# Stochastic model for barrier crossings and fluctuations in local timescale.

Rajeev Bhaskaran  Statistics and Mathematics Unit, Indian Statistical
Institute, 8th Mile Mysore Road, Bangalore 560059, India  Vijay Ganesh
Sadhasivam  Statistics and Mathematics Unit, Indian Statistical Institute, 8th
Mile Mysore Road, Bangalore 560059, India  Yusuf Hamied Department of
Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK

(March 2022)

###### Abstract

The problem of computing the rate of diffusion-aided activated barrier
crossings between metastable states is one of broad relevance in physical
sciences. The transition path formalism aims to compute the rate of these
events by analysing the statistical properties of the transition path between
the two metastable regions concerned. In this paper, we show that the
transition path process is a unique solution to an associated stochastic
differential equation (SDE), with a discontinuous and singular drift term. The
singularity arises from a local time contribution, which accounts for the
fluctuations at the boundaries of the metastable regions. The presence of
fluctuations at the local time scale calls for an excursion theoretic
consideration of barrier crossing events. We show that the rate of such
events, as computed from excursion theory, factorizes into a local time term
and an excursion measure term, which bears empirical similarity to the
transition state theory rate expression. Since excursion theory makes no
assumption about the presence of a transition state in the potential energy
landscape, the mathematical structure underlying this factorization ought to
be general. We hence expect excursion theory (and local times) to provide some
physical and mathematical insights in generic barrier crossing problems.

##  I Introduction

A systematic study of transition between metastable states in physical systems
has been of relevance in computing the rates of chemical
reactionspechukas1981transition ; truhlar1983current , protein
foldingonuchic1997theory , crystal nucleationOxtoby_1992  etc. An empirical
expression for the (temperature-dependent) rate of such events was first
provided by ArrheniusArrhenius+1889+96+116 , which identified the presence of
an ‘activation energy’ barrier that needs to be crossed for a transition to
take place. The development of potential energy surfaces(PES)
EyringPolanyi+2013+1221+1246  in the early $20^{\text{th}}$ century allowed
the interpretation of chemical reactions as a continuous (albeit rare)
progression from ‘reactants’ to ‘products’, mediated by a transition state
which is a saddle point in the PES corresponding to the aforementioned
activation energy barrier. Transition state theory (TST)eyring1935activated ,
postulated shortly after this development, provided a derivation for
Arrhenius’ expression, under the assumption of quasiequilibrium between the
reactants and transition state. While the language of transition state theory
(and much of this article) is that of chemistry, its underlying structure and
usefulness makes it relevant for understanding rare events in numerous
physical contexts.

Although transition state theory serves as a strong theoretical tool in
interpreting the rates of gas-phase molecular reactions, it is, by virtue of
its founding assumptions, incapable of describing reaction rates where the
particle dynamics is diffusive, which is the case in many condensed phase
systems. To address this problem, Kramers reconceptualized chemical reactions
as Brownian motion aided barrier escape from metastable states, driven by
thermal fluctuationskramers1940brownian . He was able to derive a formally
exact expression for reaction rates in the limits of high and low
frictionhanggi1990reaction ; kramers1940brownian , which bore a clear
structural resemblance to the TST rate expression, indicating that the basic
empirical assumptions of activated barrier crossings are still valid.

While the mathematical assumptions underlying Kramers theory allow for the
expansion of the scope of TST and help interpret reaction rates in condensed
phase systems, it hinges heavily on the presence of a transition state in the
PES. However, when a transition state cannot be unambiguously identified (for
instance, when the PES terrain is ‘rugged’, as in the protein folding
problemonuchic1997theory ), the assumption and limits that Kramers considers
do not hold anymore.

Nonetheless, modeling chemical reaction as a diffusion over a PES, that is
intrinsic to Kramers problems (and later extended by Grote and Hynes to non-
Markovian processes in 10; 11), brings the tools in the theory of stochastic
differential equations (SDE) to the discussion of reaction rates. Computing
reaction rates in such a mathematical setup can be achieved by adopting an
alternative characterization of chemical reactions involving the transition
path, i.e. the path along which a reactive transition happens by barrier
crossing. Such a viewpoint reduces the emphasis on transition states in
computing reaction rates and has been described and developed as transition
path sampling (TPS)pratt1986statistical ; bolhuis2002transition ;
bolhuis2003transition .

Transition path sampling primarily encompasses algorithmic procedures that
attempt to sample all possible routes from reactants to products and weight
them based on their relative probability of realization. Extracting useful and
relevant information about the mechanism of a reaction, presence of transition
states or other dynamical bottlenecks (if any) require a concrete mathematical
framework, the importance of which was underlined in the development of
transition path theory(TPT) by Vanden-Eijnden et alvanden2006towards ;
vanden2010transition .

The cornerstone of TPT is the definition of reactive trajectories from the
transition path between the reactants and products, and characterization of
its statistical properties. These reactive trajectories can be understood as
crossings of the diffusion process (as in Kramers theory) from the reactant
region (say $A$) to the product region (say $B$) through a transition region.
The transition path process is then the (discontinuous) semi-martingale
associated with the crossings of the transition region by the diffusion.

The semimartingale associated with such crossings of an interval $(a,b)$ by a
continuous semimartingale was systematically studied in 17. In this paper,
using the theory introduced in 17, we characterize the transition path process
as the unique solution of a singular SDE which is adapted to the original
diffusion process. We appropriately identify an open interval $(a,b)$ as the
transition region according to the specification of the PES. The rate of the
reaction $\kappa$ is obtained from the number of reactive trajectories
($N_{t}$) until time $t$, which, in our context will be the number of
upcrossings of $(a,b)$ by the diffusion until time $t$.

Each such upcrossing corresponds to a crossing of the activation energy
barrier and is hence by preceded by long-time fluctuations around the boundary
of the reactant region. A measure of the amount of such fluctuations around a
point $a$ until time $t$ is provided in semimartingale theory by the local
time $L(t,a)$ at that point which is increasing in $t$ and is defined via the
Tanaka formula. Every such fluctuation that happens during $[0,t]$ corresponds
to an excursion of the diffusion from the point $a$ with a certain height and
duration.

Of interest in this paper are the excursions of the diffusion process from the
reactant to the product region. These excursions correspond precisely to the
reactive trajectories and hence can be used to compute the reaction rate
$\kappa$, which is discussed in later sections of this paper. This procedure
factorizes the reaction rate expression into a term involving local time
fluctuations and another involving the excursion measure. It also gives us
some insight into the dependence of the excursion measure on the parameters of
the diffusion.

We also use classical results from renewal theory to compute the reaction
rate; we first show that the diffusion process describing chemical reactions
is regenerative and positive recurrent and then later compute $\kappa$ in
terms of the expected cycle time associated with a regenerative process
(compare with Kramers mean first passage time hanggi1990reaction ). By
comparing the $\kappa$ thus obtained using the two methods above, we are able
to show that the factorization of reaction rate by excursion theory
corresponds precisely to the Arrhenius/TST rate expression. Thus, transitions
between two metastable states can always be interpreted mathematically in
terms of the local time fluctuations and excursion measure.

For the sake of clarity and to illustrate the basic idea in reaction-rate
theory, we describe the excursions of the diffusion on a simple one-
dimensional PES. Much of our results extend to a more general class of one
dimensional diffusions that are positive recurrent.

The paper is organized as follows: In section II, we start with the
description of the reactive diffusion process and define the relevant notions
required to understand the local time scale intrinsic in chemical reactions.
In section III, we formulate a singular SDE that describes the transition path
process and characterize its solutions. In section IV, we provide a primer on
excursion theory and indicate its relevance in reaction rate theory. In
section V, we compute the rate expression using excursion theory. In VI, we
compare the known rate expression computed using the renewal theorem and
compare it with the one obtained from excursion theory. We conclude with a
discussion on the importance and relevance of local time scales in reaction
rate theory.

##  II Preliminaries

As in typical in discussions of transition path formalism vanden2010transition
, we characterize a chemical reaction by the trajectory of a particle
diffusing on a potential energy surface $U$ using a real valued stochastic
process $X:[0,\infty)\times\Omega\rightarrow\mathbb{R}$ defined on a
probability space $(\Omega,\mathcal{F},\text{P})$ that follows the Ito
stochastic diffusion equation (SDE):

$dX_{t}=\sigma(X_{t})\;dB_{t}+b(X_{t})dt,\;\;\;\;\;X_{0}=x_{0}$ ( II.1 )

where the process $B_{t}$ is a standard Brownian motion process on a
measurable space $\Omega$, $\sigma$ is the diffusion coefficient, $b(\cdot)$
is the drift term, and $\mathcal{F}$ is the filtration generated by the
process $B_{t}$ satisfying the usual conditions as in 18. Here and below,
following the typical convention, we shall drop the explicit dependence on the
random trajectory $\omega\in\Omega$ in our notation.

