1. Motivation and problem settings
The aim of this paper is to study some examples of a continuous-time stochastic process with deterministic behaviour between random switching times, the so-called piecewise deterministic process with continuous paths.
Let
$${({{\cal F}_t})_{t \ge 0}}$$
be a filtration and let
${{\varepsilon}} = ({{\varepsilon}}(t))_{t\geq0}$
be an arbitrary measurable and adapted process defined on
$(\Omega , \mathcal F, \mathcal F_t , {{\mathrm P}})$
with values in a finite space
$\{1,\ldots, N\}$
. Let
$\phi_1,\ldots,\phi_N$
be N deterministic flows in a phase space
$(G, \mathcal G)$
, where we assume that G is a topological space and
$\mathcal G$
is the Borel
$\sigma$
-algebra. Let
$\{\tau_n\}_{n\geq1}$
be the sequence of switching times of
${{\varepsilon}}$
. The piecewise deterministic process
${{\mathbb X}}$
is defined as
\begin{equation*}
{{\mathbb X}}(t)=\phi_{{{\varepsilon}}(\tau_n)}(t),\qquad \tau_n\leq t<\tau_{n+1}.
\end{equation*}
The family of piecewise deterministic processes was introduced in [Reference Davis4], and a subclass of piecewise linear processes was first studied in [Reference Gnedenko and Kovalenko10]. This important class of random processes was then thoroughly studied in [Reference Davis5]; see [Reference Jacobsen11] for a modern presentation. Piecewise deterministic processes are intensively exploited in biology [Reference Lasota, Mackey and Tyrcha18], insurance [Reference Embrechts and Schmidli8], storage models [Reference Boxma, Kaspi, Kella and Perry3], financial market modelling [Reference Kolesnik and Ratanov16], and in many other fields.
To simplify our presentation we restrict ourselves to switchings driven by a Markov process with only two values (states). The simplest example of such a process is a piecewise linear (telegraph) process based on the two-state Markov process
${{\varepsilon}}={{\varepsilon}}(t)\in\{0, 1\}$
:
\begin{equation}
T(t)=V(0)\int_0^t({-}\,1)^{N(\tau)}\,{{\rm d}} \tau,\qquad t>0,
\label{eqn1}
\end{equation}
driven by a homogeneous Poisson process
$N=N(t)$
. The value T(t) corresponds to the position of a particle moving on the line with velocities
$-1$
and
$+1$
alternating at Poisson times. The random starting velocity
$V(0)\in\{-1, %\;
+1\}$
is independent of N.
The theory of telegraph processes is well developed, beginning from [Reference Kac12]. Over the past few decades, many generalizations of the telegraph process have been proposed in the literature including motions characterized by arbitrary numbers of possible velocities [Reference Kolesnik13], by random velocities [Reference Stadje and Zacks24, Reference De Gregorio6], with velocity changes governed by an alternating renewal process (for instance [Reference Di Crescenzo and Martinucci7] or perturbed by jumps [Reference Ratanov23, Reference López and Ratanov19]). See also the monograph [Reference Kolesnik and Ratanov16] and the references therein for full details on the telegraph process.
The classic telegraph model (1.1) can be easily generalized to the process T(t) of inhomogeneous structure with velocities
$c_0$
and
$c_1$
,
$c_0>c_1$
, alternating with intensities
${{\lambda}}_0$
and
${{\lambda}}_1$
respectively. The distribution of the random variable T(t) is given hereafter.
Let
$$
f_i(x, t;\, n)={{\mathrm P}}\{T(t)\in{{\rm d}} x,\;
N(t)=n~|~
\!{\mid}\! {{\varepsilon}}(0)=i\}/{{\rm d}} x,\qquad n\geq1,\ quad
i\in\{0, 1\},$$
be the density function of
$T(t){\mathbf{1}}_{\{N(t)=n\}}$
. Note that
$$
{{\mathrm P}}\{T(t)\in{{\rm d}} x,\;
N(t)=0~|~
\!{\mid}\! {{\varepsilon}}(0)=i\}
={{\rm e}}^{-{{\lambda}}_it}\delta_{c_it}({{\rm d}} x),\qquad i\in\{0, 1\},$$
where
$\delta_{z}({\cdot})$
denotes Dirac’s delta-measure on a line throughout the paper.
Proposition 1.1. The distribution of T(t),
$t>0$
, is described by
\begin{equation}
\begin{split}
f_i(x, t;\, n) = q_i(\xi, t-\xi;\, n)\theta(\xi, t-\xi), \\
\xi=\xi(x)=\dfrac{x-c_1t}{c_0-c_1}, \quad t-\xi=\dfrac{c_0t-x}{c_0-c_1}.
\end{split}
\label{eqn2}
\end{equation}
Here,
$q_i(\xi, \eta;\, n)$
,
$i\in\{0, 1\}$
,
$n\geq1$
, are separately defined for even and odd n by the equalities
\begin{equation}
\begin{split}
q_0(\xi, \eta;\, 2k)=\frac{{{\lambda}}_0^k{{\lambda}}_1^k}{(k-1)!k!}\xi^k\eta^{k-1}, \qquad
q_1(\xi, \eta;\, 2k)=\frac{{{\lambda}}_0^k{{\lambda}}_1^k}{(k-1)!k!}\xi^{k-1}\eta^{k}, \\
q_0(\xi, \eta;\, 2k+1)=\frac{{{\lambda}}_0^{k+1}{{\lambda}}_1^k}{k!^2}\xi^{k}\eta^{k}, \qquad
q_1(\xi, \eta;\, 2k+1)=\frac{{{\lambda}}_0^k{{\lambda}}_1^{k+1}}{k!^2}\xi^{k}\eta^{k},
\end{split}
\label{eqn3}
\end{equation}
$$
\xi, \eta>0,$$
and
\begin{equation}
\theta(\xi,\;
\eta)\,{:\!=}\,\frac{\exp({-}\,{{\lambda}}_0\xi-{{\lambda}}_1\eta)}{c_0-c_1}{\mathbf{1}}_{\{\xi>0,\,\eta>0\}}.
