# Duality symmetry conjugates of the quantum Rabi model : effective bosonic,
fermionic and coupling-only dynamical properties

Joseph Akeyo Omolo
Department of Physics, Maseno University, P.O. Private Bag, Maseno, Kenya
e-mail<EMAIL_ADDRESS>
(01 December 2021)

###### Abstract

Symmetry transformations have proved useful in determining the algebraic
structure and internal dynamical properties of physical systems. In the
quantum Rabi model, invariance under parity symmetry transformation has been
used to obtain exact solutions of the eigenvalue equation and very good
approximations of the internal dynamics of the interacting atom-light system.
In this article, two symmetry operators, characterized as “duality” symmetry
operators, have been introduced which transform the quantum Rabi Hamiltonian
into duality conjugates. The parity and duality symmetry operators constitute
an algebraically closed set of symmetry transformation operators of the
quantum Rabi model. The closed $SU(2)$ Lie algebra provides the standard
eigenvalues and eigenstates of the parity symmetry operator. It is established
that Jaynes-Cummings and anti-Jaynes-Cummings operators are duality symmetry
conjugates. Symmetric or antisymmetric linear combinations of the Rabi
Hamiltonian and a corresponding duality conjugate yield the familiar spin-
dependent force driven bosonic , coupling-only or quantized light mode
quadrature-driven fermionic Hamiltonian. It is established that the effective
bosonic, fermionic and coupling-only Hamiltonians are exact, not approximate
forms of the quantum Rabi Hamiltonian as they have generally been interpreted.
The effective bosonic form generates the dynamics of the light mode driven by
the atomic spin-dependent force, while the fermionic form generates the
dynamics of the atomic spin driven by the quantized light mode quadrature-
dependent force, thus providing a complete picture of the quantum Rabi
dynamics. Dynamical evolution of the full quantum Rabi model is thus described
by entangled coherent light Schroedinger cat and atomic spin states of the
effective bosonic or coupling-only Hamiltonian, or, by generalized atomic spin
coherent states of the effective fermionic Hamiltonian. These nonclassical
states of the simpler effective forms of QRM Hamiltonian have revealed
fundamental quantum features observed in experiments and have been useful in
developing various quantum technology applications.

##  1 Introduction

The quantum Rabi model (QRM) of a two-level atom interacting with a quantized
light mode is generated by Hamiltonian

$H=H_{0}+H_{I}\ ;\quad\quad
H_{0}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega_{0}s_{z}\ ;\quad\quad
H_{I}=\hbar g(\hat{a}+\hat{a}^{\dagger})(s_{+}+s_{-})$ $None$

where $H_{0}$ , $H_{I}$ are the free evolution and interaction components.
Here, $\omega$ , $\hat{a}$ , $\hat{a}^{\dagger}$ are the quantized light mode
angular frequency, annihilation and creation operators, while $\omega_{0}$ ,
$s_{z}$ , $s_{+}$ , $s_{-}$ , $\sigma_{x}=s_{-}+s_{+}$ are the atomic state
transition angular frequency and operators. The light mode vacuum state energy
$\frac{1}{2}\hbar\omega$ has been ignored, but can be reintroduced as desired.

Due to its purely quantum nature, QRM has been the center of focus of both
theory and experiments in quantum optics seeking to understand fundamental
quantum mechanical properties and their potential applications to quantum
technology development. The QRM Hamiltonian $H$ in equation (1$a$) takes a
standard form applicable to a broad spectrum of physical processes based on
light-matter interactions such as atoms or electrically charged particles in
magnetic fields, cavity and circuit quantum electrodynamics, quantum dots,
trapped ions, polaritonic physics, superconducting quantum circuits, etc,
which are well described in the excellent reviews [1 , 2].

While great breakthroughs have been made in providing exact analytical
solutions of the QRM eigenvalue equations generated by $H$ [3 , 4] based on
the parity symmetry property, determining an exact general solution of the
corresponding time evolution equation remains a major challenge of theoretical
quantum optics. The spectrum of eigenvalues and eigenstates obtained in the
exact solutions in the Bargmann space in [ 3 , 4] and related subsequent work
not cited here turns out too complicated, meaning that some approximations
still have to be applied in determining the time evolving state vector [5 ,
6]. Faced with observable discrepancies between the experimental results [1 ,
2 , 7-11] and the basic Jaynes-Cummings (JC) model obtained in the rotating
wave approximation (RWA) [12 , 13 , 14], great effort has been made developing
more sophisticated effective approximations to the QRM Hamiltonian $H$
[15-19]. Even though the results of these effective approximations are closer
to experimental observations over the desired coupling ranges from weak to
ultra/deep strong coupling values, some subtle features such as excitation-
dependent damping rates are not yet properly captured.

Where, then, is the main problem of theory ? The answer to this basic question
lies in the full quantum operator form of the QRM Hamiltonian $H$ in equation
(1$a$). Since the quantized light mode and atomic spin operators have basic
algebraic properties, an accurate description of QRM must necessarily specify
the complete algebraic structure and symmetry transformation properties of the
Hamiltonian. This means defining all the composite atom-light dynamical
operators such as the composite excitation number, population inversion and
related operators, which properly characterize the algebraic symmetry and
dynamical properties generated by the QRM Hamiltonian. Specifying and using
only one or an incomplete set of these composite dynamical operators just
gives an algebraically limited approximate theory.

In particular, operators which generate symmetry transformations play a key
role in determining the algebraic structure and internal dynamical properties
of a system. Operators which commute with the Hamiltonian generate symmetry
transformations which leave the system invariant. Such operators are useful in
determining the exact eigenvalues and eigenstates of the system through
diagonalization of the Hamiltonian. Equally important are operators generating
symmetry transformations which determine the conjugates of the system. Besides
providing additional insights into the structure of the system, such symmetry
transformations often yield simpler forms of the system Hamiltonian which are
exactly solvable or much easier to handle in fairly accurate approximations.
The complete symmetry structure of a system is thus characterized by operators
which generate invariance and conjugation transformations. It is not
sufficient to define only operators which commute with the Hamiltonian and
generate invariance transformations.

The underlying problem within the current theoretical framework of QRM may now
be well explained. The great theoretical advances in studies of the symmetry
properties of QRM have focussed attention only on symmetry transformation
operators which commute with the Hamiltonian $H$ in equation (1$a$) or its
fairly more general biased asymmetric form $H+\epsilon\sigma_{x}$ [3 , 4 , 17
, 20-23]. The parity symmetry operator $\hat{\Pi}=e^{\pm
i\pi(\hat{a}^{\dagger}\hat{a}+s_{z})}$, appropriately expressed in the form
$\hat{\Pi}=\sigma_{z}{\cal P}$ (${\cal P}=e^{-i\pi\hat{a}^{\dagger}\hat{a}}$),
commutes with the basic QRM Hamiltonian $H$ [3 , 4 , 17 , 20], while the
recently discovered generalized hidden symmetry operator ${\bf J}_{\epsilon}$
($\epsilon={\frac{l}{2}}~{},~{}l=0,1,2,3,...$) commutes with the asymmetric
QRM Hamiltonian $H+\epsilon\sigma_{x}$ [21 , 22 , 23] where the biasing
parameter $\epsilon$ takes integer and half-integer values as defined. It is
established in [21 , 22 , 23] that the general form of the hidden symmetry
operator includes the basic parity symmetry operator as the $l=0$
($\epsilon=0)$ case, i.e., $\hat{\Pi}={\bf J}_{0}$. The parity and the
generalized hidden symmetry operators generate invariance transformations and
are useful in determining the exact eigenvalues and eigenstates of the QRM
system through diagonalization of the Hamiltonian $H$ , $H+\epsilon\sigma_{x}$
[3 , 4 , 20-23]. In addition, the parity symmetry has been used in [6 , 17]
and the related experiments [10 , 11] to gain insight into the internal
dynamics of QRM, revealing fundamental quantum features in the ultra/deep
strong coupling regimes.

Unfortunately, as explained earlier, the specification of only the parity and
the generalized hidden symmetry operators does not define the complete
symmetry structure and transformation properties of QRM. Other symmetry
operators which generate transformations of the QRM Hamiltonian into its
corresponding conjugates need also be specified. Indeed, it is established in
this article that the parity operator is part of an algebraically closed set
of symmetry transformation operators $\hat{\Pi}_{j}=\sigma_{j}{\cal P}$,
$j=z,y,x$, where $\hat{\Pi}_{z}$ is the parity symmetry operator, while
$\hat{\Pi}_{y}$ , $\hat{\Pi}_{x}$ are identified as “duality” symmetry
operators which generate symmetry conjugates of the QRM Hamiltonian. In
particular, the duality symmetry transformations reveal an important algebraic
property that corresponding JC and aJC operators are duality conjugates.

In progressing towards introduction of the complete set of parity and duality
symmetry operators $\hat{\Pi}_{j}=\sigma_{j}{\cal P}$, $j=z,y,x$, it becomes
important to note that the parity symmetry operator $\hat{\Pi}$ is generated
by the JC excitation number operator
$\hat{N}_{JC}=\hat{a}^{\dagger}\hat{a}+s_{z}$, generally considered to be the
only excitation number operator in QRM used extensively in both theory and
experiments [1 , 6 , 10 , 11 , 17], yet a simple symmetrization of the free
evolution component $H_{0}$ of QRM Hamiltonian in equation (1$a$) in the form
( $2(aA+bB)=(a+b)(A+B)+(a-b)(A-B)$ )

$H_{0}=\frac{1}{2}\hbar(\delta_{+}\hat{N}_{JC}+\delta_{-}\hat{N}_{aJC})\
;\quad\quad\hat{N}_{JC}=\hat{a}^{\dagger}\hat{a}+s_{z}\
;\quad\quad\hat{N}_{aJC}=\hat{a}^{\dagger}\hat{a}-s_{z}\
;\quad\quad\delta_{\pm}=\omega\pm\omega_{0}$ $None$

reveals that, besides the well known JC excitation number operator
$\hat{N}_{JC}$, the algebraic structure of QRM is also characterized by
another excitation number operator $\hat{N}_{aJC}$, identified as the anti-
Jaynes-Cummings (aJC) excitation number operator, first constructed and proved
conserved in aJC interaction in [23]. It is also proved explicitly in [23]
that the aJC excitation number operator generates the same parity symmetry
operator of QRM according to $\hat{\Pi}=e^{\pm i\pi\hat{N}_{JC}}=e^{\pm
i\pi\hat{N}_{aJC}}$. In general, $\hat{N}_{JC}$ , $\hat{N}_{aJC}$ generate the
respective $U(1)$ symmetry operators of the JC and aJC Hamiltonians. As
demonstrated below, the two commuting excitation number operators are related
by $\hat{\Pi}_{y}=\sigma_{y}{\cal P}$ , $\hat{\Pi}_{x}=\sigma_{x}{\cal P}$
symmetry transformations as duality conjugates and both must be specified in a
complete algebraic structure of QRM.