We assume that the SDE in (II.1) describes overdamped Langevin dynamics
hanggi1990reaction  and accordingly, by the fluctuation-dissipation theorem
and Einstein’s relations zwanzig2001nonequilibrium , $\sigma$ is constant with
$\sigma=\sqrt{2D}$, where $D$ is the diffusion coefficient. The drift term
$b(x)$ is obtained from the PES as:

$\displaystyle b(x)=-\frac{D}{k_{B}T}\frac{dU}{dx}$ ( II.2 )

where $T$ is the temperature and $k_{B}$ is the Boltzmann constant. For our
discussion, the PES shall be the truncated double well potential:

$\displaystyle U(x)=\begin{cases}&Q(x)\hskip 88.2037pt|x|\leq K-\epsilon\\\
&g(|x|)\hskip 85.35826pt|x|\in(K-\epsilon,K+\epsilon)\\\ &k(|x|-K)+Q(K)\hskip
26.17648pt|x|\geq K+\epsilon\end{cases}$ ( II.3 )

In (II.3), $Q(x)$ is the quartic double well potential
$Q(x)=h^{2}x^{4}-\frac{1}{2}\lambda^{2}x^{2}$, $k\;\text{and}\;K$ are positive
constants with $K\geq\left|\frac{\lambda}{h}\right|$, and $g(x)$ is a smooth
function which makes $U(x)$ differentiable at $|x|=K\pm\epsilon$, so that
$U\in C^{1}(\mathbb{R})$ with a bounded derivative. The proof of existence of
such a function is given in the supporting information. A schematic plot of
the PES is provided in figure 1.

Figure 1: Illustration of the truncated double well potential $U(x)$. The
points $a,b$ and the barrier height $V_{b}$ are as defined in the text. The
quartic and the linear of the PES are connected by the function $g$ which is
defined in an $\epsilon$ neighborhood around $K$, that makes $U\in
C^{1}(\mathbb{R})$.

The double well potential is a straightforward choice for describing
transitions between metastable states which are ubiquitous in chemistry and
biophysics onuchic1997theory . For a more realistic representation of such
transitions, the double well can be made asymmetric by the addition of a cubic
term to $Q(x)$ in (II.3) and the results of this paper will still hold. The
reason for truncating this double well potential at $K$ and appending it with
a linear term is purely mathematical - Such a modification makes the drift
term $b(x)$ Lipschitz continuous in the coordinate variable and also satisfy:

$\displaystyle\int_{0}^{x}b(y)dy\leq m|x|\hskip 28.45274pt\text{for
some}\;m<0,|x|\geq x_{0}$ ( II.4 )

The first condition allows (II.1) to have a unique strong solution $(X_{t})$
relative to the Brownian motion $(B_{t})$ that is a continuous
semimartingalekaratzas2012brownian  adapted to the filtration $\mathcal{F}$
(Theorem 2.5 in 18) and the second condition allows $(X_{t})$ to have a finite
speed measure, which shall be discussed later (Note that the point of
truncation is taken far from the minima of $Q(x)$ so as to ensure that the
linear term in $U(x)$ has negligible impact on the dynamics of $(X_{t})$).
This stochastic process $(X_{t})$ shall be the concern of this paper. We begin
first with a few definitions required for this discussion:

Let $T_{x}:=\inf\\{s:X_{s}=x\\}$. For $y\in\mathbb{R}$, we denote by $P_{y}$
the law of $X_{t}$ on $C([0,\infty))$ such that $P_{y}\\{X_{0}=y\\}=1$. Note
that $X_{t}$ is the coordinate mapping process karatzas2012brownian  on
$C([0,\infty))$, that is $X_{t}(\omega)=\omega(t)$ for $\omega\in
C([0,\infty))$. $E_{y}$ denotes the expectation under $P_{y}$. The process
$X_{t}$ is positive recurrent if $\forall x,y\in\mathbb{R}$,
$E_{y}T_{x}<\infty$.

Let $\tau_{0}:=0:=\sigma_{0}$ and define the sequence $\\{\sigma_{n};n\geq
1\\},\\{\tau_{n};n\geq 1\\}$ as follows:

$\displaystyle\sigma_{n}:=\inf\\{s>\tau_{n-1}:X_{s}=b\\}\leavevmode\nobreak\
{\rm and}\leavevmode\nobreak\ \tau_{n}:=\inf\\{s>\sigma_{n}:X_{s}=a\\}$

Here we choose $-a=b=\left|\frac{\lambda}{2h}\right|$, the points of global
minima of the quartic potential $Q(x)$ (Note that we use $b(\cdot)$ for
denoting the drift term). In the context of reaction rate theory and
metastable states, these points can be interpreted as the equilibrium
structure of the ‘reactant’ and ‘product’ metastable states.

The process $(X_{t})$ is said to be regenerative at a point $x$, if
$P(\theta_{\tau}X_{\cdot}\in A|\mathcal{F}_{\tau})=P_{x}(A)\;a.s.$ on
$\\{\tau<\infty,\;X_{\tau}=x\\}$ for every stopping time $\tau$
kallenberg1997foundations , where $\theta_{\tau}X_{\cdot}:=X_{\cdot+\tau}$.

######  Proposition 1.

The unique strong solution $(X_{t})$ to (II.1) is positive recurrent and
regenerative at the points $a$ and $b$.

###### Proof.

The fact that $(X_{t})$ is regenerative at $a$ follows from a standard
application of the strong Markov property of the diffusion $(X_{t})$ under
$P_{a}$, which can be stated as follows karatzas2012brownian ;
kallenberg1997foundations :

$\displaystyle E_{a}\\{F(X_{\tau+\cdot})|{\cal
F}_{\tau}\\}\leavevmode\nobreak\ =\leavevmode\nobreak\
E_{X_{\tau}}F(X_{\cdot}),{\rm\leavevmode\nobreak\ a.s.\leavevmode\nobreak\
}P_{a}.$ ( II.5 )

where $F:C[0,\infty)\rightarrow\mathbb{R}$ is a bounded Borel-measurable
function, $\tau<\infty\;a.s.$ is any stopping time and $X_{\tau+\cdot}$
denotes the path $s\rightarrow X_{\tau+s}$. The regenerative property follows
by taking $F=I_{A}$, where $A\subseteq C([0,\infty))$ is a Borel set and
$\tau$ such that $X_{\tau}=a$.

We first define the speed measure and the scale function. Given the diffusion
process $(X_{t})$ as in (II.1), the scale function $p$ is defined w.r.t a
point $c\in\mathbb{R}$ as karatzas2012brownian :

$\displaystyle
p(x):=\int_{c}^{x}\exp\left\\{-2\int_{c}^{y}\frac{b(y)}{\sigma^{2}(y)}dy\right\\}dx$
( II.6 )

The scale function for $(X_{t})$ is well-defined since the integrand in the
exponent is continuous. The speed measurekaratzas2012brownian  is defined as:

$\displaystyle m(dx):=\frac{2\;dx}{p^{\prime}(x)\sigma^{2}(x)}$ ( II.7 )

The positive recurrence of the diffusion $(X_{t})$ is proved as in 18 (Ex.
5.40 of Chapter 5). For completeness, we give a few details. For $x<y$, we
have positive recurrence if the RHS limit in

$\displaystyle E_{y}T_{x}=\lim\limits_{n\rightarrow\infty}E_{y}(T_{x}\wedge
T_{n})$

is finite. From eq. (5.55) and (5.59) in 18,

$\displaystyle E_{y}(T_{x}\wedge
T_{n})=-\int_{x}^{y}(p(y)-p(z))m(dz)+\frac{p(y)-p(x)}{p(n)-p(x)}\int_{x}^{n}(p(n)-p(z))m(dz)$
( II.8 )

In the limit $n\rightarrow\infty$,

$\displaystyle\lim\limits_{n\rightarrow\infty}\frac{1}{p(n)-p(x)}\int_{x}^{n}(p(n)-p(z))m(dz)=$
$\displaystyle
m([x,\infty))+\lim\limits_{n\rightarrow\infty}\frac{p(x)m(x,n)}{p(n)-p(x)}$
$\displaystyle-\lim\limits_{n\rightarrow\infty}\frac{1}{p(n)-p(x)}\int_{x}^{n}p(z)m(dz)$