\label{eqn4}
\end{equation}
For the proof, see, e.g., [Reference Kolesnik and Ratanov16, Proposition 4.1]. In the following, Proposition 1.1 will be generalized to the case of a piecewise linear process in an arbitrary linear normed space; see Section 2.1.
The paper is structured as follows. In Section 2 piecewise deterministic flows are studied. After recalling some elementary properties of basic deterministic flows, Section 2 is divided into two main parts: Section 2.1 regarding the distribution of the telegraph process
${{\mathbb T}}(t)$
,
$t\geq0$
, in a normed vector space, and Section 2.2 where we study the time-homogeneous process
${{\mathbb X}}$
defined as
${{\mathbb X}}(t)=\Phi^{-1}(\Phi(x)+{{\mathbb T}}(t))$
,
$t\geq0$
(with
$\Phi$
a continuous injection defined on the state space of the process
${{\mathbb X}}$
). In Section 3 we present two examples: a one-dimensional (1D) squared telegraph process and a two-dimensional process with alternating radial and circular movements. In Section 4 some observations concerning self-similarity are presented.
2. Piecewise deterministic flows
Consider the phase space
$(G, {{\mathcal G}})$
where G is a topological space with the Borel
$\sigma$
-algebra
${{\mathcal G}}$
. For any fixed
$x\in G$
consider a continuous flow on G,
$$
t\to\phi(t)=
\phi(t~|~
\!{\mid}\! x, s)\in G,\qquad t, s\in({-}\infty,\;
\infty),\quad
t>s,$$
starting at time s from position
$x\in G$
:
$\phi(t~|~
\!{\mid}\! x, s)|_{t\downarrow s}=x=\phi(t~|~
\!{\mid}\! x, s)|_{s\uparrow t}$
. Assume that for any s, t,
$s<t$
, the mapping
$x\to\phi^{ts}(x)=\phi(t~|~
\!{\mid}\! x, s)$
,
$t>s$
, is a homeomorphism.
Assume that
$\phi^{ts}$
as well as the inverse mapping (the reverse flow) form a two-parameter semigroup under composition; see, e.g., [Reference Jacobsen11].
In the following we will study piecewise deterministic flows consistently switching between two alternating patterns
$\phi_0(t~|~
\!{\mid}\!{\cdot}\,)$
and
$\phi_1(t~|~
\!{\mid}\!{\cdot}\,)$
at random times.
Let x denote the state of the process at initial time s, and let
$t>s$
. Consider two continuous functions
$\tau\to g_0(\tau)$
,
$\tau\to g_1(\tau)$
,
$\tau\in[s,\;
t]$
, which are defined by iterated superposition of these two flows:
\begin{equation}
g_0(\tau)=\phi_1(t~|~
\!{\mid}\!\phi_0(\tau~|~
\!{\mid}\! x, s), \tau),\qquad
g_1(\tau)=\phi_0(t~|~
\!{\mid}\!\phi_1(\tau~|~
\!{\mid}\! x, s), \tau),\qquad s\leq\tau\leq t.
\label{eqn5}
\end{equation}
These functions determine the pieces of continuous curves
$\ell_0=\ell_0(x)$
and
$\ell_1=\ell_1(x)$
on the space G,
\begin{equation}
\ell_0=\{y\in G~|~
\!{\mid}\! y=g_0(\tau),\;
\tau\in[s, t]\},\qquad
\ell_1=\{y\in G~|~
\!{\mid}\! y=g_1(\tau),\;
\tau\in[s, t]\}.
\label{eqn6}
\end{equation}
For any target point
$y\in\ell_0(x)$
, the time
$\tau_0^*(y;\, x)$
when the flow is switched from
$\phi_0$
to
$\phi_1$
exists and is unique. Indeed, the equation
$g_0(\tau)=y$
has the unique solution
$\tau=\tau_0^*(y;\, x)\in[s,\;
t]$
. Similarly,
$\tau_1^*(y;\, x)\in[s,\;
t]$
,
$y\in\ell_1(x)$
, is defined as the root of the equation
$y=g_1(\tau)$
.
Further, the stochastic switching mechanism between two deterministic flows
$\phi_0$
and
$\phi_1$
is defined by a two-state random process
${{\varepsilon}}={{\varepsilon}}(t)\in\{0, 1\}$
,
$t\in({-}\infty, \infty)$
, with independent inter-switching times.
