.1), we obtain using the boundary conditions of \(q\) and \(\widetilde{q}\) on \(\partial A\) and \(\partial B\),
This shows that \(\eta _B^+(\partial B) = 1\) and \(\nu \) is the correct normalization constant.
Let \(g\) be a positive continuous function on \(\partial B\). Define for \(x \not \in \bar{B}\),
Hence \(u\) satisfies the equation
Let \(H_{\bar{B}^c}(x, \mathrm{d}y)\) be the harmonic measure (the measure of the first hitting point on \(\bar{B}\) for the process starting at \(x\)). We have
By the maximum principle, \(u > 0\) in \(\bar{B}^{c}\). By the Harnack inequality for non-divergence form elliptic operators [18, Corollary 9.25] and the compactness of \(\partial A\), we have
where the constant \(C > 0\) only depends on the elliptic constants of \(a(x)\) and on the maximum of \(|b|\) over some compact set \(A'\) satisfying \(A \subset A' \subset \bar{B}^c\). In particular, \(C\) is independent of \(g\). Therefore, we obtain for any \(x, x' \in \partial A\), \(y \in \partial B\)
If we define
then \(\nu _B(E) > 0\) is absolutely continuous with respect to \(\sigma _B(dy)\) on \(\partial B\), and
for any \(x \in \partial A\).
Consider the Markov chain given by \(\{X_{\tau _{B, k}^+}\}_{k=0}^\infty \) on \(\partial B\). Let \(P_B\) denote its transition kernel, given by
By (3.19), \(P_B\) satisfies Doeblin’s minorization condition:
Therefore, \(P_B\) has a unique invariant measure [3, Theorem 6.1]. By Corollary 3.3, this invariant measure is given by \(\eta _B^+\). Hence, as \(N \rightarrow \infty \), \(\int _{\partial B} f(x) \,\mathrm{d}\mu _{B, N}^+(x)\) converges exponentially fast to \(\int _{\partial B} f(x) \,\mathrm{d}\eta _B^+(x)\) (see e.g. [26, Theorem 17.1.7]). The rate of the convergence depends on the sets \(A\) and \(B\). \(\square \)
Proof of Theorem 1.7
Consider the family of processes
Observe that the \(n\)th reactive trajectory \(t \mapsto Y^n_t\) is a subset of the path \(t \mapsto X^{A,n}_t\); specifically, \(Y^n_t = X^{A,n}_{t + \tau _{A,n}^- - \tau _{A,n}^+}\) for all \(t \ge 0\). The random sequence of points
corresponds to a Markov chain on the state space \(\partial A\) with transition kernel
As shown in the proof of Proposition 1.5 (reversing the role of \(B\) and \(A\)), this chain satisfies a Doeblin minorizing condition
and the chain has a unique invariant probability distribution \(\eta _A^+\) supported on \(\partial A\):
The sequence of processes \(t \mapsto X^{A,n}_t\) corresponds to a homogeneous Markov chain on the metric space \({\mathcal {X}} = C([0,\infty ))\). The transition probability \(K\) for this chain may be expressed as follows. If \(X \in C([0,\infty ))\) is such that \(\tau _B^X = \inf \{ t \ge 0 \;|\; X_t \in \partial B\}\) is finite, then for any set \(E \in {\mathcal {B}}\),
where \({\mathcal {P}}_x\) denotes the law on \(({\mathcal {X}},{\mathcal {B}})\) of the process \(t \mapsto Z_{t \wedge \tau _{B}}\) where
and \(\tau _{B}\) is the first hitting time of \(Z_t\) to \(\bar{B}\). If \(X \in C([0,\infty ))\) never hits the set \(\bar{B}\), then we define
This chain on \({\mathcal {X}}\) has a unique invariant distribution
supported on the set of paths which originate in \(\partial A\) and are constant after hitting \(\partial B\). The uniqueness of \(\bar{\mathcal {P}}\) follows from the uniqueness of \(\eta _A^+\) as an invariant distribution for the chain defined by transition kernel \(P_A\) on \(\partial A\). Since \(P_A(x,dy)\) satisfies the Doeblin condition (3.22), so does the chain on \({\mathcal {X}}\):
In particular, it is positive Harris recurrent and aperiodic, and by [26, Theorem 17.1.7], for any \(\Phi \in L^1({\mathcal {X}},{\mathcal {B}},\bar{\mathcal {P}})\) the limit
holds \({\mathbb { P}}\)-almost surely.