Since $b(x)\rightarrow\pm k$ as $x\rightarrow\pm\infty$ (from (II.3), we have,
from (II.6) that $p(x)\rightarrow\infty$ as $x\rightarrow\infty$ and
$m([x,\infty))<\infty$. So, the first term in the RHS of the equation above is
finite, and the second term vanishes. As for the third term in the RHS, we
note that

$\displaystyle\lim\limits_{n\rightarrow\infty}\frac{1}{p(n)-p(x)}\int_{x}^{n}p(z)m(dz)=\lim\limits_{n\rightarrow\infty}\frac{2}{\sigma^{2}}\frac{1}{p(n)-p(x)}\int_{x}^{n}\frac{p(z)}{p^{\prime}(z)}dz$
( II.9 )

Further note that,

$\displaystyle\lim\limits_{n\rightarrow\infty}\int_{x}^{n}\frac{p(z)}{p^{\prime}(z)}dz\rightarrow\infty$
( II.10 )

Using l’Hospitale’s rule, we evaluate the limit in (II.9),

$\displaystyle\lim\limits_{n\rightarrow\infty}\frac{2}{\sigma^{2}}\frac{1}{p(n)-p(x)}\int_{x}^{n}\frac{p(z)}{p^{\prime}(z)}dz=\lim\limits_{n\rightarrow\infty}\frac{2}{\sigma^{2}}\frac{p(n)}{p^{\prime}(n)^{2}}=0,$
( II.11 )

where the last equality follows from the estimate in (II.4) and the definition
of $p(y)$. Hence $E_{y}T_{x}<\infty$ when $x<y$. The case $y<x$ can be
similarly treated by considering
$E_{y}T_{x}=\lim\limits_{n\rightarrow\infty}E_{y}(T_{-n}\wedge T_{x})$. ∎

######  Remark 2.

Note that positive recurrence implies $E_{a}\tau_{n}<\infty$ for every $n\geq
1$. In particular, $\tau_{n}<\infty\leavevmode\nobreak\ a.s.$ Note that the
regenerative property tells us that the cycle times $\tau_{n}-\tau_{n-1}$ are
independent and identically distributed kallenberg1997foundations  under
$P_{a}$ with mean
$E_{a}(\tau_{n}-\tau_{n-1})=E_{a}\tau_{1}=E_{a}T_{b}+E_{b}T_{a}$.

Before we move on to a discussion of transition paths, we summarize the
nomenclature in transition path formalism, largely following 16:

Denoting the minima of the PES $U$ as $a$ and $b$ as before, we set
$A=(-\infty,a]$ as the reactant region and $B=[b,\infty)$ as the product
region. We define the last entrance time into and first exit time from the set
$\mathbb{R}\backslash A\cup B=(a,b)$, $\sigma_{t}$ and $\tau_{t}$ respectively
rajeev1990semi , as:

$\displaystyle\sigma_{t}$ $\displaystyle:=\sup_{s\leq
t}\left\\{s:X_{s}\notin\mathbb{R}\backslash(A\cup B)\right\\}$
$\displaystyle\tau_{t}$ $\displaystyle:=\inf_{s\geq
t}\left\\{s:X_{s}\notin\mathbb{R}\backslash(A\cup B)\right\\}$

As in 16,we denote by $R$ the times at which
$X_{t}\in\mathbb{R}\backslash(A\cup B)$,

$\displaystyle R:=\left\\{t:X_{t}\in\mathbb{R}\backslash(A\cup
B),X_{\sigma_{t}}\in A\;\text{and}\;X_{\tau_{t}}\in
B\right\\}:=\bigsqcup_{i=1}^{\infty}(a_{i},b_{i})$ ( II.12 )

where the last equality holds because $(X_{t})$ is continuous and hence $R$ is
an open set. Noting that $X_{a_{i}}=a,X_{b_{i}}=b\;\forall i$, the ensemble of
reactive trajectories (See figure 3) can be defined as:

$\displaystyle\mathcal{R}:=\bigcup_{i}\left\\{X_{t}:t\in(a_{i},b_{i})\right\\}$

where each $i$ corresponds to a reactive trajectory. Denoting the number of
reactive trajectories until time $t$ as $N_{t}$, the rate $\kappa$ of the
reaction is given as vanden2010transition :

$\displaystyle\kappa:=\lim_{t\rightarrow\infty}\frac{N_{t}}{t}$ ( II.13 )

It is easy to observe that the process $(Z_{t})$ defined as
$Z_{t}:=X_{t}-X_{\sigma_{t}}$ describes the dynamics of $(X_{t})$ in
$\mathcal{R}$, as $Z_{t}\equiv 0$ for $t\notin R$ and $Z_{t}\in(-(b-a),b-a)$
for $t\in R$ and thus can be considered the transition path process. The
dynamics of the process $(Z_{t})$ merits a separate discussion and in this
regards, 21 conceive the idea of using the Doob-h transform day1992conditional
to define an auxiliary SDE, whose solutions have the same law as that of the
reactive trajectories. We are, however, interested in associating an SDE to
the transition path process $(Z_{t})$ that focuses on crossing events or
excursions between the sets $A$ and $B$ as in 23; 17; 24 and making explicit
the role of the intrinsic local timekallenberg1997foundations  scale
associated with such events. This shall be the subject of the next section.
One advantage to this approach, despite the fact that it is not easily
extendable to higher dimensions, is that the solutions to such an SDE will be
adapted the same filtration $(\mathcal{F}_{t})$ as that of the process
$(X_{t})$. Further discussion on the insights that local times and excursion
theory provide in the context of reaction rates is reserved for later
sections.

##  III Singular SDE describing transition path

Since the definition of the transition process $(Z_{t})$ motivated in the
previous section hinges on $(X_{t})$, it is expected that the SDE describing
$(Z_{t})$ will be driven by the continuous martingale $(X_{t})$ and not by a
brownian motion as in (II.1). A detailed description of such SDEs driven by
continuous semimartingale and conditions for the existence of unique
strong/weak solutions to them has been provided in 25. However, the SDEs in 25
assume a continuous drift term.

From the discussion in 17, it is clear that the transition path process
$(Z_{t})$ will have contributions from local time terms, so the SDE describing
$(Z_{t})$ ought to have a ‘singular drift’ or a ‘local time drift’, the theory
of which is explained in 26; 27; 28. In particular, 26 also discusses the
cases in which one can expect a strong solution to such SDEs with singular
term. Using the insights in 17, in this section we define a singular SDE that
describes the transition path process, explain what a solution to it means,
and give the proof of its existence and pathwise uniqueness.

Given the continuous semimartingale $(X_{t})$ that is the strong unique
solution to (II.1), let $S:=\left\\{t:X_{t}\in(a,b)\right\\}$ and consider the
following equation:

$\displaystyle dZ_{t}$
$\displaystyle=\mathcal{V}(Z_{t})dX_{t}+d\mathcal{L}_{t}(Z_{.})$ ( III.14 )

with

$\displaystyle\mathcal{V}(x)$ $\displaystyle=I_{\left\\{0<|x|<b-a\right\\}},$
(III.15 a) $\displaystyle X_{t}\notin(a,b)$ $\displaystyle\Leftrightarrow
Z_{t}\equiv 0\;\;\text{and}$ (III.15 b)
$\displaystyle\operatorname{supp}(d\mathcal{L}_{t})\subseteq S^{c},$
$\displaystyle\;i.e.\int_{0}^{\infty}I_{S}(s)|d\mathcal{L}_{s}|=0$ (III.15 c)

Since (III.14) has a local time drift and the ‘diffusion coefficient’ is the
discontinuous function $\mathcal{V}$, this is a singular SDE, driven by a
continuous semimartingale. By a solution to (III.14), we mean a pair
$(Z_{t},\mathcal{L}_{t})$ such that the following holds:

  1. 1.

$Z_{t}$ and $\mathcal{L}_{t}$ are adapted to the filtration $\sigma(X)$, and
is right continuous with left limits(rcll), or a càdlàg.

  2. 2.

$\mathcal{L}_{t}$ has bounded variation with $\mathcal{L}_{0}=Z_{0}=0$.
Moreover, $\mathcal{L}_{t}$ is adapted to $\sigma(Z)$, namely the filtration
generated by the process $Z$ and satisfies (III.15 c).

  3. 3.