Let
$s\in({-}\infty, \infty)$
be the (fixed) starting time, and let
$\tau^s$
be the first switching time after s,
$\tau^s>s$
. Denote by
$F^s_i(t)={{\mathrm P}}_i\{\tau^s<t\}={{\mathrm P}}\{\tau^s<t~|~
\!{\mid}\!{{\varepsilon}}(s)=i\}$
the (conditional) distribution function of
$\tau^s$
under the given initial state
${{\varepsilon}}(s)=i$
,
$i\in\{0, 1\}$
. That is,
$$
\label{def:la}
{{\mathrm P}}\{{{\varepsilon}}(t')=i\,\;\forall
\text{for all}\ t'\in(s, t)~|~
\!{\mid}\!{{\varepsilon}}(s)=i\}=1-F^s_i(t)=:\overline{F^s_i}(t),\qquad t>s.$$
We study the marginal distributions of the piecewise deterministic continuous random walk
$\mathbb X=\mathbb X(t)$
on the topological space G which follows two patterns
$\phi_0$
and
$\phi_1$
alternating at switching times of
${{\varepsilon}}$
. Let
$N=N(s, t)$
count the number of switches of
${{\varepsilon}}({\cdot})$
during the time interval [s, t).
By conditioning on the first pattern’s switching, one can observe that the transition probabilities
$P_i(A, t;\,\;
n~|~
\!{\mid}\! x, s)\,{:\!=}\,{{\mathrm P}}\{\mathbb X(t)\in A,\;
N(s, t)=n~|~
\!{\mid}\!\mathbb X(s)=x,\;
{{\varepsilon}}(s)=i\}$
,
$n\geq0$
,
$i\in\{0, 1\}$
, of
${{\mathbb X}}(t),\;t>s,$
satisfy the following coupled integral Chapman–Kolmogorov equations for
$t>s$
:
\begin{equation}
\left\{
\begin{aligned}
P_0(\,{\cdot}\,, t;\,\;
n~|~
\!{\mid}\! x, s)=&
\int_s^{t}P_1(\,{\cdot}\,, t;\,\;
n-1~|~
\!{\mid}\!\phi_0(\tau~|~
\!{\mid}\! x,s), \tau)\,{{\rm d}} F^s_0(\tau),\\
P_1(\,{\cdot}\,, t;\,\;
n~|~
\!{\mid}\! x,s)=&\int_s^{t}
P_0(\,{\cdot}\,, t;\,\;
n-1~|~
\!{\mid}\!\phi_1(\tau~|~
\!{\mid}\! x,s), \tau)\,{{\rm d}} F^s_1(\tau),
\end{aligned}
\qquad n\geq1.
\right.
\label{eqn7}
\end{equation}
The distribution of
${{\mathbb X}}(t)$
with no switchings till time t is given by
\begin{equation}
\begin{aligned}
P_0(A, t;\,\;
0~|~
\!{\mid}\! x, s)=\ &(1-F^s_0(t))\ \delta_{\phi_0(t~|~
\!{\mid}\! x, s)}(A),\\
P_1(A, t;\,\;
0~|~
\!{\mid}\! x, s)=\ &(1-F^s_1(t))\ \delta_{\phi_1(t~|~
\!{\mid}\! x, s)}(A).
\end{aligned}
\label{eqn8}
\end{equation}
In the following we consider in detail the Markovian case, that is,
$$\label{def:markov}
F_i^s(t)={{\mathrm P}}_i\{\tau^s<t\}=1-{{\rm e}}^{-{{\lambda}}_i(t-s)},\qquad t\geq s,\quad
i\in\{0, 1\},
\end{equation*}
with
${{\lambda}}_0,{{\lambda}}_1>0$
.
We begin with the example of a random walk
${{\mathbb T}}(t)$
that follows a linear flow in a linear normed space.
2.1. Piecewise linear processes in a linear normed space
Let V be a linear normed vector space and
$c_0, c_1\in V$
,
$c_0\neq c_1$
. We consider the linear time-homogeneous case, where
${{\mathbb T}}={{\mathbb T}}(t)$
,
$t\geq0$
, is the piecewise linear process (the integrated telegraph process) on the space V, switching between two linear flows
$$
\phi_0(t~|~
\!{\mid}\! x, s)=x+tc_0,\qquad
\phi_1(t~|~
\!{\mid}\! x, s)=x+tc_1.$$
The current position
${{\mathbb T}}(t)$
is given by
\begin{equation}
{{\mathbb T}}(t)\,{:\!=}\,\int_0^tc_{{{\varepsilon}}(\tau)}\,{{\rm d}}\tau
=\sum_{n=0}^{N(t)-1}c_{{{\varepsilon}}_n}(\tau_{n+1}-\tau_{n})+c_{{{\varepsilon}}_{N(t)}}(t-\tau_{N(t)}),\qquad t\geq0,
\label{eqn9}
\end{equation}
where
$\tau_n$
,
$n\geq0$
, are the switching times,
$\tau_0=0$
,
${{\varepsilon}}_n={{\varepsilon}}(\tau_n)$
,
$n\geq0$
, and N(t) is the number of switchings occurring till time t,
$t>0$
,
$N(0)=0$
.
The distribution of
${{\mathbb T}}(t)$
,
$t>0$
, is supported on the straight segment
$I_t\subset V$
,
\begin{equation}
I_t=\{z\in V~|~
\!{\mid}\! z=\tau c_0+(t-\tau)c_1,\ 0\leq\tau\leq t\}.
\label{eqn10}
\end{equation}
Indeed, for any
$z\in I_t$
, we have
${{\mathbb T}}(t)=z=\tau c_0+(t-\tau)c_1$
, where
$\tau\in[0, t]$
is the time spent by the underlying Markov process
${{\varepsilon}}(u)$
,
$0\leq u\leq t$
, in state 0.