Using (3.25) we will establish the following relationship between \(\eta _A^-\) and \(\eta _A^+\):
Lemma 3.4
Let \(X_t\) satisfy the SDE (1.1) with initial distribution \(X_0 \sim \eta _A^+\) on \(\partial A\). Then for any Borel set \(U \subset \partial A\),
Proof of Lemma 3.4
Let \(f \in C({\mathbb { R}}^d)\) be bounded and non-negative. Let us recall set \(S\) introduced in the proof of Theorem 1.2. Given \(\epsilon >0\), we let \(S = \{x \in \Theta \mid q(x) < \epsilon \} \cup \bar{A}\). Then by applying (3.25) to the functional \(\Phi (X) = f(X_{\tau _{S,0}^-})\), we obtain
We also have,
holds \({\mathbb { P}}\)-almost surely, where \(N_K = |\{ k \in \{0,1,\dots , K-1\} \mid r_{B,k} < r_{A,k} \}|\). Here we have used \(\zeta _S\) to denote the unique invariant distribution (identified below) for the Markov chain defined by \(X_{r_{S,k}}\) on \(\partial S\). Therefore,
We claim that if \(f(x)\) is uniformly continuous in a neighborhood of \(\partial A\), then
First, let us identify the invariant distribution \(\zeta _S\). By applying Corollary 3.3 (replacing \(B\) by \(\bar{S}^{c}\)) we can identify \(\zeta _S\) as
where \(\widehat{n}(x)\) is the exterior normal at \(x \in \partial S\), and \(\widetilde{q}_S\) satisfies \(\widetilde{L} \widetilde{q}_S = 0\) in \(S\) with
Note that \(\nu \) is independent of \(\epsilon \). Let \(\delta > \epsilon \) be small, and suppose that \(f(x)\) is continuous on the closed set \(\{ x \in \bar{\Theta } \mid 0 \le q(x) \le \delta \}\). (This set contains both \(\partial A\) and \(\partial S\)). A computation similar to (3.12) (replacing \(B\) by \(S\)) shows that for any such function, we have
where \(u_{f,S}\) satisfies \(L u = 0\) in \(S {\setminus } \bar{A}\), and
Since \(f \ge 0\), we have \(u > 0\) in \(S {\setminus } \bar{A}\). Now, let us define
which satisfies \(L^q z = 0\) in \(S {\setminus } \bar{A}\), with \(z = f\) on \(\partial S\) (recall that \(q(x) = \epsilon \) for all \(x \in \partial S\)). By the boundary Harnack inequality (see Theorem 2 and Corollary 1 of [2], as well as [5, Theorem 2.1] and [11, Theorem 11.6]), \(z_{f,S}(x)\) is bounded and Hölder continuous on \(\bar{S} {\setminus } A\) (including \(\partial A\)). We claim that for any \(x_0 \in \partial A\), we have
Since \(\nabla u_{f,S}\), \(\nabla q\), and \(z_{f,S}\) are continuous up to \(\partial A\), this is true if and only if
Suppose \(q(x) \nabla z_{f,S}(x) \rightarrow v \ne 0\) as \(x \rightarrow x_0 \in \partial A\). Then we must have
so that \(v\) must be a multiple of \(\widehat{n}(x_0)\) (since \(u\) and \(q\) vanish on \(\partial A\)). Thus, we would have
as \(x \rightarrow x_0 \in \partial A\). If \(v \ne 0\), then \((\widehat{n}(x_0) \cdot v) \ne 0\), so (3.30) and the fact that \(q = 0\) on \(\partial A\) would contradict the boundedness of \(z_{f,S}(x)\). Therefore, (3.29) must hold.