$Z_{t}$ and $\mathcal{L}_{t}$ satisfies the following piecewise definition:

Let

$\displaystyle\tau=\inf\left\\{u>0:X_{u}\notin(a,b)\right\\}$

    1. (a)

For $0\leq t<\tau$, $Z_{t}=X_{t}-X_{0}$ and $\mathcal{L}_{t}=0$. Since
$\mathcal{V}(Z_{t})=1$ for almost every $(t,\omega)\in[0,\tau)\times\Omega$
w.r.t the measure $dt\;dP$,

$\displaystyle Z_{t}=\int_{0}^{t}\mathcal{V}(Z_{s})dX_{s}.$

    2. (b)

For $t\geq\tau$,
$Z_{t}=\int_{\tau}^{t}\mathcal{V}(Z_{s})dX_{s}+\mathcal{L}_{t}$

######  Remark 3.

Note that $Z_{t}$ and $\mathcal{L}_{t}$ are not required to be continuous
processes. Since $X_{t}$ is a continuous process, it follows from (III.14) and
(III.15 b) that $\Delta Z_{t}=\Delta\mathcal{L}_{t}$.

######  Theorem 1 (Existence-Uniqueness theorem).

There exists a unique solution to the singular stochastic differential
equation (III.14).

###### Proof.

By definition of a solution (condition 3 above) to the singular SDE, the
process $(Z_{t})=X_{t}-X_{\sigma_{t}}$ satisfies (III.14) for $t<\tau$. We
hence show that the transition path process $Z_{t}$ is a solution to (III.14)
for $t\geq\tau$. This follows right away from the Tanaka formula for a semi-
martingale, as outlined in 17. We have the following expression for $(Z_{t})$
for $t\geq\tau$:

$\displaystyle\begin{split}Z_{t}=X_{t}-X_{\sigma_{t}}&=\int_{\tau}^{t}I_{(a,b]}(X_{s})dX_{s}+\frac{1}{2}\left\\{L(t,a)-L(t,b)\right\\}-(b-a)(U(t)-D(t))\\\
&=\int_{\tau}^{t}I_{(a,b)}(X_{s})dX_{s}+\int_{\tau}^{t}I_{(X_{s}=b)}dX_{s}\\\
&+\frac{1}{2}\left\\{L(t,a)-L(t,b)\right\\}-(b-a)(U(t)-D(t))\end{split}$ (
III.16 )

where $L(t,a)$ and $L(t,b)$ are local times at $a$ and $b$ respectively of
$(X_{t})$, $U(t)$ and $D(t)$ are the number of upcrossings and downcrossings
of $(a,b)$ by $(X_{t})$ during $(0,t)$. Note that since the
$dt(\\{s:X_{s}=b\\}=0)\;a.s.$, the second integral in the RHS vanishes. It is
easy to observe that $I_{(a,b)}(X_{t})=I_{\left\\{0<|Z_{t}|<b-a\right\\}}$.

Also, the upcrossings $U(t)$ and downcrossings $D(t)$ of $(a,b)$ by $(X_{t})$
are the same as the upcrossings and downcrossings of $(0,b-a)$ and
$(-(b-a),0)$ respectively by $(Z_{t})$. In fact, $Z_{t}=(X_{t}-a)^{+}$ during
an upcrossing, $Z_{t}=(X_{t}-b)^{-}$ during a downcrossing and jumps to $0$
from $\pm(b-a)$ at the time of crossing (See figure 2 and 3). Let $L_{Z}(t,x)$
denote the local time of the process $(Z_{t})$ at $x$. It is well-known that
$L_{Z}(t,x)$ is right continuous and has left limits at every $x$
revuz2013continuous . Further, from 17 we can show that the following relation
holds:

$\displaystyle L(t,a)=L_{Z}(t,0)\;\;\;\;\;\;\;L(t,b)=L_{Z}(t,0-)$ ( III.17 )

Figure 2: Illustration of the transition path process $(Z_{t})$: One specific
realization of the diffusion $X_{t}(\omega)$ and the corresponding transition
path process $Z_{t}(\omega)$ are plotted above. Here, each
$(\tau_{i},\sigma_{i+1})$ is an upcrossing interval and
$(\sigma_{i},\tau_{i+1})$ is a downcrossing interval. As can be seen from the
inset image, $Z_{t}(\omega)$ is discontinuous at the end points of these
intervals (See figure 3 for further clarity).

The first equation in (III.17) follows from eqn. 7 in 17. The second equation
in (III.17) follows from letting $x$ increase to $0$ in eqn. 6 (We use the
notations as in 17 and observe that $L(t,b)$ has no support in the upcrossing
intervals (i.e. in $\\{s:\Theta^{u}(s)=1\\}$). This shows that

$\displaystyle\mathcal{L}_{t}(Z_{.}):=\frac{1}{2}\left\\{L(t,a)-L(t,b)\right\\}-(b-a)(U(t)-D(t))$
( III.18 )

is a well-defined functional of the process $(Z_{t})$. It is straightforward
to show that $\mathcal{L}_{t}$ has bounded variation and the support property
in (III.15 c) is satisfied. As
$X_{t}=(X_{t}-X_{\sigma_{t}})+X_{\sigma_{t}}=:Z_{t}+Y_{t}$, we have that
$(Z_{t},\mathcal{L}_{t})$ is a solution to (III.14).

We prove the uniqueness of the solution in two parts. Suppose there is another
solution $(Z_{t}^{{}^{\prime}},\mathcal{L}_{t}^{{}^{\prime}})$ to equation
(III.14). It suffices to show that $Z_{t}^{{}^{\prime}}=Z_{t}\;\;\forall
t\;a.s$. It then follows that
$\mathcal{L}^{{}^{\prime}}_{t}(Z^{\prime})=\mathcal{L}_{t}(Z)$. For $t\notin
S$, $X_{t}\notin(a,b)$ and hence $Y_{t}^{{}^{\prime}}=X_{t}=Y_{t}$ as
$Z_{t}^{{}^{\prime}}$ satisfies (III.15 b). Now, for the case of $t\in S$, we
proceed via a pathwise argument.

$\displaystyle
Z^{{}^{\prime}}_{t}=\int_{0}^{t}I_{\\{0<|Z^{{}^{\prime}}_{t}|<b-a\\}}dX_{s}+\mathcal{L}^{{}^{\prime}}_{t}(Z^{{}^{\prime}}_{t})$
( III.19 )

Figure 3: The inset image in figure 2 is zoomed further. The portions of
$X_{t}(\omega)$ highlighted in green are the reactive trajectories. Each
reactive trajectory occurs in the time interval $(a_{i},b_{i})$. The points
$\sigma_{1},\tau_{1}$ and $\sigma_{4}$ are also marked in black for reference.
The discontinuities in $Z_{t}(\omega)$ at each $\sigma_{i}$ and $\tau_{i}$ are
clearly evident.

Denoting the set $S^{{}^{\prime}}:=\\{s:0<|Z^{{}^{\prime}}_{s}|<b-a\\}$, we
write $S:=\bigcup_{i}(s_{i},t_{i})=S^{{}^{\prime}}$. For $h>0$, define

$\displaystyle D_{h}:=\inf\\{s>h:|Z^{{}^{\prime}}_{s}|\notin(0,b-a)\\}$ (
III.20 )

From the positive recurrence of $(X_{t})$, we can show that $D_{h}<\infty$
a.s. Hence we have for a fixed $h>0$, from (III.19) and the fact that
$\operatorname{supp}(d\mathcal{L}^{{}^{\prime}}_{t})\subseteq S^{c}$,

$\displaystyle
Z^{{}^{\prime}}_{h}-Z^{{}^{\prime}}_{D_{h}}=X_{h}-X_{D_{h}}\;\;\text{a.s.}$ (
III.21 )

For $\omega$ s.t. $h<D_{h}(\omega)$ and $h\in(s_{i}(\omega),t_{i}(\omega))$,
take $h_{n}\in(s_{i}(\omega),t_{i}(\omega))$, then we have from (III.21)
applied to $h$ and $h_{n}$ separately,

$\displaystyle Z^{{}^{\prime}}_{h}-Z^{{}^{\prime}}_{h_{n}}=X_{h}-X_{h_{n}}$ (
III.22 )

since $D_{h}=D_{h_{n}}$. Taking $h_{n}<h$, $h_{n}\downarrow\sigma_{h}$ and
noting that $Z^{{}^{\prime}}_{\sigma_{h}}=0$, we get

$\displaystyle Z^{{}^{\prime}}_{h}=X_{h}-X_{\sigma_{h}}\;\;\forall
h\in(s_{i}(\omega),t_{i}(\omega))$ ( III.23 )

(III.23) and the fact that $\Omega=\bigcup_{h\in
Q^{+}}\\{\omega:h<D_{h}(\omega)\\}$ completes the proof.