Due to (2.3), the distribution densities
$$
\label{def:pT}
\begin{aligned}
p_0^{{\mathbb T}}(z, t;\,\;
n)\,{:\!=}\ &{{\mathrm P}}\{{{\mathbb T}}(t)\in{{\rm d}} z,\;
N(t)=n~|~
\!{\mid}\!{{\varepsilon}}(0)=0\}/{{\rm d}} z, \\
p_1^{{\mathbb T}}(z, t;\,\;
n)\,{:\!=}\ &{{\mathrm P}}\{{{\mathbb T}}(t)\in{{\rm d}} z,\;
N(t)=n~|~
\!{\mid}\! {{\varepsilon}}(0)=1\}/{{\rm d}} z,
\end{aligned}$$
follow the coupled integral equations
\begin{equation}
\left\{
\begin{aligned}
p_0^{{\mathbb T}}(z, t;\,\;
n)=
&\int_0^t{{\lambda}}_0{{\rm e}}^{-{{\lambda}}_0\tau}p_1^{{\mathbb T}}(z-\tau c_0, t-\tau;\,\;
n-1)\,{{\rm d}}\tau, \\
p_1^{{\mathbb T}}(z, t;\,\;
n)=
&\int_0^t{{\lambda}}_1{{\rm e}}^{-{{\lambda}}_1\tau}p_0^{{\mathbb T}}(z-\tau c_1, t-\tau;\,\;
n-1)\,{{\rm d}}\tau,
\end{aligned}\qquad n\geq1.\right.
\label{eqn11}
\end{equation}
The case of no switchings, corresponding to
${{\mathbb T}}(t){\mathbf{1}}_{N(t)=0}$
, is given by
\begin{equation}
\begin{aligned}
{{\mathrm P}}\{{{\mathbb T}}(t)\in{{\rm d}} z,\;
N(t)=0~|~
\!{\mid}\!{{\varepsilon}}(0)=0\}&=\exp({-}\,{{\lambda}}_0t)\delta_{tc_0}({{\rm d}} z)\\
& =\exp({-}\,{{\lambda}}_0t)\delta(z-tc_0){{\rm d}} z,\\
{{\mathrm P}}\{{{\mathbb T}}(t)\in{{\rm d}} z,\;
N(t)=0~|~
\!{\mid}\!{{\varepsilon}}(0)=1\}&=\exp({-}\,{{\lambda}}_1t)\delta_{tc_1}({{\rm d}} z)\\
& =\exp({-}\,{{\lambda}}_1t)\delta(z-tc_1){{\rm d}} z.
\end{aligned}
\label{eqn12}
\end{equation}
In the particular case of linearly dependent vectors
$c_0, c_1\in V$
,
$c_0,c_1\neq0$
, the random process
${{\mathbb T}}={{\mathbb T}}(t)$
is one-dimensional and the distribution of
${{\mathbb T}}(t)$
is supported on the segment
$I_t$
of the straight line L with direction vector
$c_0$
(or
$c_1$
),
$I_t\subset L\subset V$
. Moreover, the density functions
$p_0^{{\mathbb T}}(\,{\cdot}\,, t;\,\;
n)$
and
$ p_1^{{\mathbb T}}(\,{\cdot}\,, t;\,\;
n)$
,
$n\geq1$
, coincide with functions
$f_0(\,{\cdot}\,, t;\,\;
n)$
and
$f_1(\,{\cdot}\,, t;\;
n)$
; see the formulae in (1.2) with
$\xi$
,
$0\leq\xi\leq t$
, defined by the equation
$z-tc_1=\xi(c_0-c_1)$
,
$z\in L$
.
In general, the segment
$I_t$
given in (2.6) floats in V (with constant velocity
$\frac12(c_0+c_1)$
). By solving the equations in (2.7), the density functions
$p_0^{{\mathbb T}}(z, t;\,\;
n)$
and
$ p_1^{{\mathbb T}}(z, t;\,\;
n)$
,
$n\geq1$
, can be shown to satisfy formulae similar to (1.2) with
$\xi\in[0, t]$
, which is defined as the (unique) solution
$\xi=\varphi(z, t)$
of the equation
\begin{equation}
z-tc_1=\xi(c_0-c_1),\qquad z\in I_t.
\label{eqn13}
\end{equation}
Proposition 2.1. The density functions
$p_0^{{\mathbb T}}(z, t;\,\;
n)$
and
$ p_1^{{\mathbb T}}(z, t;\,\;
n)$
,
$n\geq1$
, are given by
$p_i^{{\mathbb T}}(z, t;\, \;
n)=q_i(\xi, t-\xi;\,\;
n)\theta(\xi, t-\xi)$
, where
$q_i(\xi, \eta;\, \;
n)$
are defined by (1.3), and the function
$\theta$
is
\begin{equation}
\theta(\xi,\;
\eta)\,{:\!=}\,\frac{1}{\|c_0-c_1\|} \exp({-}\,{{\lambda}}_0\xi-{{\lambda}}_1\eta)
{\mathbf{1}}_{\{\xi>0, \eta>0\}}.
\label{eqn14}
\end{equation}
Here,
$\xi=\varphi(z, t)\in[0, t]$
,
$z\in I_t$
is the solution of (2.9) and
$\eta=t-\xi$
.
See the proof in Appendix B.