For completeness, the interaction Hamiltonian $H_{I}$ in equation (1$a$) may
also decomposed into a sum of JC and aJC components in the familiar form

$H_{I}=H_{IJC}+H_{IaJC}\ ;\quad\quad H_{IJC}=\hbar
g(\hat{a}^{+}s_{-}+\hat{a}s_{+})\ ;\quad\quad H_{IaJC}=\hbar
g(\hat{a}s_{-}+\hat{a}^{+}s_{+})$ $None$

so that substituting equations (1$b$) , (1$c$) into equation (1$a$) provides a
decomposition of the QRM Hamiltonian into JC and aJC components in the form

$H=H_{JC}+H_{aJC}\ ;\quad\quad
H_{JC}=\frac{1}{2}\hbar(\delta_{+}\hat{N}_{JC}+H_{IJC}\ ;\quad\quad
H_{aJC}=\frac{1}{2}\hbar(\delta_{-}\hat{N}_{aJC}+H_{IaJC}$ $None$

In general, the JC and aJC components are found to be duality conjugates under
the symmetry transformations generated by the operators $\hat{\Pi}_{y}$ ,
$\hat{\Pi}_{x}$.

It is now noted that, in addition to the excitation number operators which
generate the parity symmetry operator, two other symmetry transformation
operators which determine QRM “duality” symmetry conjugates exist as defined
in this article. The “duality” symmetry transformations by both
$\hat{\Pi}_{y}$ and $\hat{\Pi}-x$ map the JC , aJC excitation number operators
into each other. For the JC , aJC interaction Hamiltonians, the operator
$\hat{\Pi}_{y}$ generates a _symmetric_ duality transformation $H_{IJC}\to
H_{IaJC}$ , $H_{IaJC}\to H_{JC}$, while the operator $\hat{\Pi}_{x}$ generates
an _antisymmetric_ duality transformation $H_{IJC}\to-H_{IaJC}$ ,
$H_{IaJC}\to-H_{IJC}$. It then follows from equation (1$c$) that the duality
symmetry transformations either map the full QRM interaction Hamiltonian
$H_{I}$ onto itself ($H_{I}\to H_{I}$) or onto its mirror image ($H_{I}\to-
H_{I}$), thus leading to characterization as symmetric or antisymmetric
transformations. Based on the standard property that the sum or difference of
an operator and its conjugate yields an effective operator, it is established
that the sum (symmetric linear combination) of the QRM Hamiltonian $H$ and its
symmetric duality conjugate yields an effective _bosonic_ Hamiltonian, while
the difference (antisymmetric linear combination) with the antisymmetric
duality conjugate yields an effective _fermionic_ Hamiltonian. A product of
the duality symmetry operators provides a transform of the QRM Hamiltonian by
changing only the sign of the interaction Hamiltonian $H_{I}$, so that the
difference of the Hamiltonian and the transform provides a useful effective
_coupling-only_ QRM Hamiltonian. It emerges that the effective bosonic,
fermionic and coupling-only QRM Hamiltonians take familiar forms usually
obtained as approximations in the general theoretical methods [16-19 , 25].

This article is organized as follows. QRM symmetry operators are introduced in
section $2$, where the transformation properties on the basic quantized light
mode and atomic spin operators are presented. In section $3$, the symmetry
transformations are applied on the QRM Hamiltonian $H$ to determine invariance
and duality conjugation properties. Basic dynamical properties of the
symmetric and antisymmetric forms of QRM Hamiltonian are discussed briefly.
Section $4$ contains the Conclusion.

##  2 Symmetry transformation operators

A set of symmetry transformation operators $\hat{N}_{j}$, $j=z,y,x$ are
introduced, defined in terms of the light mode excitation number
$\hat{a}^{\dagger}\hat{a}$ and the atomic spin operators
$s_{j}=\frac{1}{2}\sigma_{j}$ in the form

$\hat{N}_{z}=\hat{a}^{\dagger}\hat{a}+s_{z}\
;\quad\quad\hat{N}_{y}=\hat{a}^{\dagger}\hat{a}+s_{y}\
;\quad\quad\hat{N}_{x}=\hat{a}^{\dagger}\hat{a}+s_{x}$ $None$

where $\hat{N}_{z}$ is just the JC excitation number operator $\hat{N}_{JC}$
defined in the Introduction. These operators generate symmetry transformation
operators of the general form

$U_{j}(\theta)=e^{\pm i\theta\hat{N}_{j}}\ ;\quad j=z,y,x$ $None$

where $\theta$ is taken real in these definitions. The special case
$\theta=\pi$ provides the basic symmetry operators of interest in this
article, which after substituting $\hat{N}_{j}$ from equation (2$a$) and
reorganizing take the form

$\hat{\Pi}_{j}=e^{\pm
i\pi\hat{N}_{j}}\quad\quad\Rightarrow\quad\quad\hat{\Pi}_{j}=\sigma_{j}{\cal
P}\ ;\quad\quad{\cal P}=e^{\pm i\hat{a}^{\dagger}\hat{a}}\ ;\quad j=z,y,x$
$None$

in the conventional representation adopted in [17 , 21 , 22 , 23], noting that
for $j=z$, the operator $\hat{\Pi}_{z}$ is just the parity symmetry operator
$\hat{\Pi}$ defined earlier. It is easily established that the operators
$\hat{\Pi}_{j}$ as defined in equation (2$c$) satisfy a closed $SU(2)$
symmetry group algebra

$\\{\hat{\Pi}_{j}~{},~{}\hat{\Pi}_{k}\\}=2\delta_{jk}\
;\quad\quad[~{}\hat{\Pi}_{j}~{},~{}\hat{\Pi}_{k}]=2i\epsilon_{jkl}\hat{\Pi}_{l}{\cal
P}$ $None$

noting that ${\cal P}^{2}=I$ in the computational space. By introducing
$\hat{\Pi}_{\pm}=\sigma_{\pm}{\cal P}$ ($\sigma_{\pm}=\sigma_{x}\pm
i\sigma_{y}$), the $SU(2)$ Lie algebra of the closed set
$\\{~{}\hat{\Pi}_{z}~{},~{}\hat{\Pi}_{+}~{},~{}\hat{\Pi}_{-}\\}$ easily
provides the standard eigenvalues and eigenstates of the parity symmetry
operator $\hat{\Pi}_{z}$.

The operators $\hat{\Pi}_{j}$ generate symmetry transformations of the light
mode and atomic spin operators in the form

$\hat{\Pi}_{z}^{\dagger}\hat{a}\hat{\Pi}_{z}=-\hat{a}\
;\quad\quad\hat{\Pi}_{z}^{\dagger}\hat{a}^{\dagger}\hat{\Pi}_{z}=-\hat{a}^{\dagger}\
;\quad\quad\hat{\Pi}_{z}^{\dagger}s_{z}\hat{\Pi}_{z}=s_{z}\
;\quad\quad\hat{\Pi}_{z}^{\dagger}s_{\mp}\hat{\Pi}_{z}=-s_{\mp}$ $None$
$\hat{\Pi}_{y}^{\dagger}\hat{a}\hat{\Pi}_{y}=-\hat{a}\
;\quad\quad\hat{\Pi}_{y}^{\dagger}\hat{a}^{\dagger}\hat{\Pi}_{y}=-\hat{a}^{\dagger}\
;\quad\quad\hat{\Pi}_{y}^{\dagger}s_{z}\hat{\Pi}_{y}=-s_{z}\
;\quad\quad\hat{\Pi}_{y}^{\dagger}s_{\mp}\hat{\Pi}_{y}=-s_{\pm}$ $None$
$\hat{\Pi}_{x}^{\dagger}\hat{a}\hat{\Pi}_{x}=-\hat{a}\
;\quad\quad\hat{\Pi}_{x}^{\dagger}\hat{a}^{\dagger}\hat{\Pi}_{x}=-\hat{a}^{\dagger}\
;\quad\quad\hat{\Pi}_{x}^{\dagger}s_{z}\hat{\Pi}_{x}=-s_{z}\
;\quad\quad\hat{\Pi}_{x}^{\dagger}s_{\mp}\hat{\Pi}_{x}=s_{\pm}$ $None$

The transformations in equation (2$e$) confirm that $\hat{\Pi}_{z}$ is the
standard parity symmetry operator of QRM , which leaves the Hamiltonian $H$ in
equation (1$a$) invariant. New QRM symmetry transformation properties are
generated by the operators $\hat{\Pi}_{y}$ , $\hat{\Pi}_{x}$, which according
to the actions on the atomic spin operators $s_{\mp}$ in equations (2$f$) ,
(2$g$), may be interpreted as “duality” symmetry operators.

##  3 QRM parity and duality symmetry conjugates

Applying the operator $\hat{\Pi}_{z}$ symmetry transformation, equivalent to
the parity symmetry transformation, on the JC , aJC operators and the QRM
Hamiltonian in equations (1$b$)-(1$d$), then using the relations obtained in
equation (2$e$) leaves the operators invariant according to

$\hat{\Pi}_{z}^{\dagger}~{}\hat{N}_{JC}~{}\hat{\Pi}_{z}=\hat{N}_{JC}\
;\quad\hat{\Pi}_{z}^{\dagger}~{}\hat{N}_{aJC}~{}\hat{\Pi}_{z}=\hat{N}_{aJC}\
;\quad\quad\hat{\Pi}_{z}^{\dagger}~{}H_{IJC}~{}\hat{\Pi}_{z}=H_{IJC}\
;\quad\hat{\Pi}_{z}^{\dagger}~{}H_{IaJC}~{}\hat{\Pi}_{z}=H_{IaJC}$
$\hat{\Pi}_{z}^{\dagger}~{}H_{0}~{}\hat{\Pi}_{z}=H_{0}\
;\quad\quad\hat{\Pi}_{z}^{\dagger}~{}H_{I}~{}\hat{\Pi}_{z}=H_{I}\
;\quad\quad\hat{\Pi}_{z}^{\dagger}~{}H~{}\hat{\Pi}_{z}=H$ $None$

which provides the standard parity symmetry of QRM. The associated commutation
relation $[~{}\hat{\Pi}_{z}~{},~{}H~{}]=0$ follows easily from the invariance
relation.