∎

Thus we have characterized the transition path process as the unique solution
of an associated singular stochastic differential equation. The fact that the
transition path process $(Z_{t})$ is described by a singular SDE involving
local time drift suggests that there are local timescales associated with
$(Z_{t})$ that one needs to take cognizance of. In the next section, we will
explore this connection in detail and provide an excursion theoretic
conceptualization of reactive trajectories.

##  IV Excursions of transition paths and reaction rates

The transition path process $(Z_{t})$ characterizes the shuttling action of
the process $(X_{t})$ into and outside the set $\mathbb{R}\backslash(A\cup
B)$, which is $(a,b)$ in our discussion. Ito excursion theory ito1972poisson
describes the Poisson point process of excursions from a point associated with
the continuous recurrent diffusion such as $(X_{t})$, using the local time at
that point. In the context of this paper, we need a theory of excursions of
the semimartingale $(X_{t})$ into the set $(a,b)$ starting from the boundary
$\left\\{a,b\right\\}$. This need a suitable modification of Ito’s theory to
include excursions from more than one point. For a generalization of Ito’s
excursion theory, we refer the reader to Maissonneuve’s discussion on Exit
systems maisonneuve1975exit .

For the specific case of the set $B$ being $(a,b)$ and $(X_{t})$ being a
Brownian motion, 32 had described the associated excursions of $(X_{t})$ in
terms of Ito point excursion processes and local times at the boundary points
of the interval $(a,b)$. It was shown that the Maissonneuve excursion of
$(X_{t})$ into the set $(a,b)$ is ‘decomposable’ into Ito excursions of
$(X_{t})$ from the points $a$ and $b$. This section shall focus on extending
the arguments in bhaskaran2022asymptotic  to describe the excursions of the
continuous diffusion $(X_{t})$ about $a$ and $b$, which can be equivalently
formulated in terms of the excursions of the transition path $(Z_{t})$ about
the point $0$. We use the excursion theory presented in 20 for regenerative
processes.

We begin this section with the definitions of excursion space and excursion
processes and then proceed to give an expression for the rate constant in
terms of excursions of the processes $(X_{t})$ or equivalently that of
$(Z_{t})$.

The space of excursions of $(X_{t})$ into the set $(a,b)$ is defined as
follows:

$\begin{split}\mathcal{E}(a,b):=\\{u|\;&u:[0,R(u))\rightarrow[a,b],\;u\text{
is continuous on}\;[0,R(u)),\\\ &a<u(t)<b\;\forall
t\in(0,R(u)),\;u(0)\in\left\\{a,b\right\\}\;\text{and}\;u(R(u)-)\in\left\\{a,b\\}\right\\}.\end{split}$

$\mathcal{E}(a,b)$ can be identified as a measurable subset of the space
$C([0,\infty))$ with the induced topology by extending each function
$u\in\mathcal{E}(a,b)$ as a constant beyond $R(u)$. Note that
$\mathcal{E}(a,b)=\mathcal{E}_{a}\bigcup\mathcal{E}_{b}$, where, for
$x\in\\{a,b\\}$,

$\displaystyle\mathcal{E}_{x}:=\\{u\in\mathcal{E}(a,b)|\;u(0)=x\\}.$

The same space can be expressed in terms of excursions of the transition path
process $Z_{t}$:

$\begin{split}\mathcal{E}_{Z}(\mathcal{S}):=\\{u|\;&u:[0,R(u))\rightarrow\mathcal{S}\cup\\{0\\},\;\text{u
is continuous in}\;[0,R(u)),\\\
&u(0)=0,\;u(R(u)-)\in\left\\{0,b-a,-(b-a)\right\\},u(t)\in\mathcal{S}\;\forall
t\in(0,R(u))\;\\},\end{split}$

with $\mathcal{S}:=(-(b-a),0)\cup(0,b-a)$. As in the case of
$\mathcal{E}(a,b)$, we can write
$\mathcal{E}_{Z}(\mathcal{S})=\mathcal{E}_{0}^{+}\cup\mathcal{E}_{0}^{-}$,
where $\mathcal{E}_{0}^{+}$ and $\mathcal{E}_{0}^{-}$ are the positive and
negative excursions from $0$.

For the continuous, non decreasing, $\mathcal{F}_{t}$-adapted process
$L_{t}:=L(t,a)+L(t,b)$, the right continuous inverse $\tau_{t}$ can be defined
as follows:

$\displaystyle\tau_{t}:=\inf\\{s>0:L_{s}>t\\}$

The right continuous inverse $\tau_{t}^{a}$ and $\tau_{t}^{b}$ of $L(t,a)$ and
$L(t,b)$ respectively and the right continuous inverse $\tilde{\tau}_{t}$ of
the local time process at $0$ for the process $(Z_{t})$, namely $L_{Z}(t,0)$,
can be defined similarly.

######  Proposition 4.

$L_{Z}(t,0)+L_{Z}(t,0-)=L(t,a)+L(t,b)=L_{t}$.

###### Proof.

The proof follows from 17 (See the proof of theorem 1 above). ∎

We now define the excursion process; setting
$D:=\\{s>0:\tau_{s}\neq\tau_{s-}\\}$, for $t\in D$, we define:

$\displaystyle\widetilde{e}_{t}(\omega)(s):=X_{\tau_{t-}+\;s\wedge(\tau_{t}-\tau_{t-})}(\omega).$

For $t\notin D$, $\widetilde{e}_{t}(\omega)\equiv\delta$. The above definition
of excursion includes all the excursions starting from $a$ and $b$. However,
we only need those excursions starting from $a$ or $b$ into the set $(a,b)$.
Hence we discard the excursions below $a$ and above $b$ by redefining the
excursion process as:

$\displaystyle
e_{t}(\omega):=I_{\mathcal{E}(a,b)}(\widetilde{e}_{t}(\omega))\;\widetilde{e}_{t}(\omega)+\delta
I_{(\mathcal{E}(a,b))^{c}}\;(\widetilde{e}_{t}(\omega)).$

where $I_{A}$ denotes the indicator function of the set $A$. In terms of
$(Z_{t})$, we can redefine the excursion process as:

$\displaystyle
e_{t}^{Z}(w):=I_{\mathcal{E}_{Z}(\mathcal{S})}(\widetilde{e}_{t}^{Z}(\omega))\;\widetilde{e}_{t}^{Z}(\omega)+\delta
I_{(\mathcal{E}_{Z}(\mathcal{S}))^{c}}\;(\widetilde{e}_{t}^{Z}(\omega))$

where $\widetilde{e}_{t}^{Z}(\omega)$ is defined similar to
$\widetilde{e}_{t}(\omega)$, by replacing $(X_{t})$ by $(Z_{t})$.

Let $\mathcal{E}^{\prime}_{a}$ be the set of excursions from $a$, that is,

$\displaystyle\mathcal{E}^{\prime}_{a}=\\{u:[0,R(u))\rightarrow\mathbb{R},u\;\text{is
continuous on}\;[0,R(u))\;$ $\displaystyle\text{and}\;u(t)\neq
a\;\text{for}\;t\in(0,R(u)),\;u(0)=u(R(u)-)=a\\}$

and

$\displaystyle(\mathcal{E}^{\prime}_{a})^{+}=\\{u\in\mathcal{E}^{\prime}_{a}|u(t)>a\;\;\text{for}\;t\in(0,R(u))\\}$

is the set of positive excursions starting from $a$. Defining
$(\mathcal{E}^{\prime}_{a})^{-}$ similarly, we have
$\mathcal{E}_{a}^{\prime}=(\mathcal{E}^{\prime}_{a})^{+}\bigcup(\mathcal{E}^{\prime}_{a})^{-}$.
Similar definitions hold for the set of excursions starting from $b.$

Define the map $t_{b}(u)$ for $u\in(\mathcal{E}_{a}^{\prime})^{+}$ as follows:

$\displaystyle t_{b}(u):=\inf\;\\{t|\;u(t)=b\\}.$

Now define

$\displaystyle\eta_{a}:=\;(\mathcal{E}_{a}^{\prime})^{+}$
$\displaystyle\rightarrow\mathcal{E}_{a}$ $\displaystyle u$
$\displaystyle\mapsto\eta_{a}(u):=u(t_{b}\wedge\cdot)$

We can similarly define $t_{a}$ and $\eta_{b}$.