2.2. Time-homogeneous piecewise deterministic process
${{\mathbb X}}$
Consider the time-homogeneous case, so that the deterministic pattern
$\phi(t~|~
\!{\mid}\! x, s)$
depends on s, t through
$t-s$
only. Assume that the flow
$\phi$
is defined by
\begin{equation}
t\to\phi(t~|~
\!{\mid}\! x, s)=\Phi^{-1}(\Phi(x)+c(t-s)),\qquad t\geq s,
\label{eqn15}
\end{equation}
where
$\Phi\,{:}\, G\to V$
is a continuous injective map from G to a topological vector space V and
$c\in V$
is a constant. The reverse flow is defined by
$s\to\Phi^{-1}(\Phi(y)-c(t-s))$
,
$s\leq t$
.
In the following we will use the shortened notation
$$\label{eq:notations}
\phi(t;\, x)\,{:\!=}\,\phi(t~|~
\!{\mid}\! x, 0).$$
Remark 2.1. Let
$G={{\mathbb R}}^d$
,
$V={{\mathbb R}}^d$
, and
$\Phi\,{:}\, {{\mathbb R}}^d\to{{\mathbb R}}^d$
be a diffeomorphism. Therefore, the trajectory of
$\phi$
defined by (2.11) is differentiable,
$\Phi(\phi(t;\, x))=\Phi(x)+ct$
, and
$$
\frac{{\rm d}}{{{\rm d}} t}[\Phi(\phi(t;\, x))]\equiv c,\qquad t>0.$$
This means that
$\phi$
follows the differential equation
\begin{equation}
\frac{{{\rm d}} \phi(t;\, x)}{{{\rm d}} t}=a( \phi(t;\, x)),\qquad t>0,
\label{eqn16}
\end{equation}
with the initial condition
$ \phi(t;\, x)|_{t\downarrow 0}=x$
, where
$a(y)=[\Phi'(y)]^{-1}c$
.
The mapping
$\Phi$
acts as a rectifying diffeomorphism for equation (2.12); see [Reference Arnold1].
In the case when the time-homogeneous flows
$\phi_0$
and
$\phi_1$
are defined by (2.11) with
$c_0, c_1\,{\in}\, V$
,
$c_0\neq c_1$
, and are characterized by a common rectifying mapping
$\Phi\,{:}\,G\to V$
, that is,
$$
\phi_0(t~|~
\!{\mid}\! x, s)=\Phi^{-1}(\Phi(x)+c_0(t-s)),\qquad
\phi_1(t~|~
\!{\mid}\! x, s)=\Phi^{-1}(\Phi(x)+c_1(t-s)),
\qquad t\geq s,$$
the mappings
$g_0$
and
$g_1$
defined by (2.1) become
$$\label{eq:lg0}
\begin{aligned}
g_0(\tau)
&=\Phi^{-1}(\Phi(x)+c_0\tau+c_1(t-\tau)),\qquad \tau\in[0,\;
t],\\
g_1(\tau)
&=\Phi^{-1}(\Phi(x)+c_1\tau+c_0(t-\tau)),\qquad \tau\in[0,\;
t].
\end{aligned}$$
Hence, the curves
$\ell_0$
and
$\ell_1$
defined in (2.2) identify
$$\ell\,{:\!=}\,\ell_0=\ell_1=\Phi^{-1}(\Phi(x)+I_{t}),$$
where
$I_{t}$
is the straight segment (2.6).
Let the time-homogeneous flows
$\phi_0=\phi_0(t;\, x)$
and
$\phi_1=\phi_1(t;\, x)$
,
$0\leq t<\infty$
, be defined by (2.11) with a common diffeomorphism
$\Phi\,{:}\, G\to V$
from the open subset G of a linear normed space into a linear normed space V, and with constant ‘velocities’
$c_0, c_1\in V$
,
$c_0\neq c_1$
. Therefore, the corresponding piecewise deterministic time-homogeneous continuous process
${{\mathbb X}}^x={{\mathbb X}}^x(t)\in G$
starting from point x is defined by
\begin{equation}
{{\mathbb X}}^x(t)=\Phi^{-1}(\Phi(x)+{{\mathbb T}}(t)),\quad 0\leq t<\infty;\qquad
{{\mathbb X}}^x(0)=x.
\label{eqn17}
\end{equation}
Here,
${{\mathbb T}}={{\mathbb T}}(t)$
,
$t\geq0$
, is the telegraph process defined by (2.5) with the two velocities
$c_0, c_1\in V$
alternating with switching intensities
${{\lambda}}_0, {{\lambda}}_1>0$
.
For any fixed
$t>0$
, the distribution of
${{\mathbb T}}(t)$
is supported on the straight segment
$I_t\subset V$
; see Proposition 2.1. Hence, the distribution of
$\mathbb X^x(t)$
is supported on the segment of the continuous curve
$\ell=\ell_{t,x}$
,
$\ell\subset G$
,
$\ell=\Phi^{-1}(\Phi(x)+I_{t })$
; see Figure 1.
Flows
$\phi_0(\,{\cdot}\,;\,\;
x)$
and
$\phi_1(\,{\cdot}\,;\,\;
x)$
with common mapping
$\Phi\,:\, G\to V$
; a sample path of
${{\mathbb X}}^x(t)$
.