Applying the operators $\hat{\Pi}_{y}$ , $\hat{\Pi}_{x}$ transformations
obtained in equations (2$f$) , (2$g$) on the excitation number operators and
the free evolution Hamiltonian provides the duality symmetry transformations

$j=y~{},~{}x~{}:\quad\quad\hat{\Pi}_{j}^{\dagger}~{}\hat{N}_{JC}~{}\hat{\Pi}_{j}=\hat{N}_{aJC}\
;\quad\quad\hat{\Pi}_{j}^{\dagger}~{}\hat{N}_{aJC}~{}\hat{\Pi}_{j}=\hat{N}_{JC}$
$\hat{\Pi}_{j}^{\dagger}~{}H_{0}~{}\hat{\Pi}_{j}=\overline{H}_{0}\
;\quad\quad\hat{\Pi}_{j}^{\dagger}~{}\overline{H}_{0}~{}\hat{\Pi}_{j}=H_{0}\
;\quad\quad\overline{H}_{0}=\frac{1}{2}\hbar(\delta_{+}\hat{N}_{aJC}+\delta_{-}\hat{N}_{JC})$
$None$

revealing that the JC and aJC excitation number operators $\hat{N}_{JC}$ ,
$\hat{N}_{aJC}$ are duality symmetry conjugates, while $\overline{H}_{0}$ is
the duality symmetry conjugate of the free evolution Hamiltonian $H_{0}$.

Similarly, applying the operators $\hat{\Pi}_{y}$ , $\hat{\Pi}_{x}$
transformations on the JC , aJC interaction Hamiltonians provides the duality
symmetry transformations

$\hat{\Pi}_{y}^{\dagger}~{}H_{IJC}~{}\hat{\Pi}_{y}=H_{IaJC}\
;\quad\quad\hat{\Pi}_{y}^{\dagger}~{}H_{IaJC}~{}\hat{\Pi}_{y}=H_{IJC}$
$\hat{\Pi}_{x}^{\dagger}~{}H_{IJC}~{}\hat{\Pi}_{x}=-H_{IaJC}\
;\quad\quad\hat{\Pi}_{x}^{\dagger}~{}H_{IaJC}~{}\hat{\Pi}_{x}=-H_{IJC}$ $None$

The JC and aJC interaction Hamiltonians, which have generally been interpreted
as the rotating and counter-rotating components of the QRM Hamiltonian, are
here identified as duality symmetry conjugates. It is then important to note
that, just as the free evolution Hamiltonian $H_{0}$ is a sum of duality
symmetry conjugates according to equations (1$b$) , (3$b$), the interaction
Hamiltonian $H_{I}$ is also a sum of duality symmetry conjugates according to
equations (1$c$) , (3$c$).

It emerges here that both duality symmetry operators $\hat{\Pi}_{y}$ ,
$\hat{\Pi}_{x}$ transform $H_{0}$ , $\overline{H}_{0}$ directly into each
other ($H_{0}\leftrightarrow\overline{H}_{0}$) without sign differences
($\pm$). However, it follows from the relations in equations (1$c$) , (3$c$)
that $\hat{\Pi}_{y}$ maps the interaction Hamiltonian $H_{I}$ onto itself
($H_{I}\to H_{I}$), while $\hat{\Pi}_{x}$ generates the mirror image
transformation $H_{I}\to-H_{I}$. This property leads to an interpretation that
$\hat{\Pi}_{y}$ generates symmetric and $\hat{\Pi}_{x}$ antisymmetric duality
symmetry conjugations of QRM, which are treated separately.

###  3.1 Symmetric QRM duality conjugation

Applying the $\hat{\Pi}_{y}$ transformation on $H_{I}$ in equation (1$c$) and
using the corresponding relations from equation (3$c$) generates the symmetric
duality transformation

$\hat{\Pi}_{y}^{\dagger}~{}H_{I}~{}\hat{\Pi}_{y}=H_{I}\quad\quad\Rightarrow\quad\quad[~{}\hat{\Pi}_{y}~{},~{}H_{I}]=0$
$None$

In a standard interpretation, the symmetric duality transformation, governed
by the specified commutation relation, leaves the interaction Hamiltonian
$H_{I}$ invariant. It follows from equations (3$b$) , (4$a$) that application
of $\hat{\Pi}_{y}$ transforms the QRM Hamiltonian $H$ in equation (1$a$) into
its symmetric duality conjugate $\overline{H}_{+}$ according to

$\hat{\Pi}_{y}^{\dagger}~{}H~{}\hat{\Pi}_{y}=\overline{H}_{+}\
;\quad\quad\hat{\Pi}_{y}^{\dagger}~{}\overline{H}_{+}~{}\hat{\Pi}_{y}=H\
;\quad\quad\overline{H}_{+}=\overline{H}_{0}+H_{I}$ $None$

####  3.1.1 Symmetric QRM Hamiltonian : effective bosonic dynamics

Taking a symmetric linear combination of the QRM Hamiltonian $H$ in equation
(1$a$) and its symmetric duality conjugate $\overline{H}_{+}$ in equation
(4$b$), then using the definitions from equations (1$a$) , (1$b$) , (3$b$) ,
(4$b$), provides the symmetric QRM Hamiltonian $H_{+}$ obtained as

$H_{+}=\frac{1}{2}(H+\overline{H}_{+})\quad\quad\Rightarrow\quad\quad
H_{+}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar
g(\hat{a}+\hat{a}^{\dagger})\sigma_{x}$ $None$

after introducing the Pauli spin operator $\sigma_{x}=s_{-}+s_{+}$ to give a
familiar form. This form shows that the symmetric QRM Hamiltonian $H_{+}$ is
an effective bosonic Hamiltonian describing the dynamics of the quantized
light mode driven by the atomic spin-dependent force. Notice that $H_{+}$ is
invariant under the symmetric $\hat{\Pi}_{y}$ duality symmetry transformation
according to

$\hat{\Pi}_{y}^{\dagger}~{}H_{+}\hat{\Pi}_{y}=H_{+}\quad\quad\Rightarrow\quad\quad[~{}\hat{\Pi}_{y}~{},~{}H_{+}~{}]=0$
$None$

The effective bosonic Hamiltonian $H_{+}$ first arose as an exactly solvable
degenerate spin state ($\omega_{0}=0$) approximation of QRM in [17] and has
inspired in-depth theoretical and experimental investigations of QRM dynamics
under spin-dependent forces over the entire coupling parameter range [1 , 2 ,
4 , 10 , 11 , 18 , 19 , 25-31]. The simple derivation through symmetric
duality transformations in equations (4$b$) , (5$a$) reveals the important
physical property that the effective bosonic Hamiltonian $H_{+}$ is an exact
symmetric QRM Hamiltonian, entirely independent of the atomic spin state
transition angular frequency $\omega_{0}$, thus applicable to both degenerate
($\omega_{0}=0$) and non-degenerate ($\omega_{0}\neq 0$) spin state cases.

The bosonic nature of $H_{+}$ is easily demonstrated by introducing composite
hermitian conjugate operators $\hat{b}$ , $\hat{b}^{\dagger}$ satisfying
bosonic algebra according to

$\hat{b}=\hat{a}\sigma_{x}\
;\quad\hat{b}^{\dagger}=\hat{a}^{\dagger}\sigma_{x}\
;\quad\sigma_{x}^{2}=I~{}:\quad\quad[~{}\hat{b}~{},~{}\hat{b}^{\dagger}~{}]=1\
;\quad\quad[~{}\hat{b}~{},~{}\hat{b}~{}]=0\
;\quad\quad[~{}\hat{b}^{\dagger}~{},~{}\hat{b}^{\dagger}~{}]=0$ $None$

where $I$ is the $2\times 2$ identity matrix. The Hamiltonian in equation
(5$a$) now takes the form

$H_{+}=\hbar\omega\hat{b}^{\dagger}\hat{b}+\hbar g(\hat{b}+\hat{b}^{\dagger})$
$None$

Application of the Heisenberg equation of motion for the operator $\hat{b}$
gives time evolution reorganized in the appropriate form

$\hat{b}(t)=e^{-i\omega t}\left(\hat{b}+\frac{g}{\omega}(1-e^{i\omega
t})\right)\ ;\quad\quad\hat{b}=\hat{b}(0)$ $None$

which is a displaced bosonic state annihilation operator easily obtained
through a time evolution operator $U(t)$ according to

$U(t)=e^{-i\omega t\hat{b}^{\dagger}\hat{b}}D(\beta(t))\ ;\quad
D(\beta(t))=e^{\beta(t)\hat{b}^{\dagger}-\beta^{*}(t)\hat{b}}\
;\quad\beta(t)=\frac{g}{\omega}(1-e^{i\omega t})\
;\quad\quad\hat{b}(t)=U^{\dagger}(t)~{}\hat{b}~{}U(t)$ $None$

noting

$U^{\dagger}(t)=D^{\dagger}(\beta(t))e^{i\omega
t\hat{b}^{\dagger}\hat{b}}~{}:\quad\quad D^{\dagger}(\beta(t))e^{i\omega
t\hat{b}^{\dagger}\hat{b}}\hat{b}e^{-i\omega
t\hat{b}^{\dagger}\hat{b}}D(\beta(t))=e^{-i\omega
t}D^{\dagger}(\beta(t))\hat{b}D(\beta(t))$ $None$

Determination of the time evolution operator $U(t)$ in equation (5$f$)
provides the exact solution of the dynamics generated by the bosonic QRM
Hamiltonian $H_{+}$. Taking the atom initially in the excited state
$|e\rangle$, the light mode in the vacuum state $|0\rangle$ and introducing
the $\sigma_{x}$ eigenstates
$|\pm\rangle=\frac{1}{\sqrt{2}}(|e\rangle\pm|g\rangle)$ with
$\sigma_{x}|\pm\rangle=\pm|\pm\rangle$, the general time evolving state
$|\Psi(t)\rangle=U(t)|e0\rangle$ is obtained as an entangled atom-light state
in the form

$|\Psi(t)\rangle=|\beta_{+}(t)\rangle|e\rangle+|\beta_{-}(t)\rangle|g\rangle\
;\quad\quad|\beta_{\pm}(t)\rangle=\frac{1}{\sqrt{2}}(|\beta(t)\rangle\pm|-\beta(t)\rangle)$
$None$

where $|\beta_{\pm}(t)\rangle$ as defined is the coherent light mode
Schroedinger cat state. The time evolving state $|\Psi(t)\rangle$ of the
effective bosonic Hamiltonian is therefore an entanglement of the coherent
light Schroedinger cat and atomic spin states. The physical interpretation is
that in the effective bosonic dynamics driven by the atomic spin-dependent
force, the dynamical evolution of the full quantum Rabi model is described by
the general time evolving entangled coherent light Schroedinger cat and atomic
spin state $|\Psi(t)\rangle$ in equation (5$h$). The appropriate order
parameter for studying the dynamical evolution is the mean value of the
bosonic excitation number operator
$\hat{b}^{\dagger}\hat{b}=\hat{a}^{\dagger}\hat{a}$ in the state
$|\Psi(t)\rangle$. The fundamental quantum properties and practical
applications of QRM dynamics described by these nonclassical bosonic atom-
light entangled states are well established in the theoretical and
experimental studies [1 , 4 , 6 , 10 , 17 , 19 , 26] where details which need
not be repeated here can be found.