Let $n^{\prime}_{a}$, $n^{\prime}_{b}$ be the characteristic measures on
$\mathcal{E}_{a}^{\prime}$ and $\mathcal{E}_{b}^{\prime}$ respectively of the
Poisson point process $N_{a}$ and $N_{b}$ associated with the excursions of
$(X_{t})$ starting from $a$ and $b$ respectively. Define $n(x,A)$ on
$\mathcal{E}(a,b)=\mathcal{E}_{a}\cup\mathcal{E}_{b}$ as follows:

$\displaystyle
n(x,A):=n^{\prime}_{x}(\eta_{x}^{-1}(A\cap\mathcal{E}_{x})),\;\text{where}\;x\in\\{a,b\\}\;\text{and}\;A\subseteq\mathcal{E}(a,b)$
( IV.24 )

Using these definitions of excursion process, we can now characterize reactive
trajectories. First we set:

$\displaystyle\Lambda$
$\displaystyle:=\\{u\in\mathcal{E}(a,b):u(0)=a,\underset{0<t<R(u)}{\sup}u(t)=b\\}\subseteq\mathcal{E}_{a}$
$\displaystyle\Lambda_{Z}$
$\displaystyle:=\\{u\in\mathcal{E}_{Z}(\mathcal{S}):\underset{0<t<R(u)}{\sup}u(t)=b-a\\}\subseteq\mathcal{E}_{0}^{+}$

Then, we define:

$\displaystyle\Lambda^{\prime}:=\\{u\in(\mathcal{E}_{a}^{\prime})^{+}|u(0)=a,\underset{0<t<R(u)}{\sup}u(t)\geq
b\\}=\eta_{a}^{-1}(\Lambda)$

With this, we define the quantity $N(t,A)$ for $\;A\subseteq\mathcal{E}(a,b)$:

$\displaystyle N(t,A):=\\#\\{s\leq t:\tau_{s-}\neq\tau_{s},e_{s}(\omega)\in
A\\}=N_{a}(t,\eta_{a}^{-1}(A))+N_{b}(t,\eta_{b}^{-1}(A))$

$N_{Z}(t,A)$ can be defined similarly. For computing the rate of the reaction
we need to calculate the number of reactive trajectories, $N_{t}$ until time
$t$.

######  Proposition 5.

The number $N_{t}$ of reactive trajectories until time $t$, is given as
follows:

$\displaystyle N_{t}=N(L_{t},\Lambda)$
$\displaystyle=N_{a}(L(t,a),\Lambda^{\prime}),$ $\displaystyle
where\;\;L_{t}=L(t,a)$ $\displaystyle+L(t,b),\;\text{ as defined above.}$

###### Proof.

By construction, the support of the local time process $dL_{t}$,

$\operatorname{supp}(dL_{t})\subseteq\left\\{t:X_{t}=a\;\;or\;\;X_{t}=b\right\\}$.

Note that $L_{t}$ is constant during the excursion interval
$(\tau_{t-},\tau_{t})$. The number of reactive trajectories until the time $t$
is, by definition, equal to the number of upcrossings of $(X_{t})$ completed
before $t$. So, each upcrossing of $(a,b)$ by $s\mapsto X_{s}(\omega)$
completed before $t$ corresponds to an excursion $e_{s}(\omega)$ of $(X_{t})$
into the set $\Lambda$, with $\tau_{s}\leq t$. Since $\tau_{t}$ is the right
inverse of the local time process $L_{t}$, we have that:

$\displaystyle N_{t}=N({L_{t}},{\Lambda})$

Clearly the number of excursions from $a$ of height $b-a$ counted via $L_{t}$
and $L(t,a)$ must be the same. So we have:

$\displaystyle N_{t}=N_{a}({L(t,a)},\Lambda^{\prime})$

∎

##  V Computing reaction rate using excursion theory

The rate $\kappa$ of the reaction mentioned in (II.13) can be expressed as:

$\displaystyle\kappa=\underset{t\rightarrow\infty}{\lim}\frac{N_{t}}{t}=\underset{t\rightarrow\infty}{\lim}\frac{N(L_{t},\Lambda)}{t}=\underset{t\rightarrow\infty}{\lim}\frac{N_{a}(L(t,a),\Lambda^{\prime})}{t}$

This can be rewritten as:

$\kappa=\underset{t\rightarrow\infty}{\lim}\frac{N(L_{t},\Lambda)}{L_{t}}\;\underset{t\rightarrow\infty}{\lim}\frac{L_{t}}{t}=\underset{t\rightarrow\infty}{\lim}\frac{N_{a}(L(t,a),\Lambda^{\prime})}{L(t,a)}\;\underset{t\rightarrow\infty}{\lim}\frac{L(t,a)}{t}$
( V.25 )

provided both the limit exists (Note that $N(t,A)\leq\infty$ for an arbitrary
set $A\in\mathcal{E}(a,b)$. However, $N_{a}(t,A)<\infty$ a.e. iff
$n^{\prime}_{a}(A)<\infty$). We show that is the indeed the case.

######  Proposition 6.

$\displaystyle\underset{t\rightarrow\infty}{\lim}\frac{L(t,a)}{t}=\frac{E_{a}L(\tau_{1},a)}{E_{a}\tau_{1}}=\frac{-2E_{a}\int_{0}^{\tau_{1}}I_{\\{X_{s}>a\\}}b(X_{s})ds}{E_{a}\tau_{1}}<\infty$

###### Proof.

The proof is given in 2 steps:
Step 1: To show

$\displaystyle\underset{n\rightarrow\infty}{\lim}\frac{L(\tau_{n},a)}{n}=E_{a}L(\tau_{1},a)<\infty$
( V.26 )

where $\tau_{n}$ is as defined in section II.

Step 2: To show

$\displaystyle\underset{t\rightarrow\infty}{\lim}\frac{L(t,a)}{t}=\frac{E_{a}L(\tau_{1},a)}{E_{a}\tau_{1}}$
( V.27 )

Proof of step 1: That the limit in the LHS exists follows from a standard ‘Law
of large numbers’ argument; denoting $\Xi_{k}:=L(\tau_{k},a)-L(\tau_{k-1},a)$,
we have, by the regenerative property:

$\displaystyle L(\tau_{n},a)=\sum_{k=1}^{n}\Xi_{k}$

where $\Xi_{k}$ are independent and identically distributed with

$\displaystyle E_{a}\Xi_{1}$ $\displaystyle=E_{a}L(\tau_{1},a)$ (V.28 a)
$\displaystyle=2E_{a}\left((X_{\tau_{1}}-a)^{+}-(X_{0}-a)^{+}-\int_{0}^{\tau_{1}}I_{\\{X_{s}>a\\}}dX_{s}\right)$
(V.28 b)
$\displaystyle=-2E_{a}\int_{0}^{\tau_{1}}I_{\\{X_{s}>a\\}}b(X_{s})ds=-\frac{2}{\sigma^{2}}\int_{a}^{\infty}b(y)\;E_{a}L(\tau_{1},y)dy$
(V.28 c)

In the expression above, the second equality follows from the Tanaka formula
kallenberg1997foundations  and the last equality follows from the ‘Occupation
density formula’(See Chapter 3, theorem 7.1(iii) in 18) for the semimartingale
$(X_{t})$ and a bounded Borel function. Note that the expectations are finite
because $E_{a}\tau_{1}<\infty$ and $b$ is a bounded function. This completes
the proof of step 1.

Proof of step 2: Using a standard argument, we can write:

$\displaystyle\frac{L(\tau_{n},a)}{\tau_{n+1}}$
$\displaystyle\leq\frac{L(t,a)}{t}\leq\frac{L(\tau_{n+1},a)}{\tau_{n}}$

for $\tau_{n}\leq t\leq\tau_{n+1}$.

Now the proof follows using step 1 and the fact observed earlier in the proof
of proposition 1 and remark 2 that

$\displaystyle\lim_{n\rightarrow\infty}\frac{\tau_{n}}{n}\rightarrow
E_{a}\tau_{1}$

∎

######  Remark 7.

Considering (V.28) for an arbitrary point $z\in\mathbb{R}$, we get:

$\displaystyle
E_{a}L(\tau_{1},z)=-\frac{2}{\sigma^{2}}\int_{z}^{\infty}b(y)E_{a}L(\tau_{1},y)dy$

Setting $f(z):=E_{a}L(\tau_{1},z)$, this yields an ordinary differential
equation, whose general solution is given as
$f(z)\sim\exp(\nicefrac{{2}}{{\sigma^{2}}}{\int_{0}^{z}b(x)dx})$, thereby
yielding:

$\displaystyle E_{a}L(\tau_{1},z)=Ce^{-\beta(U(z)-U(0))}$ ( V.29 )

where $\beta=\nicefrac{{1}}{{k_{B}T}}$. Thus, the expected value of local time
is highest near the minima of the PES $U(x)$, which are the points $a$ and $b$
and explicitly:

$\displaystyle E_{a}L(\tau_{1},a)\sim e^{-\beta(U(a)-U(0))}=e^{\beta V_{b}}$ (
V.30 )

where $V_{b}$ is the height of the potential energy barrier in $U(x)$, as $0$
corresponds to a local maxima of $U(x)$. In the above expression, $a\sim b\iff
a=Cb$ for some constant $C$.