Combining (3.28) and (3.29) we obtain
Therefore, as \(\epsilon \rightarrow 0\),
This establishes (3.27) and completes the proof of Lemma 3.4. \(\square \)
Now we continue with the proof of Theorem 1.7. We will apply Theorem 1.2. Suppose that \(F \in L^1({\mathcal {X}},{\mathcal {B}},{\mathcal {Q}}_{\eta _A^-})\), and define the functional
Combining Theorem 1.2 and Lemma 3.4 we see that \(\Phi \in L^1({\mathcal {X}},{\mathcal {B}},\bar{\mathcal {P}})\), since
Therefore,
By (3.25) and Theorem 1.2, we now conclude that the limit
holds \({\mathbb { P}}\)-almost surely. This completes the proof of Theorem 1.7. \(\square \)
4 Reaction rate, density and current of transition paths
4.1 Reaction rate
Proof of Proposition 1.8
Denote \(\tau _B\) the first hitting time of \(X_t\) to \(\bar{B}\). Consider the mean first hitting time
which satisfies the equation
By definition of \(\eta _A^+\), we have
Observe that
Using (3.1) with \(D = A\), \(\phi (x) = 1\) and \(\psi (x) = u_B\), we obtain
where \(\widehat{n}\) is the interior normal vector at \(\partial A\). Apply (3.1) again with \(D = \Theta \), \(\phi = \widetilde{q}\) and \(\psi = u_B\),
Combining the two with (4.2), we get
Similarly, defining \(u_A(x)\) to be the mean first hitting time of \(X_t\) to \(\bar{A}\) starting at \(x\), we have
Add the integrals together to obtain
On the other hand, observe that
As \(N \rightarrow \infty \), we have
and similarly
Therefore
or equivalently \(\nu = \nu _R\).
From Theorem 1.7 it follows immediately that
Indeed, the functional \(F: Y \rightarrow \tau ^Y_B\) is in \(L^1({\mathcal {X}},{\mathcal {B}},{\mathcal {Q}}_{\eta _A^-})\) by Proposition 2.7. The function \(v_B(x) = \widehat{{\mathbb {E}}}[ \tau _B^Y \mid Y_0 = x]\) satisfies
with \(v(x) = 0\) for \(x \in \partial B\). Hence, the function \(w(x) = q(x) v_B(x)\) satisfies \(L w = -q\) for \(x \in \Theta \) with boundary condition \(w(x) = 0\) for \(x \in \partial \Theta \). Moreover, for \(x_0 \in \partial A\), we have
Therefore,
Now applying (3.1) with \(D = \Theta \), \(\phi = \tilde{q}\) and \(\psi = w\), we have
It remains to show that
Using integration by parts, we have
The first term on the right hand side vanishes as
where we have used that \(q^2 - q = 0\) on \(\partial A \cup \partial B\). The conclusion then follows from Lemma 1.3, \(q = 0\) on \(\partial A\), and \(q = 1\) on \(\partial B\). \(\square \)
4.2 Density of transition paths
We define the Green’s function \(G_{\Theta }\) of the operator \(L\) in \(\Theta \) with Dirichlet boundary condition on \(\partial \Theta \):
The existence of the Green’s function is guaranteed by the ergodicity of \(X_t\) in \({\mathbb { R}}^d\), which implies that \(X_t\) is transient in \(\Theta \) (see e.g. [28, Section 4.2]).