###  3.2 Antisymmetric QRM duality conjugation

Applying the $\hat{\Pi}_{x}$ transformation on $H_{I}$ in equation (1$c$) and
using the corresponding relations from equation (3$c$) generates the
antisymmetric duality transformation

$\hat{\Pi}_{x}^{\dagger}~{}H_{I}~{}\hat{\Pi}_{x}=-H_{I}\quad\quad\Rightarrow\quad\quad\\{~{}\hat{\Pi}_{x}~{},~{}H_{I}\\}=0$
$None$

In an interpretation, the antisymmetric duality transformation, governed by
the specified anticommutation relation, leaves the operator form of the
interaction Hamiltonian $H_{I}$ invariant, but maps it onto its mirror image.
It follows from equations (3$b$) , (6$a$) that application of $\hat{\Pi}_{x}$
transformation on the QRM Hamiltonian $H$ in equation (1$a$) generates the
antisymmetric duality conjugate $\overline{H}_{-}$ according to

$\hat{\Pi}_{x}^{\dagger}~{}H~{}\hat{\Pi}_{x}=\overline{H}_{-}\
;\quad\quad\hat{\Pi}_{x}^{\dagger}~{}\overline{H}_{-}~{}\hat{\Pi}_{x}=H\
;\quad\quad\overline{H}_{-}=\overline{H}_{0}-H_{I}$ $None$

####  3.2.1 Antisymmetric QRM Hamiltonian : effective fermionic dynamics

Taking an antisymmetric linear combination of the QRM Hamiltonian $H$ in
equation (1$a$) and its antisymmetric duality conjugate $\overline{H}_{-}$ in
equation (6$b$), then using the definitions from equations (1$a$) , (1$b$) ,
(3$b$) , (6$b$), provides the antisymmetric QRM Hamiltonian $H_{-}$ obtained
as

$H_{-}=\frac{1}{2}(H-\overline{H}_{-})\quad\quad\Rightarrow\quad\quad
H_{-}=\hbar\omega_{0}s_{z}+\hbar g(\hat{a}+\hat{a}^{\dagger})\sigma_{x}$
$None$

This form shows that the antisymmetric QRM Hamiltonian $H_{-}$ is an effective
fermionic Hamiltonian describing the dynamics of the atomic spin driven by the
quantized light mode quadrature-dependent force. The QRM fermionic Hamiltonian
$H_{-}$ maps onto its mirror image under the antisymmetric $\hat{\Pi}_{x}$
duality transformation according to

$\hat{\Pi}_{x}^{\dagger}~{}H_{-}~{}\hat{\Pi}_{x}=-H_{-}\quad\quad\Rightarrow\quad\quad\\{~{}\hat{\Pi}_{x}~{},~{}H_{-}~{}\\}=0$
$None$

It is remarkable that the QRM fermionic Hamiltonian $H_{-}$ in equation (7$a$)
is similar to the one-dimensional Dirac Hamiltonian for a fermion in
relativistic quantum mechanics, which has inspired quantum simulations of the
Dirac equation with trapped ions in quantum optics [1 , 31 , 32].

Unlike the corresponding bosonic Hamiltonian $H_{+}$ in equation (5$a$) which
is expressible in terms of composite operators satisfying bosonic algebra
according to equations (5$c$) , (5$d$), the QRM fermionic Hamiltonian $H_{-}$
with a free evolution spin-only component in equation (7$a$) is not
expressible in terms of composite atom-light operators satisfying fermionic
algebra characterizing the antisymmetric duality transformation in equations
(6$a$) , (7$b$). Consequently, exact analytical solutions of the dynamical
evolution generated by $H_{-}$ have proved too difficult to determine and only
approximate solutions, exemplified by the detailed analysis in [25], have been
provided in the quantum optics literature.

Corresponding to the bosonic case $H_{+}$ characterized by $\omega\neq 0$ ,
$\omega_{0}=0$ in [25] where the driving spin operator $\sigma_{x}$ is
averaged in its eigenstate $|\pm\rangle$ and replaced by an eigenvalue, the
authors applied a similar procedure in the analysis of the dynamical features
of the ferminic Hamiltonian $H_{-}$ characterized there by $\omega=0$ ,
$\omega_{0}\neq 0$ by replacing the driving light mode quadrature operator
$\hat{x}=\hat{a}+\hat{a}^{\dagger}$ with a mean value $x$ in an eigenstate of
the light mode. The resulting time evolution equation is then exactly solvable
if $x$ is time-independent, but remains challenging if $x$ is time-dependent.

In the present article, the property that the equations of dynamics generated
by the correlated bosonic and fermionic forms $H_{+}$ , $H_{-}$ (
$[~{}H_{+}~{},~{}H_{-}~{}]\neq 0$ ) of QRM are solved simultaneously means
that the driving light mode quadrature operator $\hat{a}+\hat{a}^{\dagger}$ in
$H_{-}$ may be replaced with its mean value evaluated in the entangled state
$|\Psi(t)\rangle$ in the bosonic dynamics generated by $H_{+}$ in equation
(5$h$), so that, as in [25], the fermionic Hamiltonian takes the effective
form

${\cal H}_{-}=\hbar\omega_{0}s_{z}+2\hbar gx(t)\sigma_{x}\ ;\quad\quad
x(t)=\langle\Psi(t)|\hat{a}+\hat{a}^{\dagger}|\Psi(t)\rangle$ $None$

The general time evolving state vector describing the dynamics of the
fermionic system may be obtained through diagonalization of the effective
Hamiltonian ${\cal H}_{-}$, carefully taking account of the time-dependence of
the light mode mean quadrature $x(t)$. Details of such analysis can be found
in [25].

Another possible method of solution is based on the important property that
the fermionic Hamiltonian $H_{-}$ in equation (7$a$) is invariant under the
parity symmetry transformation. The elaborate Braak methods developed in [3 ,
4 , 20 , 37] can be applied to obtain exact solutions of the eigenvalue
equation for $H_{-}$ in the Bargmann space, which is simpler, but has not been
considered at all, essentially due to the integrability of the full QRM
Hamiltonian $H$. The resulting eigenvalue and eigenstate spectrum can be used
to determine dynamical evolution generated by $H_{-}$ using methods developed
in [5 , 6].

In general, the time evolving state of the effective fermionic Hamiltonian
$H_{-}$ is a generalized atomic spin coherent state. The physical
interpretation then follows that, in the effective fermionic dynamics driven
by the light mode quadrature-dependent force, the dynamical evolution of the
full quantum Rabi model is described by time evolving generalized atomic spin
coherent states.

###  3.3 $\hat{\Pi}_{yx}$ symmetry transformation : effective coupling-only
dynamics

Another interesting symmetry transformation operator is obtained as a product
of the operators $\hat{\Pi}_{y}$ , $\hat{\Pi}_{x}$ in either order with
$\hat{\Pi}_{y}$ to the left or right of $\hat{\Pi}_{x}$ denoted by
$\hat{\Pi}_{yx}$ or $\hat{\Pi}_{xy}$ according to

$\hat{\Pi}_{yx}=\hat{\Pi}_{y}\hat{\Pi}_{x}\
;\quad\quad\hat{\Pi}_{xy}=\hat{\Pi}_{x}\hat{\Pi}_{y}$ $None$

Applying $\hat{\Pi}_{yx}$ or $\hat{\Pi}_{xy}$ symmetry transformation on the
QRM Hamiltonian $H$ and using the relations obtained in equations (3$b$) ,
(4$a$) , (4$b$) , (6$a$) , (6$b$) provides the transform $\widetilde{H}$ of
the QRM Hamiltonian in the form

$\hat{\Pi}_{yx}^{\dagger}~{}H~{}\hat{\Pi}_{yx}=\widetilde{H}\
;\quad\quad\hat{\Pi}_{yx}^{\dagger}~{}\widetilde{H}~{}\hat{\Pi}_{yx}=H\
;\quad\quad\widetilde{H}=H_{0}-H_{I}$ $None$

noting that the $\hat{\Pi}_{xy}$ symmetry transformation gives the same
result. Substituting $H_{0}$ , $H_{I}$ as defined in equation (1$a$) reveals
that the QRM Hamiltonian transform $\widetilde{H}$ has been used as an
alternative form for describing QRM dynamics in [34] where the quantum phase
transition phenomenon is presented. It follows from equation (8$b$) that the
two forms of QRM Hamiltonian are related by symmetry transformation.

Taking an antisymmetric linear combination of $H$ in equation (1$a$) and its
transform $\widetilde{H}$ in equation (8$b$) provides an effective coupling-
only Hamiltonian $H_{RI}$ in the form

$H_{RI}=\frac{1}{2}(H-\widetilde{H})\quad\quad\Rightarrow\quad\quad
H_{RI}=H_{I}$ $None$

where the QRM interaction Hamiltonian $H_{I}$ is defined in equation (1$a$).
The exactly solvable dynamics of the coupling-only QRM Hamiltonian $H_{RI}$
has been widely used to generate Schroedinger cat states [27 , 35 , 36], where
it is characterized as $\omega=0$ , $\omega_{0}=0$ approximation of QRM
Hamiltonian in the ultra/deep strong coupling regime. The time evolution
operator for dynamics generated by $H_{RI}$ follows as a simple solution of
the time-dependent Schroedinger equation in the form

$U_{RI}(t)=e^{\overline{\beta}(t)(\hat{b}+\hat{b}^{\dagger})}\
;\quad\quad\overline{\beta}(t)=-igt$ $None$

where the conjugate composite atom-light bosonic operators $\hat{b}$ ,
$\hat{b}^{\dagger}$ are defined in equation (5$c$). The time evolving state of
the effective coupling-only Hamiltonian is thus an entanglement of the
coherent light Schroedinger cat and atomic spin states. The time evolving
states of the effective bosonic and coupling-only Hamiltonians differ only in
the form of the respective displacement parameters $\beta(t)$ in equation
(5$f$) and $\overline{\beta}(t)$ in equation (8$d$). The physical
interpretation follows that, in the effective coupling-only dynamics generated
only by the coupled light mode quadrature and atomic spin forces, the
dynamical evolution of the full quantum Rabi model is described by a general
time evolving entangled coherent light Schroedinger cat and atomic spin state.
General dynamical features described by the Schroedinger cat states generated
by $U_{RI}$ with displacement variable $\overline{\beta}(t)$ exactly in the
form defined in equation (8$d$) are discussed in [35], agreeing also with the
results in [27 , 36] for phase $\phi=0$. Details which need not be repeated
here can be found in the cited articles.