Let
$p_0^{{\mathbb X}}(y, t;\, n~|~
\!{\mid}\! x)$
and
$p_1^{{\mathbb X}}(y, t;\, n~|~
\!{\mid}\! x)$
be the transition densities of
${{\mathbb X}}(t)$
,
$t>s$
:
$$
\label{def:density}
p_i^{{\mathbb X}}(y, t;\, n~|~
\!{\mid}\! x)\,{{\rm d}} y\,{:\!=}\,{{\mathrm P}}\{{{\mathbb X}}^x(t)\in{{\rm d}} y,\;
N(t)=n~|~
\!{\mid}\!{{\varepsilon}}(0)=i\},\,\,
\qquad i\in\{0, 1\},\,\,
n=0, 1, 2,\ldots.$$
In the case of no switchings,
$n=0$
, by (2.4) we have
$$\begin{aligned}
p_0^{\mathbb X}(y, t;\, 0~|~
\!{\mid}\! x)
={{\rm e}}^{-{{\lambda}}_0t}\delta(y-\phi_0(t;\, x)),\qquad
p_1^{\mathbb X}(y, t;\, 0~|~
\!{\mid}\! x)
={{\rm e}}^{-{{\lambda}}_1t}\delta(y-\phi_1(t;\, x)).
\end{aligned}$$
Theorem 2.1. The transition densities
$p_i^{{\mathbb X}}(y, t;\, n~|~
\!{\mid}\! x)$
,
$n\geq1$
, for each positive t are given by Proposition 2.1 with
$\xi=\varphi(\Phi(y)-\Phi(x),\;
t)$
, see (2.9), and with
$\theta$
given by
$$\label{def:theta0}
\begin{aligned}
\theta&=k(y)\exp\{\left(
-{{\lambda}}_0\xi-{{\lambda}}_1(t-\xi)
\} \\ \right)\\
&=
k(y)\exp\{\left(
-{{\lambda}}_0\varphi(\Phi(y)-\Phi(x),\;
t)-{{\lambda}}_1(t-\varphi(\Phi(y)-\Phi(x),\;
t))
\},\right),
\end{aligned}$$
where
$k(y)=\dfrac{\|\Phi'(y)\|}{\|c_0-c_1\|}{\mathbf{1}}_{\{y\in\ell\}}$
.
Further,
\begin{equation}
\begin{aligned}
p_0^{\mathbb X}(y, t~|~
\!{\mid}\! x)= {{\rm e}}^{-{{\lambda}}_0t}\delta(y-\phi_0(t;\, x))
& +\theta\mathcal P_0(\xi, t-\xi;\, t),\\
\label{eq:p1bessel}
p_1^{\mathbb X}(y, t~|~
\!{\mid}\! x)= {{\rm e}}^{-{{\lambda}}_1t}\delta(y-\phi_1(t;\, x))
& + \theta \mathcal P_1(\xi, t-\xi;\, t),
\end{aligned}
\label{eqn18}
\end{equation}
where
\begin{equation}
\begin{aligned}
\mathcal P_0(\xi, \eta;\, t)&={{\lambda}}_0I_0(2\sqrt{{{\lambda}}_0{{\lambda}}_1\xi{\cdot}\eta})
+\sqrt{{{\lambda}}_0{{\lambda}}_1\xi/\eta}I_1(2\sqrt{{{\lambda}}_0{{\lambda}}_1\xi{\cdot}\eta}), \\
\mathcal P_1(\xi, \eta;\, t)&={{\lambda}}_1I_0(2\sqrt{{{\lambda}}_0{{\lambda}}_1\xi{\cdot}\eta})
+\sqrt{{{\lambda}}_0{{\lambda}}_1\eta/\xi}I_1(2\sqrt{{{\lambda}}_0{{\lambda}}_1\xi{\cdot}\eta}).
\end{aligned}
\label{eqn19}
\end{equation}
Proof. By (2.13),
$$
{{\mathrm P}}_i\{{{\mathbb X}}(t)\in{{\rm d}} y~|~
\!{\mid}\!{{\mathbb X}}(0)=x\}={{\mathrm P}}_i\{\Phi(x)+{{\mathbb T}}(t)\in\Phi({{\rm d}} y)\},
\qquad i\in\{0, 1\}.$$
The proof follows from the result of Proposition 2.1. Summing over n one can obtain (2.14).
The next section is related to other examples.
3. Examples
3.1. Squared telegraph process
First, we present the important example of the squared telegraph process,
$${{\mathbb X}}(t)={{\mathbb X}}^x(t)=(\sqrt{x}+T(t))^2,\qquad t>0,$$
${{\mathbb X}}^x(0)=x$
, where the underlying telegraph process
$T=T(t)$
is determined by velocities
$c_0, c_1$
,
$c_0>c_1$
, and switching intensities
${{\lambda}}_0, {{\lambda}}_1$
(see (1.1)). Such a process can be obtained by (2.13), with
$\Phi(x)=\sqrt{x}$
,
$x\geq0$
.
Although
$x\to\Phi^{-1}(x)=x^2$
,
$x\in({-}\infty, \infty)$
, is not a diffeomorphism, Theorem 2.1 can be applied.
The density functions
$p_i(\,{\cdot}\,, t;\, n~|~
\!{\mid}\! x)$
,
$n\geq1$
, of
${{\mathbb X}}^x(t)$
can be expressed using
$f_0(x, t;\, n)$
and
$f_1(x, t;\, n)$
defined in (1.2)–(1.4). The explicit expressions for
$p_i(\,{\cdot}\,, t;\, n~|~
\!{\mid}\! x)$
,
$n\geq1$
, are different for the following four cases, defined by the four possible relationships between the parameters and the time value t,
$t>0$
.