######  Proposition 8.

The excursion measure of $\Lambda^{\prime}$, namely
$n^{\prime}_{a}(\Lambda^{\prime})$, is finite and in particular,
$(N_{a}(t,\Lambda^{\prime}))_{t\geq 0}$ is a Poisson process with

$\displaystyle EN_{a}(t,\Lambda^{\prime})=t\;n^{\prime}_{a}(\Lambda^{\prime})$

###### Proof.

For a Poisson random measure $\xi$, we have $\xi(A)<\infty\;a.s.$ if and only
if $\mu(A):=E\xi(A)$ is finite. This follows from the ‘exponential formula’
(Refer to lemma 12.2 part (i) in 20):

$\displaystyle Ee^{-\xi(A)}=e^{-\int(1-e^{-I_{A}})d\mu}$

Let $\Lambda^{a}_{h}$ be the set of excursions of length $h$ from $a$, with
$\Lambda^{a}_{h}=\Lambda_{h}^{a,+}\bigcup\Lambda_{h}^{a,-}$ where
$\Lambda_{h}^{a,+}$ and $\Lambda_{h}^{a,-}$ are the positive and negative
excursions from $a$. Note that $n^{\prime}_{a}(\Lambda^{a}_{h})<\infty$ for
$h>0$ (Refer 20 for proof). We define Poisson random measures
$\Xi_{h}(\cdot):=N_{a}(t,\cdot\cap\Lambda_{h}^{a})$ for $h>0$
kallenberg1997foundations  and note that
$\Xi_{h}(\Lambda^{\prime})<\infty\;a.s.$, as
$n^{\prime}_{a}(\Lambda_{h}^{a})<\infty$ for each $h>0$. Defining a sequence
$\Xi_{h_{n}}$ of Poisson random measures with $h_{n}\downarrow 0$, we have:

$\displaystyle
E\>\Xi_{h_{n}}(\Lambda^{\prime})=t\;n^{\prime}_{a}(\Lambda^{\prime}\cap\Lambda^{a}_{h_{n}})$

The sequence $\Xi_{h_{n}}(\Lambda^{\prime})$ converges $a.s.$ by the monotone
convergence theorem, to a random variable $\Xi(\Lambda^{\prime})$ for which,

$\displaystyle Ee^{-\Xi(\Lambda^{\prime})}$
$\displaystyle=\lim_{n\rightarrow\infty}Ee^{-\Xi_{h_{n}}(\Lambda^{\prime})}=\lim_{n\rightarrow\infty}e^{-\int(1-e^{-I_{\Lambda^{\prime}}})d\mu_{h_{n}}}$
( V.31 )
$\displaystyle=\lim_{n\rightarrow\infty}\exp{(-\mu_{h_{n}}(\Lambda^{\prime})(1-e^{-1}))}=\exp{(-\mu(\Lambda^{\prime})(1-e^{-1}))}$

Now, $\Xi(\Lambda^{\prime})=N_{a}(t,\Lambda^{\prime})$ by definition. So, by
the continuity of the trajectories of $(X_{t})$,

$\displaystyle N_{a}(t,\Lambda^{\prime})$ $\displaystyle=\text{Number of
upcrossings of $(a,b)$ by the process $(X_{s})$}$ during the interval
$(0,\tau_{t})$ $\displaystyle\Rightarrow N_{a}(t,\Lambda^{\prime})$
$\displaystyle<\infty\;\textit{a.s.}$

Since the LHS in (V.31) is positive, it follows that
$\mu(\Lambda^{\prime})<\infty$. Thus, the sequence
$\Xi_{h_{n}}(\Lambda^{\prime})$ converges in distribution to
$\Xi(\Lambda^{\prime})$. That $\Xi(\Lambda^{\prime})$ is a Poisson random
variable follows from (V.31).

Also, we have:

$\displaystyle\mu(\Lambda^{\prime})\equiv
E\>\Xi(\Lambda^{\prime})=\lim_{n\rightarrow\infty}E\>\Xi_{h_{n}}(\Lambda^{\prime})=t\;\lim_{n\rightarrow\infty}n^{\prime}_{a}(\Lambda^{\prime}\cap\Lambda^{h_{n}}_{a})=t\;n^{\prime}_{a}(\Lambda^{\prime})<\infty$

From the independent increment property of
$(N_{a}(t,\Lambda^{\prime}\cap\Lambda^{a}_{h}))_{t\geq 0}$, the corresponding
property of $(N_{a}(t,\Lambda^{\prime}))_{t\geq 0}$ follows by letting
$h\rightarrow 0$. ∎

######  Corollary 9.

The first limit in (V.25) exists with

$n^{\prime}_{a}(\Lambda^{\prime})=\underset{t\rightarrow\infty}{\lim}\frac{N_{a}{(t,\Lambda^{\prime}})}{t}=\underset{t\rightarrow\infty}{\lim}\frac{N_{a}(L(t,a),\Lambda^{\prime})}{L(t,a)},\;\;a.s.$
( V.32 )

###### Proof.

It follows from the independent increment property of the process
$(N_{a}(t,\Lambda^{\prime}))_{t\geq 0}$ and the law of large numbers that

$\displaystyle\lim_{k\rightarrow\infty}\frac{N_{a}(k,\Lambda^{\prime})}{k}=n^{\prime}_{a}(\Lambda^{\prime}),\;\;a.s.$

The first equality in the statement follows by the usual interpolation
argument as in the proof of proposition 6. The second equality in the
statement follows from the first equality and the fact that
$L(t,a)\rightarrow\infty$ monotonically as $t\rightarrow\infty$, $a.s$. The
null set in the statement is obtained by the intersection of the null sets in
the above two steps. ∎

######  Remark 10.

The factorization of the reaction rate expression (VI.36) for
$\nicefrac{{N_{t}}}{{t}}$ as $t\rightarrow\infty$ in the positive recurrent
case contrasts sharply with the null recurrent case of Brownian motion (see
33).

Thus, (V.25) factorizes the reaction rate into a product of a local time term
and an excursion measure term $n^{\prime}_{a}(\Lambda^{\prime})$. This
suggests that the local time at $a$ represents the fluctuations of the process
$(X_{t})$ around $a$, and the excursion measure captures the ‘transition
probability’ of going from $a$ to $b$, which is consistent with the expression
for reaction rate. In the next section, we follow a alternate procedure using
renewal theory, familiar in the theory of discrete Markov processes, to obtain
the rate of the reaction. Using the expression thus obtained, we obtain an
explicit expression for the excursion measure
$n^{\prime}_{a}(\Lambda^{\prime})$.

##  VI Transition path excursions as renewal events

Regenerative processes, such as $(X_{t})$, can be decomposed into independent,
identically distributed blocks with varying path length, and an analysis that
exploits this structure of a regenerative process is provided by renewal
theory resnick1992adventures . Such blocks satisfy the renewal limit theorem,
which allows us to calculate the reaction rate in (II.13) which shall be the
concern of this section. We first begin with a brief introduction to the
theory of discrete renewal processes following 34.

Consider a sequence $\\{Y_{n}\\}_{n\geq 0}$ of independent, positive, real
valued random variables such that $\\{Y_{n}\\}_{n\geq 1}$ is identically
distributed with a distribution $F$, with $F(0-)=0$ and $F(0)<1$. Define the
renewal sequence, for $n\geq 0$,

$\displaystyle S_{n}=\sum_{i=0}^{n}Y_{i}$

The counting function, which counts the number of renewals until a given time
$t$, is:

$\displaystyle C(t):=\sum_{n=0}^{\infty}I_{[0,t]}(S_{n})$

The expectation of the counting function is the renewal function,

$\displaystyle\mu(t):=EC(t)$

If the distribution of $Y_{n}=S_{n}-S_{n-1},n\geq 1$ has finite mean $\mu$,
then we have the following Renewal theorem:

######  Theorem 2.

Renewal theorem
If the distribution $F$ of $Y_{i},i\geq 1$ has finite mean, i.e.
$EY_{1}=\mu<\infty$, then we have the following:

$\displaystyle\underset{t\rightarrow\infty}{\lim}\frac{C(t)}{t}=\underset{t\rightarrow\infty}{\lim}\frac{\mu(t)}{t}=\frac{1}{\mu}$

An important consequence of the regenerative property of the process $(X_{t})$
is the following ‘cycle decomposition’, which is sometimes taken as an
alternative definition of regenerative processes resnick1992adventures .