##  4 Conclusion

A complete set of symmetry transformation operators comprising parity and
duality symmetry operators have been defined within the general algebraic
structure of QRM. The operators satisfy a closed $SU(2)$ symmetry group
algebra. The duality symmetry operators transform the JC and aJC excitation
number operators into each other as duality symmetry conjugates, while the
corresponding JC and aJC interaction Hamiltonians transform into each other as
symmetric or antisymmetric duality symmetry conjugates. The parity symmetry
transformation leaves the QRM Hamiltonian invariant, while the duality
symmetry transformations generate symmetric and antisymmetric conjugates of
the Hamiltonian. The transformation generated by the product of the two
duality symmetry operators provides a QRM Hamiltonian transform differing only
in the sign of the interaction component. The closed $SU(2)$ Lie algebra of
the symmetry operators is useful in determining the standard eigenvalues and
eigenstates of the parity symmetry operator. Symmetric or antisymmetric linear
combinations of the QRM Hamiltonian and the respective symmetric or
antisymmetric conjugates produce exact effective bosonic , fermionic or
coupling-only Hamiltonians, which provide a clear physical picture of the
internal dynamics of QRM. The emerging physical interpretation is that, the
effective bosonic or coupling-only Hamiltonian describes the dynamics of the
quantized light mode driven by a spin-dependent force, while the fermionic
Hamiltonian describes the dynamics of the atomic spin driven by a quantized
light mode quadrature-dependent force. The effective bosonic or coupling-only
Hamiltonian generates a time evolving entangled coherent light Schroedinger
cat and atomic spin state, while the effective fermionic Hamiltonian generates
a time evolving generalized atomic spin coherent state. The important physical
interpretation which arises is that the general dynamical evolution of the
full quantum Rabi model is described by the nonclassical time evolving
entangled coherent light Schroedinger cat and atomic states or generalized
atomic spin coherent states. These nonclassical states have been prepared and
used in developing quantum technology applications in a number of experiments.
The remarkable achievement of the present work is that the effective bosonic,
fermionic or coupling-only QRM Hamiltonians generated through the symmetry
transformations are exact, not involving approximations based on the atomic
spin and light mode angular frequencies or coupling strength as generally
assumed in the earlier theoretical and experimental studies of QRM.
Consequently, the brief exact results provided in this article may be
considered to apply over the entire frequency and coupling parameter ranges.
It is interesting that within the framework of the duality symmetry
transformations, the full QRM dynamics has been developed without
decomposition into coupling-strength related Jaynes-Cummings and anti-Jaynes-
Cummings interaction mechanisms, each of which usually requiring serious
approximations.

##  5 Acknowledgement

I thank Maseno University for providing facilities and a conducive work
environment during the preparation of the manuscript. My colleague Chris
Mayero has provided useful technical support.

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$\big{(}L_{v}(\xi(\theta),\theta,\dot{\xi}(\theta))\big{)}_{\theta}=L_{x}(\xi(\theta),\theta,\dot{\xi}(\theta)).$

Equivalently, if $p(\theta)=L_{v}(\xi(\theta),\theta,\dot{\xi}(\theta))$, then
the pair $(\xi,p)$ satisfies the Hamiltonian ODE

(5.4) $\dot{\xi}(\theta)=-H_{\rho}(\xi(\theta),\theta,p(\theta)),\ \ \ \
\dot{p}(\theta)=H_{x}(\xi(\theta),\theta,p(\theta)).$

We now take $x_{1}<x_{2}$, and write $\xi^{1},\xi^{2}:[s,t]\to{\mathbb{R}}$
for the maximizing paths in (1.21) for $x=x_{1}$ and $x=x_{2}$ respectively.
We wish to show that $\xi^{1}(\theta)\neq\xi^{2}(\theta)$ for every
$\theta\in(s,t)$. We argue by contradiction. Suppose to the contrary
$\xi^{1}(\theta_{0})=\xi^{2}(\theta_{0})$ for some $\theta_{0}\in(s,t)$. We
define

(5.5) $\eta^{2}(\theta)=\begin{cases}\xi^{1}(\theta),\ \ \ \
&\theta\in[s,\theta_{0}],\\\ \xi^{2}(\theta),\ \ \ \
&\theta\in[\theta_{0},t],\end{cases},\ \ \ \ \ \
\eta^{1}(\theta)=\begin{cases}\xi^{2}(\theta),\ \ \ \
&\theta\in[s,\theta_{0}],\\\ \xi^{1}(\theta),\ \ \ \
&\theta\in[\theta_{0},t].\end{cases}$

Since $\xi^{i}$ maximizes the action, and $\eta^{i}$ is weakly differentiable
with square integrable derivative for $i=1,2$, we learn

$\displaystyle\int_{s}^{t}L\big{(}\eta^{1}(\theta),\theta,\dot{\eta}^{1}(\theta)\big{)}\
d\theta$
$\displaystyle\leq\int_{s}^{t}L\big{(}\xi^{1}(\theta),\theta,\dot{\xi}^{1}(\theta)\big{)}\
d\theta,$ (5.6)
$\displaystyle\int_{s}^{t}L\big{(}\eta^{2}(\theta),\theta,\dot{\eta}^{2}(\theta)\big{)}\
d\theta$
$\displaystyle\leq\int_{s}^{t}L\big{(}\xi^{2}(\theta),\theta,\dot{\xi}^{2}(\theta)\big{)}\
d\theta.$

Expressing the integrals on the left in terms of $\xi^{i}$ would lead to

$\int_{s}^{\theta_{0}}L\big{(}\xi^{1}(\theta),\theta,\dot{\xi}^{1}(\theta)\big{)}\
d\theta=\int_{s}^{\theta_{0}}L\big{(}\xi^{2}(\theta),\theta,\dot{\xi}^{2}(\theta)\big{)}\
d\theta.$

This in turn implies that we have equality in (5.6). As a result, $\eta^{i}$
is also a maximizing path with $\eta^{i}(t)=x_{i}$. Hence by (2) above,
$\eta^{i}$ must be $C^{1}$. This means that we must have
$\dot{\xi}^{1}(\theta_{0})=\dot{\xi}^{2}(\theta_{0})$. Since we also have
$\xi^{1}(\theta_{0})=\xi^{2}(\theta_{0})$, we may use the uniqueness of the
solutions to Euler-Lagrange equation (5.3), to deduce that $\xi^{1}=\xi^{2}$
on $[s,t]$. This contradicts $x_{1}=\xi^{1}(t)<x_{2}=\xi^{2}(t)$. As a result,
$\xi^{1}$ and $\xi^{2}$ cannot intersect in $(s,t]$. Since $x_{1}<x_{2}$, we
must have $\xi^{1}(\theta)<\xi^{2}(\theta)$ for $\theta>s$.

(ii) We take $y_{1}<y_{2}$, and write $\xi^{1},\xi^{2}:[s,t]\to{\mathbb{R}}$
for the maximizing paths in (1.21) for $z=(y_{1},s)$ and $z=(y_{2},s)$
respectively. We wish to show that $\xi^{1}(\theta)\neq\xi^{2}(\theta)$ for
every $\theta\in(s,t)$. We again argue by contradiction. Suppose to the
contrary $\xi^{1}(\theta_{0})=\xi^{2}(\theta_{0})$ for some
$\theta_{0}\in(s,t)$. We define $\eta^{1}$ and $\eta^{2}$ as in (5.5). Again,
since $\xi^{1}$ (respectively $\xi^{2}$) is a maximizer in (1.21), and that
$\eta^{2}$ (respectively $\eta^{1}$) is weakly differentiable with square
integrable derivative, we learn

$\displaystyle\int_{s}^{t}L\big{(}\eta^{2}(\theta),\theta,\dot{\eta}^{2}(\theta)\big{)}\
d\theta$
$\displaystyle\leq\int_{s}^{t}L\big{(}\xi^{1}(\theta),\theta,\dot{\xi}^{1}(\theta)\big{)}\
d\theta,$ (5.7)
$\displaystyle\int_{s}^{t}L\big{(}\eta^{1}(\theta),\theta,\dot{\eta}^{1}(\theta)\big{)}\
d\theta$
$\displaystyle\leq\int_{s}^{t}L\big{(}\xi^{2}(\theta),\theta,\dot{\xi}^{2}(\theta)\big{)}\
d\theta.$

Expressing the integrals on the left in terms of $\xi^{i}$ would lead to

$\int_{\theta_{0}}^{t}L\big{(}\xi^{1}(\theta),\theta,\dot{\xi}^{1}(\theta)\big{)}\
d\theta=\int_{\theta_{0}}^{t}L\big{(}\xi^{2}(\theta),\theta,\dot{\xi}^{2}(\theta)\big{)}\
d\theta.$

This in turn implies that we have equality in (5.7), and that $\eta^{1}$
(respectively $\eta^{2}$) is also a maximizing path with $\eta^{1}(s)=y_{2}$
(respectively $\eta^{2}(s)=y_{1}$). Hence by (2) above, $\eta^{i}$ must be
$C^{1}$ for $i=1,2$. This means that we must have
$\dot{\xi}^{1}(\theta_{0})=\dot{\xi}^{2}(\theta)$. By the uniqueness of the
corresponding Euler-Lagrange equation, we must have $\xi^{1}=\xi^{2}$ on
$[s,t]$. This contradicts $y_{1}=\xi^{1}(s)<y_{2}=\xi^{2}(s)$. As a result,
$\xi^{1}$ and $\xi^{2}$ cannot intersect in $(s,t]$. Since $y_{1}<y_{2}$, we
must have $\xi^{1}(\theta)<\xi^{2}(\theta)$ for $\theta\leq t$. This in
particular implies that $\dot{\xi}^{1}(t)\geq\dot{\xi}^{2}(t)$. Moreover
$\dot{\xi}^{1}(t)=\dot{\xi}^{2}(t)$ would imply $\dot{\xi}^{1}=\dot{\xi}^{2}$
by the uniqueness of the corresponding (5.3). As a result, we must have
$\dot{\xi}^{1}(t)>\dot{\xi}^{2}(t)$. This, (1.22), and the strict concavity of
$L$ in $v$ imply the desired inequality $M(x,t;y_{1},s)<M(x,t;y_{2},s)$.