(A)
${0\leq\sqrt{x}+c_1t<\sqrt{x}+c_0t}$
:
The distribution of
${{\mathbb X}}^x(t)$
is supported on the segment
$$\Delta_{\rm A}\,{:\!=}\,[(\sqrt{x}+c_1t)^2,\;
(\sqrt{x}+c_0t)^2]\subset{{\mathbb R}}^1_+,$$
the equation
$(\sqrt{x}+z)^2=y$
,
$y\in\Delta_{\rm A}$
, has the unique solution
$z=\sqrt{y}-\sqrt{x}$
, and
\begin{equation}
p_i(y, t;\, n~|~
\!{\mid}\! x)=\frac{1}{2\sqrt{y}\,}f_i(\sqrt{y}-\sqrt{x}, t;\, n),
\qquad n\geq1,\quad
i\in\{0, 1\},\qquad
y\in\Delta_{\rm A}.;
\label{eqn20}
\end{equation}
(B)
${\sqrt{x}+c_1t<0<-\sqrt{x}-c_1t\leq\sqrt{x}+c_0t}$
:
The distribution of
${{\mathbb X}}^x(t)$
is supported on
$$\Delta_{\rm B}\,{:\!=}\,[0,\;
(\sqrt{x}+c_0t)^2]\subset{{\mathbb R}}^1_+.$$
For all y,
$0<y\leq(\sqrt{x}+c_1t)^2$
, the equation
$(\sqrt{x}+z)^2=y$
has two roots
$z=\pm\sqrt{y}-\sqrt{x}$
; if
$(\sqrt{x}+c_1t)^2<y\leq(\sqrt{x}+c_0t)^2$
this equation has the unique solution
$z=\sqrt{y}-\sqrt{x}$
between
$c_1t$
and
$c_0t$
. Hence, for
$n\geq1$
,
$i\in\{0, 1\}$
, the density function
$p_i(y, t;\, n~|~
\!{\mid}\! x)$
is given by
\begin{equation}
\frac{1}{2\sqrt{y}}\left\{
\begin{aligned}
&f_i({-}\,\sqrt{y}-\sqrt{x}, t;\, n)+f_i(\sqrt{y}-\sqrt{x}, t;\, n),\quad
& &
0<y<(\sqrt{x}+c_1t)^2,\\
& f_i(\sqrt{y}-\sqrt{x}, t;\, n), & & \qquad
(\sqrt{x}+c_1t)^2<y\leq(\sqrt{x}+c_0t)^2 . ;
\end{aligned}
\right.
\label{eqn21}
\end{equation}
(C)
${\sqrt{x}+c_1t\leq-\sqrt{x}-c_0t<0<\sqrt{x}+c_0t}$
:
The distribution of
${{\mathbb X}}^x(t)$
is supported on
$$\Delta_{\rm C}\,{:\!=}\,[0,\;
(\sqrt{x}+c_1t)^2]\subset{{\mathbb R}}^1_+.$$
For all y,
$0<y\leq(\sqrt{x}+c_0t)^2$
, the equation
$(\sqrt{x}+z)^2=y$
has two roots
$z=\pm\sqrt{y}-\sqrt{x}$
; if
$(\sqrt{x}+c_0t)^2<y\leq(\sqrt{x}+c_1t)^2$
, this equation has the unique solution
$z=-\sqrt{y}-\sqrt{x}$
between
$c_1t$
and
$c_0t$
. Hence, for
$n\geq1$
,
$i\in\{0, 1\}$
, the density function
$p_i(y, t;\, n~|~
\!{\mid}\! x)$
is given by
\begin{equation}
\frac{1}{2\sqrt{y}}\left\{
\begin{aligned}
& f_i({-}\,\sqrt{y}-\sqrt{x}, t;\, n)+f_i(\sqrt{y}-\sqrt{x}, t;\, n), & & \quad
y<(\sqrt{x}+c_0t)^2,\\
& f_i({-}\,\sqrt{y}-\sqrt{x}, t;\, n), & & \quad
(\sqrt{x}+c_0t)^2<y\leq(\sqrt{x}+c_1t)^2,
\end{aligned}
\right.
\label{eqn22}
\end{equation}
$$\qquad n\geq1,\quad i\in\{0, 1\}.$$
(D)
${\sqrt{x}+c_1t<\sqrt{x}+c_0t\leq0}$
:
The distribution of
${{\mathbb X}}^x(t)$
is supported on the segment
$$\Delta_{\rm D}\,{:\!=}\,[(\sqrt{x}+c_0t)^2,\;
(\sqrt{x}+c_1t)^2]\subset{{\mathbb R}}^1_+,$$
the equation
$(\sqrt{x}+z)^2=y$
,
$y\in\Delta_{\rm D}$
, has the unique root
$z=-\sqrt{y}-\sqrt{x}$
. Thus
\begin{equation}
p_i(y, t;\, n~|~
\!{\mid}\! x)=\frac{1}{2\sqrt{y}}f_i({-}\,\sqrt{y}-\sqrt{x}, t;\, n),
\qquad n\geq1,\quad
i\in\{0, 1\},\qquad
y\in\Delta_{\rm D}.
\label{eqn23}
\end{equation}
As a result, the distribution of
${{\mathbb X}}(t)$
depends on the signs of velocities.
First, if both velocities are positive,
$c_0>c_1\geq0$
, then T(t) is a subordinator and the distribution of
${{\mathbb X}}^x(t)=(\sqrt{x}+T(t))^2$
fits case (A).
Second, let
$c_0\geq0>c_1$
. For sufficiently small times,
$0<t\leq \sqrt{x}/({-}\,c_1)$
, the value
$\sqrt{x}+T(t)$
remains positive. Hence the density functions
$p_i(y, t;\, n~|~
\!{\mid}\! x)$
,
$i\in\{0, 1\}$
, are again given by (3.1) (case (A)).