Lemma 4.1
Let \(G_{\Theta }\) be the Green’s function of \(L\) in \(\Theta \) with Dirichlet boundary condition on \(\partial \Theta \). We have
In particular, for \(x \in \partial A\), \(y \in \Theta \)
Proof
Fix \(y \in \Theta \). For \(x \in \Theta \), (4.4) follows from [28, Proposition 4.2.2]. Specifically, the function \(G_{\Theta }^{q}(x, y)\) defined by
is related to the Green’s function (4.3) by the formula
Because of the regularity of the coefficients \(a(x)\) and \(b(x)\), Schauder-type interior and boundary estimates imply that \(G(\cdot ,y) \in C^{2,\alpha }(\bar{\Theta } {\setminus } \{y\})\). Since \(G(x,y) = q(x) = 0\) for \(x \in \partial A\), the Hopf Lemma implies that for all \(x \in \partial A\), \(\nabla _x G(x,y)\) is a nonzero multiple of \(\widehat{n}(x)\). That is, for all \(x \in \partial A\), \(\nabla _x G(x,y) = r(x)\widehat{n}(x)\) for some continuous \(r(x) < 0\). The same is true for \(q\). Therefore, \(G_{\Theta }^{q}(x, y)\) is continuous in \(x\) up to the boundary \(\partial \Theta \) and for \(x_0 \in \partial A\),
It remains to show that for \(x_0 \in \partial A\),
Let \(\varphi \ge 0\) be smooth and compactly supported in \(\Theta \). By Proposition 2.6, we have
Moreover,
By Proposition 2.7, for any \(R > 0\), there a function \(h_R \in L^1(0,+\infty )\) such that \({\mathbb {Q}}( Y_t \in \Theta \mid Y_0 = x) \le h_R(t)\) for all \(x \in \Theta \), \(|x| < R\), \(t \ge 0\). Therefore, we have \(\widehat{{\mathbb {E}}}[ \varphi (Y_t)\mid Y_0 = x] \le ||\varphi ||_\infty h_R(t)\) so the dominated convergence theorem implies that
On the other hand, we also have
Therefore, by combining (4.7) and (4.8) we conclude
Since \(\varphi \) is arbitrary, this implies (4.6).\(\square \)
Proof of Proposition 1.9
Using Lemma 4.1 and (1.38),
Recall the explicit formula of \(\eta _A^-\) in terms of \(q\) (1.23), we obtain for \(z \in \Theta \)
Apply (3.1) by taking \(\psi (x) = G_{\Theta }(x, y)\) and \(\phi (x) = \widetilde{q}(x)\), we conclude that
Here to get the second equality, we have used that \(\widetilde{L}\widetilde{q} = 0\) in \(\Theta \) and \(\psi (x) = 0\) on \(\partial \Theta \).\(\square \)
4.3 Current of transition paths
Proof of Proposition 1.10
It follows from a direct calculation from the definition of \(J_R\) as (1.41), noticing that \(q = 0, \widetilde{q} = 1\) on \(\partial A\), and \(q = 1, \widetilde{q} = 0\) on \(\partial B\).\(\square \)
Proof of Corollary 1.11
By Proposition 1.10, we have
Hence, it suffices to show that
which follows from the fact that \(J_R\) is divergence free in \(\Theta \) (see (1.40)).\(\square \)
Proof of Proposition 1.12
Using Proposition 1.10 for the left hand side of (1.45), we obtain
where \(\widehat{n}\) is the unit normal exterior to \(\Theta \). Equation (1.45) then follows from the divergence theorem.
Now fix any \(g \in C^1(\partial B)\), we extend \(g\) to \(\bar{\Theta }\) using the flow (1.43): for any \(x \in \bar{\Theta }\), we define
In particular, for \(x \in \partial A\), we have \( g(x) = g(\Phi _{J_R}(x))\), in other words,
By the construction (4.10), for any \(x \in \Theta \), \(J_R \cdot \nabla g = 0\). Combining with the first part of the Proposition and (4.11), we obtain
Therefore, \(\Phi _{J_R, *}(\eta _A^-) = \eta _B^+\).