(iii) Recall that the pair $(\xi,p)$ satisfies (5.4), and the boundary
conditions

(5.8) $\xi(s)=y,\ \ \ \ \xi(t)=x,\ \ \ \ p(t)=M(x,t;y,s).$

From (5.3) and Hypothesis 1.2(i), we learn

$|\dot{p}(\theta)|=\big{|}\big{(}L_{v}(\xi(\theta),\theta,\dot{\xi}(\theta))\big{)}_{\theta}\big{|}=\big{|}L_{x}(\xi(\theta),\theta,\dot{\xi}(\theta))\big{|}\leq
c_{1},$

which in turn implies

(5.9) $|p(\theta)-p(t)|\leq c_{1}(t-s),$

for $\theta\in[s,t]$. This, and (5.4) imply

$|\dot{\xi}(\theta)|=\big{|}H_{\rho}(\xi(\theta),\theta,p(\theta))\big{|}\leq
c_{0}c_{2}^{-1}+c_{2}^{-1}|p(\theta)|\leq c(1+|p(t)|),$

for a constant $c=c(s,T)$. Here we used

(5.10) $-c_{0}+c_{2}\big{|}H_{\rho}(x,\theta,\rho)\big{|}\leq|\rho|\leq
c_{0}+c_{1}\big{|}H_{\rho}(x,\theta,\rho)\big{|},$

which follows from Hypothesis 1.2(i). As a result,

$|x-y|=|\xi(t)-\xi(s)|\leq c^{\prime}(1+|p(t)|),$

for a positive constant $c^{\prime}=c^{\prime}(s,T)$. On the other hand, if
$|y|\leq(1-\delta)|x|$, then we deduce

(5.11) $|x|\leq c^{\prime}\delta^{-1}(1+|p(t)|).$

We next claim that there exists a constant $C_{0}$ such that

(5.12) $|x|\geq C_{0}\ \ \ \implies\ \ \ xp(t)<0.$

To see this, observe that by (5.10) and the monotonicity of $\rho\mapsto
H_{\rho}(x,\theta,\rho)$, we can find a constant $c^{\prime\prime}$ such that

(5.13) $|\rho|\geq c^{\prime\prime}\ \ \ \implies\ \ \
H_{\rho}(x,\theta,\rho)\rho\geq 0,$

for all $(x,\theta)$. Let us assume that $x\geq C_{0}$, for a positive
constant $C_{0}$ (to be determined later). Suppose contrary to (5.12), we have
$p(t)\geq 0$. From (5.11) we deduce

$p(t)\geq{(c^{\prime})^{-1}}{\delta}x-1.$

This and (5.9) imply

(5.14)
$p(\theta)\geq{(c^{\prime})^{-1}}{\delta}x-1-c_{1}(t-s)\geq{(c^{\prime})^{-1}}{\delta}C_{0}-1-c_{1}(T-s),$

for all $\theta\in(s,t)$. Choose $C_{0}$ large enough so that the right-hand
side of (5.14) is at least $c^{\prime\prime}$. This would guarantee

$H_{\rho}(\xi(\theta),\theta,p(\theta))\geq 0,\ \ \ {\text{ for }}\
\theta\in[s,t],$

by (5.13). From this and (5.4) we deduce that $\dot{\xi}(\theta)\leq 0$ for
$\theta\in[s,t]$. As a result, $x-y=\xi(t)-\xi(s)\leq 0$. But this is
impossible if $y\leq(1-\delta)x$. Hence the condition $x\geq C_{0}$ implies
that $p(t)>0$. In the same fashion, we can show that the condition $x\leq-
C_{0}$ implies that $p(t)<0$. This completes the proof of (5.12). From this,
(5.8), and (5.11), we can readily deduce (5.2). $\Box$

We next give a recipe for the law of the process ${\bf{y}}_{t}$.

Definition 5.2(i) We set

$\Gamma(a,b,t,\rho)=\int_{a}^{b}\hat{A}(g)(z,t,y)\ dz,\ \ \ \
\Gamma({\mathbb{q}},t)=\sum_{i=0}^{n}\Gamma(x_{i},x_{i+1},t,y_{i}).$

(ii) We define a measure $\mu(d{\mathbb{q}},t)$ on the set $\Delta$ that is
our candidate for the law of ${\mathbb{q}}(t)$. The restriction of $\mu$ to
$\Delta_{n}$ is given by

$\mu^{n}(d{\mathbb{q}},t):=\ell(t,dy_{0})\exp\left\\{-\Gamma({\mathbb{q}},t)\right\\}\prod_{i=1}^{n}\
g\big{(}x_{i},t,y_{i-1},{y_{i}})\ dx_{i}dy_{i},$

where $g$ solves (1.25) and $\ell$ solves (1.26). To simplify our
presentation, we assume that $\ell(t,dy_{0})=\ell(t,y_{0})\ dy_{0}$ is
absolutely continuous with respect to the Lebesgue measure. Such an assumption
would allow us to express $\mu^{n}(d{\mathbb{q}},t):=\mu^{n}({\mathbb{q}},t)\
d{\mathbb{q}}$, where

$d{\mathbb{q}}=dy_{0}\ \prod_{i=1}^{n}dx_{i}dy_{i},\ \ \ \
\mu^{n}({\mathbb{q}},t)=\ell(t,y_{0})\exp\left\\{-\Gamma({\mathbb{q}},t)\right\\}\prod_{i=1}^{n}\
g\big{(}x_{i},t,y_{i-1},{y_{i}}).$

$\Box$

###### Proposition 5.3

Let $g$ be a solution of (1.25). Then $\hat{A}(g^{1})_{t}=\hat{A}(g^{2})_{x}$.

Proof From integrating both sides of (1.25) with respect $y_{+}$ we learn

(5.15)
$\hat{A}(g^{1})_{t}-\hat{A}(g^{2})_{x}=\hat{A}\big{(}\hat{Q}^{+}(g)\big{)}-\hat{A}\big{(}g\hat{J}(g)\big{)}.$

On the other hand,

$\displaystyle\hat{A}\big{(}\hat{Q}^{+}(g)\big{)}(y_{-})=$ $\displaystyle\int
g^{1}(y_{-},y_{*})\hat{A}(g^{2})(y_{*})\ dy_{*}-\int
g^{2}(y_{-},y_{*})\hat{A}(g^{1})(y_{*})\ dy_{*},$
$\displaystyle\hat{A}\big{(}g\hat{J}(g)\big{)}(y_{-})=$ $\displaystyle\int
g^{1}(y_{-},y_{*})\hat{A}(g^{2})(y_{*})\
dy_{*}-\hat{A}(g^{2})(y_{-})\hat{A}(g^{1})(y_{-})$ $\displaystyle-\int
g^{2}(y_{-},y_{*})\hat{A}(g^{1})(y_{*})\
dy_{*}+\hat{A}(g^{1})(y_{-})\hat{A}(g^{2})(y_{-}).$

This implies that the right-hand side of (5.15) is $0$. $\Box$

We are now ready to present the proof of Theorem 5.1, which is similar to the
proof of Theorem 2.1.

Proof of Theorem 5.1 We wish to establish the analog of (4.1) in our setting.
Theorem 3.1, and a repetition of the first step of the proof of Theorem 2.1
allow us to reduce the proof of Theorem 5.1 to the verification of an analog
of (4.4), namely

$\displaystyle\int_{\Delta_{n}}\hat{G}^{n}({\mathbb{q}},s)\
\mu^{n}_{s}(d{\mathbb{q}},s)=$
$\displaystyle\int_{\Delta_{n}}\hat{G}^{n}({\mathbb{q}},s)\big{(}\widehat{\mathcal{L}}_{0n}^{s*}\mu^{n}\big{)}({\mathbb{q}},s)\
d{\mathbb{q}}$ (5.16)
$\displaystyle+\int_{\hat{\partial}\Delta_{n+1}}\hat{G}^{n+1}({\mathbb{q}},s)\mu^{n+1}({\mathbb{q}},s)({\mathbb{b}}_{n+1}\cdot{\mathbb{N}}_{n+1})\
\sigma(d{\mathbb{q}}),$

with $\hat{G}$ as in (3.5). For a more tractable expression for the left hand
side of (5.16), we write

(5.17) $\mu_{s}^{n}=X^{n}\mu^{n},$

where

(5.18)
$X^{n}=-\Gamma_{s}({\mathbb{q}},s)+\frac{\ell_{s}(s,y_{0})}{\ell(s,y_{0})}+\sum_{i=1}^{n}\frac{g_{s}(x_{i},s,y_{i-1},y_{i})}{g(x_{i},s,y_{i-1},y_{i})}.$

On the other hand, by Proposition 5.3,

$\displaystyle\Gamma_{s}({\mathbb{q}},s)=$
$\displaystyle\sum_{i=0}^{n}\int_{x_{i}}^{x_{i+1}}\hat{A}(g^{1})_{s}(z,s,y_{i})\
dz=\sum_{i=0}^{n}\int_{x_{i}}^{x_{i+1}}\hat{A}(g^{2})_{z}(z,s,y_{i})\ dz$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\big{(}\hat{A}(g^{2})(x_{i+1},s,y_{i})-\hat{A}(g^{2})(x_{i},s,y_{i})\big{)}$
$\displaystyle=$
$\displaystyle\hat{A}(g^{2})(x_{n+1},s,y_{n})-\hat{A}(g^{2})(x_{0},s,y_{0})-\sum_{i=1}^{n}\big{(}\hat{A}(g^{2})(x_{i},s,y_{i})-\hat{A}(g^{2})(x_{i},s,y_{i-1})\big{)}.$