######  Proposition 11.

The process $\\{X_{t}\\}_{t\geq 0}$ has the following cycle decomposition:
There exists a renewal sequence $\\{S_{n}\\}_{n\geq 0}$, such that
$S_{n}\rightarrow\infty$, and for every $k,0<t_{1}<...<t_{k}$,
$B\in\mathcal{B}(\mathbb{R}_{+}^{k})$,
$A\in\mathcal{B}(\mathbb{R}_{+}^{\infty})$, $\forall n\geq 0$, we have:

$\displaystyle P_{a}[(X(S_{n}+t_{i}),i=1,2...,k)\in B,\\{S_{n+i}-S_{n},i\geq
1\\}\in A|\;S_{0},...S_{n}]=$ $\displaystyle P_{a}[(X(t_{i}),i=1,2...,k)\in
B,\\{S_{i}-S_{0},i\geq 1\\}\in A]$

###### Proof.

We take $S_{n}:=\tau_{n}$. In the strong Markov property, we take $F$ as
follows:

$\displaystyle F:=I_{\\{(X_{t_{1}},X_{t_{2}}...X_{t_{k}})\in
B,\;(S_{1},S_{2}...)\in A\\}}$ ( VI.33 )

Now, we note that

$\displaystyle
F\circ\theta_{\tau_{n}}=I_{\\{(X_{S_{n}+t_{1}},X_{S_{n}+t_{2}}...X_{S_{n}+t_{k}})\in
B,\;(S_{n+1}-S_{n},S_{n+2}-S_{n}...)\in A\\}}$ ( VI.34 )

Integrating (II.5) over a set
$G\in\sigma{\\{S_{0},S_{1}...S_{n}\\}}\subset\mathcal{F}_{\tau_{n}}$, we get
the equation in the proposition.

∎

Thus, $(X_{t})$ allows a cycle decomposition with
$\left\\{{\tau_{n}}\right\\}_{n\geq 0}$ as the renewal sequence. From section
II, it is clear that $\tau_{n}$ correspond to the time of the $n^{\text{th}}$
downcrossing of $(X_{t})$ from $b$ to $a$, so

$\displaystyle C^{(d)}(t):=\sum_{n=1}^{\infty}I_{[0,t]}(\tau_{n})$

becomes the counting function $C(t)$ corresponding to this renewal sequence,
for which the renewal limit theorem holds. For computing the reaction rate
$\kappa$ in (II.13), we need the number $N_{t}$ of reactive trajectories until
time $t$, which will correspond to the number $C^{(u)}(t)$ of upcrossings of
$(X_{t})$ from $a$ to $b$ until time $t$. But, since there cannot be a
downcrossing event without a corresponding upcrossing, we have:

$\displaystyle|{C^{(u)}(t)-C^{(d)}(t)}|\leq 1$

Denoting $\mu=E(\tau_{1}-\tau_{0})$, we have:

$\displaystyle\underset{t\rightarrow\infty}{\lim}\frac{N_{t}}{t}$
$\displaystyle=\underset{t\rightarrow\infty}{\lim}\frac{C^{(u)}(t)}{t}$
$\displaystyle=\underset{t\rightarrow\infty}{\lim}\frac{C(t)}{t}$
$\displaystyle=\frac{1}{\mu}$

Thus, from the renewal theorem, we get the required expression for the rate of
the reaction:

$\displaystyle\kappa=\lim_{t\rightarrow\infty}\frac{N_{t}}{t}=\frac{1}{\mu}=\frac{1}{E_{a}\tau_{1}}$
( VI.35 )

where the last equality follows from remark 2. The expression for reaction
rate obtained from excursion theory (V.25) and renewal theory should clearly
be the same, so we have:

$\displaystyle\kappa=\frac{1}{\mu}=\frac{1}{E_{a}\tau_{1}}=n^{\prime}_{a}(\Lambda^{\prime})\;\;\frac{E_{a}L(\tau_{1},a)}{E_{a}\tau_{1}}$
( VI.36 )

This expression implies that the product of excursion measure
$n^{\prime}_{a}(\Lambda^{\prime})$ and the expected local time at $a$ is
unity. From (V.29), we hence obtain:

$\displaystyle n^{\prime}_{a}(\Lambda^{\prime})\sim
e^{-\beta(U(0)-U(a))}=e^{-\beta V_{b}}$

Thus, renewal theory allows the computation of an explicit expression for the
excursion measure of the set $\Lambda^{\prime}$ corresponding to transition
path excursions from $a$ to $b$.

The forward and backward rate constant, $k_{A,B}$ and $k_{B,A}$ respectively,
can be computed from (VI.35) and remark 2 using eqn. (34) and (35) in 16. In
the context of our paper, it follows as in the proof of proposition 6 that
$\rho_{A}$, the proportion of time spent by $(X_{t})$ in upcrossings viz.
$\nicefrac{{\mathcal{T}_{U}(t)}}{{t}}$ as $t\rightarrow\infty$ is given as:

$\rho_{A}:=\lim_{t\rightarrow\infty}\frac{\mathcal{T}_{U}(t)}{t}=\frac{E_{a}T_{b}}{E_{a}\tau_{1}}$
( VI.37 )

where

$\displaystyle\mathcal{T}_{U}(t):=\int_{0}^{t}\sum_{i=1}^{\infty}I_{(\tau_{i-1},\sigma_{i})}(s)ds$
( VI.38 )

is the time spent in upcrossings during $(0,t)$. This yields the following
expression for the forward rate constant:

$\displaystyle
k_{A,B}=\frac{\kappa}{\rho_{A}}=\frac{1}{E_{a}T_{b}}=n^{\prime}_{a}(\Lambda^{\prime})\;\;\frac{E_{a}L(\tau_{1},a)}{E_{a}T_{b}}$
( VI.39 )

Moreover, in the context of Kramers reaction rate theory, $E_{a}T_{b}$ has
been interpreted as the mean first passage time hanggi1990reaction , that is,
the average time taken by the diffusion $(X_{t})$ to leave the potential well
at $a$ in $U(x)$ and reach $b$. In the overdamped regime that we consider, an
explicit expression has been obtained for the same hanggi1990reaction :

$\displaystyle E_{a}T_{b}=\frac{1}{D}\frac{2\pi
k_{B}T}{m\omega_{a}\omega_{0}}e^{\beta V_{b}}$ ( VI.40 )

where $\omega_{0}=m^{-1}U^{\prime\prime}(0)$ and
$\omega_{a}=m^{-1}U^{\prime\prime}(a)$ are constants that depend on the second
derivative of $U(x)$ at the origin and $a$ respectively. Comparing the above
equation with (VI.39), we get:

$\displaystyle\frac{E_{a}L(\tau_{1},a)}{E_{a}T_{b}}\sim
D\frac{m\omega_{a}\omega_{0}}{2\pi k_{B}T}$ ( VI.41 )

as $E_{a}L(\tau_{1},a)\sim e^{\beta V_{b}}$.

##  VII Conclusion

The factorization of the reaction rate into an excursion measure term and a
local time fluctuation term, as in the RHS of (VI.36) and (VI.39), allows a
direct comparison with the Kramers (and TST) rate expression in the
Smoluchowski (high friction/overdamped) limit. We emphasize that the
factorization of the rate expression is expected to hold even in cases where
an explicit transition state cannot be identified in the PES, the type of
problems for which TPS and TPT are useful; the existence of the limits in
(VI.36) is only conditional upon $(X_{t})$ being positive recurrent and
regenerative, which is a reasonable assumption for systems with discernible
‘reactants’ and ‘products’ metastable states. Since the expression for
reaction rate remains structurally the same for all elementary chemical
reaction, we expect this factorization to hold even for general
multidimensional potential energy surfaces. Hence we believe that further work
using an excursion-theoretic interpretation of chemical reactions can
potentially unravel the relevant fluctuation timescales intrinsic to the
underlying PES.

###### Acknowledgements.

We thank Stuart Althorpe for reading through the introduction of the
manuscript and providing useful suggestions. V.G.S. acknowledges support from
St. John’s College, University of Cambridge, through a Dr. Manmohan Singh
Scholarship. Both V.G.S. and R.B. acknowledge funding from a SERB matrix grant
(#MTR/2017/000750).

## Conflict of interest

The authors have no conflicts to disclose.

## Data Availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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