References
Athreya, S.R., Barlow, M.T., Bass, R.F., Perkins, E.A.: Degenerate stochastic differential equations and super-Markov chains. Probab. Theory Relat. Fields 123, 484–520 (2002)
Athanasopoulos, I., Caffarelli, L.A.: A theorem of real analysis and its application to free boundary problems. Commun. Pure Appl. Math. 38, 499–502 (1985)
Athreya, K.B., Ney, P.: A new approach to the limit thoery of recurrent markov chains. Trans. Am. Math. Soc. 245, 493–501 (1978)
Bakhtin, Y.: Gumbel distribution in exit problems (2013, preprint) [arXiv:1307.7060]
Bauman, P.: Positive solutions of elliptic equations in nondivergence form and their adjoints. Ark. Mat. 22, 153–173 (1984)
Bolhuis, P.G., Chandler, D., Dellago, C., Geissler, P.L.: Transition path sampling: throwing ropes over rough mountain passes, in the dark. Annu. Rev. Phys. Chem. 53, 291–318 (2002)
Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. 6, 399–424 (2004)
Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. 7, 69–99 (2005)
Berman, K.A., Konsowa, M.H.: Random paths and cuts, electrical networks, and reversible Markov chains. SIAM J. Discrete Math. 3, 311–319 (1990)
Cerou, F., Guyader, A., Lelievre, T., Malrieu, F.: On the length of one-dimensional reactive paths (2012, preprint) [arXiv:1206.0949]
Caffarelli, L., Salsa, S.: Geometric Approach to Free Boundary Problems. American Mathematical Society, Providence (2005)
Dellago, C., Bolhuis, P.G., Geissler, P.L.: Transition path sampling. Adv. Chem. Phys. 123 (2002)
Dean, T., Dupuis, P.: The design and analysis of a generalized RESTART/DPR algorithm for rare event simulation. Ann. Oper. Res. 189, 63–102 (2011)
DeBlassie, D.: Uniqueness for diffusions degenerating at the boundary of a smooth bounded set. Ann. Probab. 32, 3167–3190 (2004)
Weinan, E., Vanden-Eijnden, E.: Toward a theory of transition paths. J. Stat. Phys. 123, 503–523 (2006)
Weinan, E., Vanden-Eijnden, E.: Transition path theory and path-finding algorithms for the study of rare events. Annu. Rev. Phys. Chem. 61, 391–420 (2010)
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1984)
Gilberg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998)
Haussmann, U.G., Pardoux, É.: Time reversal of diffusions. Ann. Probab. 14, 1188–1205 (1986)
Hairer, M., Stuart, A.M., Voss, J.: Analysis of SPDEs arising in path sampling part II: the nonlinear case. Ann. Appl. Probab. 17, 1657–1706 (2007)
Hairer, M., Stuart, A.M., Voss, J., Wiberg, P.: Analysis of SPDEs airisng in path sampling part I: the Gaussian case. Commun. Math. Sci. 3, 587–603 (2005)
Hummer, G.: From transition paths to transition states and rate coefficients. J. Chem. Phys. 120, 516–523 (2004)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)
Metzner, P., Schütte, C., Vanden-Eijnden, E.: Illustration of transition path theory on a collection of simple examples. J. Chem. Phys. 125, 084110 (2006)
Meyer, P.A., Smythe, R.T., Walsh, J.B.: Birth and death of markov processes. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III, pp. 295–305 (1972)
Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge (2009)
Prinz, J.-H., Held, M., Smith, J.C., Noé, F.: Efficient computation, sensitivity, and error analysis of committor probabilities for complex dynamical processes. Multiscale Model. Simul. 9, 545–567 (2011)
Pinsky, R.G.: Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics, vol. 45. Cambridge University Press, Cambridge (1995)
Reznikoff, M.G., Vanden-Eijnden, E.: Invariant measures of stochastic partial differential equations and conditioned diffusions. C. R. Acad. Sci. Paris Ser. I 340, 305–308 (2005)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, New York (1999)
Stuart, A.M., Voss, J., Wiberg, P.: Conditional path sampling of SDEs and the Langevin MCMC method. Commun. Math. Sci. 2, 685–697 (2004)
Sznitman, A.-S.: Brownian Motion, Obstacles and Random Media. Springer, New York (1998)
Vanden-Eijnden, E.: Transition path theory (2013, preprint)
Veretennikov, A.Yu.: On polynomial mixing bounds for stochastic differential equations. Stochastic Process Appl. 70, 115–127 (1997)
Vanden-Eijnden, E., Venturoli, M., Ciccotti, G., Elber, R.: On the assumptions underlying milestoning. J. Chem. Phys. 129, 174102 (2008)
Vanden-Eijnden, E., Weare, J.: Rare event simulation of small noise diffusions. Commun. Pure Appl. Math. 65, 1770–1803 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
We are grateful to Weinan E, Jonathan Mattingly, and Eric Vanden-Eijnden for helpful discussions. The work of JL was supported in part by the Alfred P. Sloan foundation and the National Science Foundation under Grant No. DMS-1312659. The work of JN was supported by NSF Grant DMS-1007572.
Rights and permissions
About this article
Cite this article
Lu, J., Nolen, J. Reactive trajectories and the transition path process. Probab. Theory Relat. Fields 161, 195–244 (2015). https://doi.org/10.1007/s00440-014-0547-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-014-0547-y