From this, (5.18), (1.26), and (1.25) we deduce

$\displaystyle X^{n}=$
$\displaystyle-\hat{A}(g^{2})(a_{+},s,y_{n})+\frac{(\ell*g^{2})(a_{-},s,y_{0})}{\ell(s,y_{0})}+\sum_{i=1}^{n}\frac{\hat{Q}^{+}(g)(x_{i},s,y_{i-1},y_{i})}{g(x_{i},s,y_{i-1},y_{i})}$
(5.19)
$\displaystyle+\sum_{i=1}^{n}\hat{v}(x_{i},s,y_{i-1},y_{i})\big{(}\hat{A}(g)(x_{i},s,y_{i})-\hat{A}(g)(x_{i},s,y_{i-1})\big{)}.$

We can rewrite the right-hand side of (5.16) as

$\int_{\Delta_{n}}\hat{G}^{n}({\mathbb{q}},s)Y^{n}({\mathbb{q}})\mu^{n}({\mathbb{q}},s)\
d{\mathbb{q}},$

where $Y^{n}=Y^{n}_{1}+Y^{n}_{2}$, with $Y^{n}_{1}$ and $Y^{n}_{2}$
corresponding to two terms on the right-hand side of (5.16). Indeed, an
integration by parts yields

$\displaystyle Y^{1}_{n}({\mathbb{q}})=$
$\displaystyle-\sum_{i=1}^{n}\hat{v}_{i}(x_{i},s,y_{i-1},y_{i})\Gamma(s,{\mathbb{q}})_{x_{i}}$
(5.20) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\hat{v}_{i}(x_{i},s,y_{i-1},y_{i})\big{(}\hat{A}(g)(x_{i},s,y_{i})-\hat{A}(g)(x_{i},s,y_{i-1})\big{)}.$

As for $Y^{n}_{2}$, we write $Y^{n}_{2}=Y^{n}_{2-}+Y^{n}_{2*}+Y^{n}_{2+}$,
where the terms $Y^{n}_{2-},\ Y^{n}_{2*}$, and $Y^{n}_{2+}$ correspond to the
boundary contributions associated with the conditions $x_{1}=a_{-}$,
$x_{i}=x_{i+1},$ with $i\in\\{1,\dots,n\\}$, and $x_{n+1}=a_{+}$,
respectively. More precisely,

$\displaystyle Y^{n}_{2-}({\mathbb{q}})=$
$\displaystyle\frac{(\ell*g^{2})(a_{-},s,y_{0})}{\ell(s,y_{0})},\ \ \ \ \
Y^{n}_{2+}({\mathbb{q}})=-\hat{A}(g^{2})(a_{+},s,y_{n}),$ $\displaystyle
Y^{n}_{2*}({\mathbb{q}})=$
$\displaystyle\sum_{i=1}^{n}\frac{\hat{Q}^{+}(g)(x_{i},s,y_{i-1},y_{i})}{g(x_{i},s,y_{i-1},y_{i})}.$

This, (5.17), (5.19), and (5.20) complete the proof of (5.16). $\Box$

##  6 Proofs of Proposition 1.1 and Theorem 1.3

Proof of Proposition 1.1 Let us write

${\mathcal{K}}(g)=\nabla\cdot(\nu g)-\hat{Q}^{+}(g)+\hat{Q}^{-}(g).$

where $\nu=(-\hat{v},1)$, and $\nabla=(\partial_{x},\partial_{t})$. To ease
the notation, we do not display the dependence of $g,\ \hat{g},\eta$, and $h$
on $(x,t)$. We certainly have

$\displaystyle\frac{\hat{Q}^{+}(\hat{g})}{\hat{g}}-\frac{\hat{Q}^{+}(g)}{g}=0,$
$\displaystyle\left(\frac{\nabla\cdot(\nu\hat{g})}{\hat{g}}-\frac{\nabla\cdot(\nu
g)}{g}\right)(y_{-},y_{+})=\left(\nu\cdot\frac{\nabla\eta}{\eta}\right)(y_{-},y_{+})=\nu\cdot\frac{\nabla
h}{h}(y_{+})-\nu\cdot\frac{\nabla h}{h}(y_{-}),$
$\displaystyle\frac{\hat{Q}^{-}(g)}{g}(y_{-},y_{+})={\hat{A}(\hat{v}g)}(y_{+})-\hat{A}(\hat{v}g)(y_{-})-\hat{v}(y_{-},y_{+})\left({\hat{A}(g)}(y_{+})-\hat{A}(g)(y_{-})\right)$
$\displaystyle\frac{\hat{Q}^{-}(\hat{g})}{\hat{g}}(y_{-},y_{+})=\frac{\hat{A}(\hat{v}g\otimes
h)}{h}(y_{+})-\frac{\hat{A}(\hat{v}g\otimes h)}{h}(y_{-})$ $\displaystyle\ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\hat{v}(y_{-},y_{+})\left(\frac{\hat{A}(g\otimes
h)}{h}(y_{+})-\frac{\hat{A}(g\otimes h)}{h}(y_{-})\right).$

Hence

$\displaystyle\left(\frac{{\mathcal{K}}(\hat{g})}{\hat{g}}-\frac{{\mathcal{K}}(g)}{g}\right)(y_{-},y_{+})=$
$\displaystyle\nu(y_{-},y_{+})\cdot\left(\frac{\nabla
h}{h}(y_{+})-\frac{\nabla
h}{h}(y_{-})\right)+\left(\frac{\hat{Q}^{-}(\hat{g})}{\hat{g}}-\frac{\hat{Q}^{-}(g)}{g}\right)(y_{-},y_{+})$
$\displaystyle=$
$\displaystyle\frac{h_{t}+{\mathcal{B}}^{2}h}{h}(y_{+})-\frac{h_{t}+{\mathcal{B}}^{2}h}{h}(y_{-})$
$\displaystyle-\hat{v}(y_{-},y_{+})\left[\frac{h_{x}+{\mathcal{B}}^{1}h}{h}(y_{+})-\frac{h_{x}+{\mathcal{B}}^{1}h}{h}(y_{-})\right].$

The right-hand side is $0$, when $h$ satisfies (1.31). This completes the
proof because $g$ (respectively $\hat{g}$) solves (1.25) if and only if
${\mathcal{K}}(g)=0$ (respectively ${\mathcal{K}}(\hat{g})=0$). $\Box$

The proof of Theorem 1.3, uses Doob’s $h$-transform that we now recall.

###### Proposition 6.1

Let ${\mathbb{P}}$ be the law of Markov jump process $\big{(}{\bf{y}}(x):\
x\in[a_{-},a_{+}]\big{)}$, with the jump kernel density $g(x,y_{-},y_{+})$,
and the generator ${\mathcal{L}}_{x}$. Assume that $g$ is $C^{1}$ in $x$. Let
$U$ be an interval, and let $\widehat{\mathbb{P}}$ denote the law of
${\mathbb{P}}$, conditioned on the event ${\bf{y}}(x)\in U$ for all
$x\in[a_{-},a_{+}]$. Then $\widehat{\mathbb{P}}$ is the law of a Markov jump
process with a jump kernel density $\hat{g}$, given by

(6.1) $\hat{g}(x,y_{-},y_{+})=\frac{h(x,y_{+})}{h(x,y_{-})}\
g(x,y_{-},y_{+}),\ \ \ \ \ x\in[a_{-},a_{+}),\ \ y_{\pm}\in U,$

where

(6.2) $h(x,y)={\mathbb{P}}\big{(}{\bf{y}}(a)\in U\ \ {\text{ for }}\
a\in[x,a_{+}]\ |\ {\bf{y}}(x)=y\big{)}.$

Moreover, $h$ is $C^{1}$ in $x$, and satisfies

(6.3) $h_{x}+{\mathcal{L}}_{x}h=0.$

Proof (Step 1) We can write

(6.4) $h(x,y)=\sum_{n=0}^{\infty}h_{n}(x,y),$

where

(6.5) $h_{n}(x,y)=\int_{X_{n}(x,y)}\mu^{n}({\mathbb{q}},x,y)\
d{\mathbb{q}}_{n}\\\ $

where for $n\geq 1$,

$\displaystyle{\mathbb{q}}_{n}=(x_{1},y_{1},\dots,x_{n},y_{n}),\ \ \ \ \
d{\mathbb{q}}_{n}=\prod_{i=1}^{n}dx_{i}dy_{i},$
$\displaystyle\mu^{n}({\mathbb{q}}_{n},x,y)=\exp\left\\{-\Gamma({\mathbb{q}}_{n},x,y)\right\\}\prod_{i=1}^{n}\
g\big{(}x_{i},y_{i-1},{y_{i}}),\ \ {\text{ with }}\ \ y_{0}=y,$
$\displaystyle\Gamma({\mathbb{q}}_{n},x,y)=\sum_{i=0}^{n}\Gamma(x_{i},x_{i+1},y_{i}),\
\ \ \ \ {\text{with }}\ \ \ \ x_{0}=x,\ y_{0}=y,$
$\displaystyle\Gamma(a,b,y)=\int_{a}^{b}(\hat{A}g)(z,y)\ dz,$

and the set $X_{n}(x,y)$ consists of ${\mathbb{q}}$, satisfying

$x<x_{1}<\dots<x_{n}<a_{+},\ \ \ \ y_{1},\dots,y_{n}\in U.$

When $n=0$, we simply have
$h_{0}(x,y)=\exp\big{\\{}-\Gamma(x,a_{+},y)\big{\\}}.$ It is straightforward
to verify continuous differentiability of $h$, and deduce (6.3) from (6.4) and
(6.5).

(Step 2) The law $\hat{P}$ is simply given by

$\hat{P}=\sum_{n=1}^{\infty}\hat{\mu}^{n},$

where $\hat{\mu}_{n}(d{\mathbb{q}}_{n})=\hat{\mu}_{n}({\mathbb{q}}_{n})\
d{\mathbb{q}}_{n},$ with

(6.6) $\hat{\mu}_{n}({\mathbb{q}}_{n})=h(a_{-},y)^{-1}\
\mu_{n}({\mathbb{q}}_{n})\ 1\\!\\!1\big{(}y_{1},\dots,y_{n}\in U\big{)}.$

We wish to show that $\hat{P}$ is the law of a jump process associated with
the jump density $\hat{g}$. To achieve this, we rewrite $\hat{\mu}_{n}$ using
the fact that $h$ satisfies (6.3). Indeed, (6.3) implies

(6.7) $e^{-\int_{a}^{b}\hat{A}(g)(z,y_{-})dz}\
\frac{h(b,y_{+})}{h(a,y_{-})}=e^{-\int_{a}^{b}\hat{A}(\hat{g})(z,y_{-})dz}\
\frac{h(b,y_{+})}{h(b,y_{-})}.$

This is equivalent to asserting

(6.8) $\displaystyle\frac{h(b,y_{-})}{h(a,y_{-})}=$
$\displaystyle\exp\left(-\int_{a}^{b}\big{(}\hat{A}(\hat{g})-\hat{A}(g)\big{)}(z,y_{-})\
dz\right)$ $\displaystyle=$
$\displaystyle\exp\left(-\int_{a}^{b}\left(\frac{\hat{A}(g\otimes
h)-\hat{A}(g)h}{h}\right)(z,y_{-})\ dz\right)$ $\displaystyle=$
$\displaystyle\exp\left(\int_{a}^{b}\frac{-({\mathcal{L}}_{z}h)(z,y_{-})}{h(z,y_{-})}\
dz\right)=\exp\left(\int_{a}^{b}\frac{h_{z}(z,y_{-})}{h(z,y_{-})}\ dz\right)$
$\displaystyle=$ $\displaystyle\exp\left(\int_{a}^{b}(\log h)_{z}(z,y_{-})\
dz\right),$

which is evidently true. We set, $x_{0}=a_{-},\ y_{0}=y$ as before. Observe
that $\hat{\mu}_{n}({\mathbb{q}}_{n})$ of (6.6) can be written as

$\displaystyle\frac{1}{h(a_{+},y_{n})}\
e^{-\int_{x_{n}}^{a_{+}}\hat{A}(g)(z,y_{n})dz}\
\frac{h(a_{+},y_{n})}{h(x_{n},y_{n})}\
\prod_{i=1}^{n}e^{-\int_{x_{i-1}}^{x_{i}}\hat{A}(g)(z,y_{i-1})dz}\
\frac{h(x_{i},y_{i})}{h(x_{i-1},y_{i-1})}g(x_{i},y_{i-1},y_{i})$
$\displaystyle\ \ \ \ \ \ \ =\frac{1}{h(a_{+},y_{n})}\
e^{-\int_{x_{n}}^{a_{+}}\hat{A}(\hat{g})(z,y_{n})dz}\prod_{i=1}^{n}e^{-\int_{x_{i-1}}^{x_{i}}\hat{A}(\hat{g})(z,y_{i-1})dz}\
\frac{h(x_{i},y_{i})}{h(x_{i},y_{i-1})}g(x_{i},y_{i-1},y_{i})$ $\displaystyle\
\ \ \ \ \ \
=e^{-\int_{x_{n}}^{a_{+}}\hat{A}(\hat{g})(z,y_{n})dz}\prod_{i=1}^{n}e^{-\int_{x_{i-1}}^{x_{i}}\hat{A}(\hat{g})(z,y_{i-1})dz}\
\hat{g}(x_{i},y_{i-1},y_{i}),$

where we used (6.8) and (6.7) for the first equality, and for the last
equality we used the definition of $\hat{g}$, and $h(a_{+},y_{n})=1$, which
follows from the definition of $h$. The right-hand side is the law of a Markov
jump process associated with the kernel density $\hat{g}$, as desired. $\Box$

Proof of Theorem 1.3 (Step 1) Recall that $\rho(x,t)$ is the solution of (1.2)
with the initial condition $\rho(x,t_{0})=\rho(x,t_{0};{\bf{y}}_{t_{0}},s)$,
where ${\bf{y}}_{t_{0}}$ is a jump process associated with the kernel
$g(x,t_{0},y_{-},y_{+})$. We wish to show that
$\rho(x,t)=\rho(x,t;{\bf{y}}_{t},s)$ for
$(x,t)\in[a_{-},\infty)\times[t_{0},T]$. It suffices to verify this for
$(x,t)\in[a_{-},a_{+}]\times[t_{0},T]$, where $a_{+}$ is any large number in
$(a_{-},\infty)$.

Pick any $\delta\in(0,1)$. From Proposition 5.2(iii), and (5.13), we learn
that there exist constants $C_{0}=C_{0}(\delta),$ $C_{1}=C_{1}(\delta)$, and
$C_{2}$ such that

(6.9) $\displaystyle|y_{+}|\leq(1-\delta)a_{+},\ \ \ a_{+}\geq C_{0}\ \ $
$\displaystyle\implies\ \ M(a_{+},t_{0};y_{+},s)<-C_{1}a_{+},$ (6.10)
$\displaystyle\ (x,\theta)\in{\mathbb{R}}\times[s,T],\ \ \ \ |\rho|\geq C_{2}\
\ $ $\displaystyle\implies\ \ \ \rho H_{\rho}(x,t,\rho)>0,$

for every $t\in[t_{0},T]$. Note that (6.9) and Proposition 5.2(ii) imply that
if $a_{+}\geq C_{0}$, then

$y_{-}<y_{+},\ \ |y_{+}|<(1-\delta)a_{+},\ \ \ \implies\ \ \
M(a_{+},t_{0};y_{-},s)<M(a_{+},t_{0};y_{+},s)<-C_{1}a_{+}.$

From this, and (6.2) we can readily deduce

(6.11) $Y_{-}\leq y_{-}<y_{+}\leq(1-\delta)a_{+},\ \ \ a_{+}\geq C_{3}\ \ \
\implies\ \ \ \hat{v}(x,t,y_{-},y_{+})>0,$

for every $t\in[t_{0},T]$, where

$C_{3}=\max\big{\\{}C_{0},C_{1}^{-1}C_{2},|Y_{-}|\big{\\}}.$

We pick any $a_{+}\geq\max\\{a_{-},C_{3}\\}$.

(Step 2) We write ${\mathbb{W}}$ for the law of the Markov process
$({\mathbb{w}}(t):\ t\in[t_{0},T])$, associated with the generator
${\mathcal{B}}^{2}_{a_{-},t}$, such that ${\mathbb{w}}(t_{0})=y^{0}$. We also
define a family of probability measures
$\big{(}{\mathbb{P}}_{t}:t\in[t_{0},T]\big{)}$ with the following recipe: For
each $t$, ${\mathbb{P}}_{t}$ is the law of the Markov process
${\bf{y}}_{t}:[a_{-},a_{+}]\to[Y_{-},\infty),$ associated with the generator
${\mathcal{B}}^{1}_{x,t}$, satisfying the initial condition
${\bf{y}}_{t}(a_{-})={\mathbb{w}}(t)$. We define $\theta_{\delta}$ to be the
smallest $t>t_{0}$ such that ${\mathbb{w}}(t)\notin U(\delta)$, where

$U(\delta):=[Y_{-},(1-\delta)a_{+})=:[Y_{-},Y^{\delta}).$

We also define $\tau_{\delta}(t)$ to be the smallest $x>a_{-}$ such that
${\bf{y}}_{t}(x)\notin U(\delta)$. We set

${\mathbb{W}}^{\delta}(A)={\mathbb{W}}(A\ |\ \theta_{\delta}>T),\ \ \ \
{\mathbb{P}}_{t}^{\delta}(A)={\mathbb{P}}_{t}(A\ |\ \tau_{\delta}(t)>a_{+}).$

We write ${\mathbb{w}}^{\delta}(t),\ t\in[t_{0},T]$ for the jump process that
is distributed according to ${\mathbb{W}}^{\delta}$. For each $t\in[t_{0},T]$,
we write ${\bf{y}}_{t}^{\delta}$ for the jump process that is distributed
according to ${\mathbb{P}}_{t}^{\delta}$. By Proposition 6.1, the process
$t\to{\mathbb{w}}^{\delta}(t)$, and the processes
$x\mapsto{\bf{y}}_{t}^{\delta}(x),\ t\in[t_{0},T]$ are again Markov jump
processes. We set

$\displaystyle h^{\delta}(x,t,y)$
$\displaystyle={\mathbb{P}}^{\delta}_{t}\big{(}\tau_{\delta}(t)>a_{+}\ |\
{\bf{y}}_{t}(x)=y\big{)},$ $\displaystyle\ell^{\delta}(t,y)$
$\displaystyle={\mathbb{W}}\big{(}\theta_{\delta}>T\ |\
{\mathbb{w}}(t)=y\big{)}.$

By (6.3), we have the following equations for $h^{\delta}$ and
$\ell^{\delta}$:

(6.12) $\displaystyle
h^{\delta}_{x}(x,t,y)+\big{(}{\mathcal{B}}^{1}_{x,t}h^{\delta}\big{)}(x,t,y)=0,\
\ \ \ $ $\displaystyle x\in(a_{-},a_{+}),\ y\in U(\delta),\ t\in[t_{0},T],$
(6.13)
$\displaystyle\ell^{\delta}_{t}(t,y)+\big{(}{\mathcal{B}}^{2}_{a_{-},t}\ell^{\delta}\big{)}(s,y)=0,\
\ \ \ $ $\displaystyle t\in(t_{0},T),\ y\in U(\delta).$

Since, $h^{\delta}(a_{-},t,y)=\ell^{\delta}(t,y)$, the equations (6.12) and
(6.13) allow us to apply Proposition 1.2 to assert that $h^{\delta}$ satisfies

(6.14)
$h^{\delta}_{x}(x,t,y)+\big{(}{\mathcal{B}}^{1}_{x,t}h^{\delta}\big{)}(x,t,y)=0,\
\ \ \
h^{\delta}_{t}(x,t,y)+\big{(}{\mathcal{B}}^{2}_{x,t}h^{\delta}\big{)}(x,t,y)=0.$

(Step 3) Since the jump process ${\bf{y}}^{\delta}_{t_{0}}$ takes value in
$[Y_{-},(1-\delta)a_{+})$, our Theorem 1.2 or Theorem 5.1 is applicable. More
precisely, if the initial data of (1.2) is given by
$\rho(x,t_{0};{\bf{y}}^{\delta}_{t_{0}},s)$, for some $s<t_{0}$, and with
$y^{\delta}_{t_{0}}$ a Markov process distributed according to
${\mathbb{P}}_{t_{0}}^{a_{+},\delta}$, then the solution $\rho(x,t)$ at a
later time $t>t_{0}$ is given by ${\mathbb{P}}_{t}^{\delta}$. As a
consequence, Theorem 1.3 is true when the initial process satisfies
${\bf{y}}_{t_{0}}(a_{+})\in U(\delta)$. This condition is true with
probability density $h^{\delta}(a_{-},t_{0},y^{0})$. The condition (1.29)
would implies that $h^{\delta}(a_{-},t_{0},y^{0})\to 1$, and
${\bf{y}}^{\delta}_{t}\to{\bf{y}}_{t}$ in small $\delta$ limit, when
restricted to the interval $[a_{-},a_{+}]$. This completes the proof. $\Box$

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