For large t the solution depends on the relation between
$c_0$
and
$|c_1|$
.
If
$c_0+c_1<0$
and
$\sqrt{x}/({-}\,c_1)<t\leq2\sqrt{x}/({-}\,c_0-c_1)$
or
$c_0+c_1\geq0$
and
$t>\sqrt{x}/({-}\,c_1)$
, then
$\sqrt{x}+c_1t<0<-\sqrt{x}-c_1t<\sqrt{x}+c_0t$
, which corresponds to case (B). Hence, the formula (3.2) holds.
If
$c_0+c_1<0$
and
$t\geq2\sqrt{x}/({-}\,c_0-c_1),$
then
$\sqrt{x}+c_1t<-\sqrt{x}-c_0t<0<\sqrt{x}+c_0t$
, which is case (C), and (3.3) holds.
Third, let both velocities be negative,
$0>c_0>c_1$
. The distribution of
${{\mathbb X}}^x(t)$
is given separately for the different time intervals:
$$
\begin{aligned}
0<t\leq \sqrt{x}/({-}\,c_1)& \Rightarrow \qquad
\text{case (A) and formula (\ref{eqn20}); }\\
\sqrt{x}/({-}\,c_1)<t\leq2\sqrt{x}/({-}\,c_0-c_1)& \Rightarrow \qquad
\text{case (B) and formula (\ref{eqn21});}\\
2\sqrt{x}/({-}\,c_0-c_1)\leq t<\sqrt{x}/({-}\,c_0)& \Rightarrow \qquad
\text{case (C) and formula (\ref{eqn22});}\\
t>\sqrt{x}/({-}\,c_0)& \Rightarrow \qquad
\text{case (D) and formula (\ref{eqn23}).}
\end{aligned}$$
If
$t=2\sqrt{x}/({-}\,c_0-c_1)$
(with
$c_0+c_1<0$
), case (B) coincides with case (C) and
$p_i(y, t;\, n~|~
\!{\mid}\! x)=\dfrac{1}{2\sqrt{y}}
[ f_i({-}\,\sqrt{y}-\sqrt{x}, t;\, n)+f_i(\sqrt{y}-\sqrt{x}, t;\, n)]$
,
$0<y<(\sqrt{x}+c_1t)^2$
.
A slightly different approach is given in [Reference Martinucci and Meoli20].
3.2. Process in the plane and polar coordinates
The piecewise deterministic process in the plane has been studied in the past in various contexts [Reference Garra, Orsingher and Ratanov9, Reference Kolesnik14, Reference Kolesnik and Orsingher15, Reference Orsingher21, Reference Orsingher and Ratanov22]. Here we present an example of planar motion in the spirit of our construction (2.13).
Let
$\Phi({{x}})=(r({{x}}),\;
\alpha({{x}}))$
,
${{x}}=(x_1, x_2)\in{{\mathbb R}}^2$
,
${{x}}\neq{{\mathbf 0}}$
, be the operator setting the polar coordinates
$r({{x}})=|{{x}}|=\sqrt{x_1^2+x_2^2}>0$
and
$\alpha({{x}})\in S^1$
for any
${{x}}=(x_1, x_2)\in{{\mathbb R}}^2$
,
${{x}}\neq{{\mathbf 0}}$
. The mapping
$\Phi$
is the (local) diffeomorphism from
${{\mathbb R}}^2\setminus\{{{\mathbf 0}}\}$
to the semi-cylinder
$(0,\;
+\infty)\times S^1$
.
Let
$\mathcal J\,{:}\, {{\mathbb C}}\to{{\mathbb R}}^2$
be defined by
$$
\mathcal J(z)=(r\cos(\alpha),\;
r\sin(\alpha))^\top,\qquad z=r{{\rm e}}^{{\rm i}\alpha}\in{{\mathbb C}}.$$
Consider the two basic deterministic flows
$\phi_0(t;\, {{x}})$
and
$\phi_1(t;\, {{x}})$
defined by (2.13) with
${{c}}={{c}}_0=(c_0, 0)^\top$
and
${{c}}={{c}}_1=(0, c_1)^\top$
respectively. Here,
$c_0>0$
is the velocity of a radial flight and
$c_1>0$
is the constant angular velocity.
The flow
$$\phi_0(t;\, {{x}})=\widehat r_{c_0t}({{x}})={{x}}+c_0t{{x}}/|{{x}}|=(1+c_0t/|{{x}}|){{x}}$$
is the radial movement starting from point
${{x}}\in {{\mathbb R}}^2$
,
${{x}}\neq{{\mathbf 0}}$
, and the flow
$\phi_1(t;\, {{x}})$
is the circular motion defined by rotation of
${{x}}$
:
$$\phi_1(t;\, {{x}})=\widehat \omega_{c_1t}({{x}})
=\mathcal J(r({{x}}){{\rm e}}^{{\rm i}(\alpha+c_1t)}),\qquad t\geq0.$$
The process
${{\mathbb X}}^{{x}}$
is defined by the radial-circular motion, switching from radial to circular motion with intensity
${{\lambda}}_0$
and vice versa with intensity
${{\lambda}}_1$
.
The distribution of
${{\mathbb X}}^{{x}}(t)$
is supported on the segment
$\ell=\ell(t, {{x}})$
of the Archimedean spiral,
${{y}}\in\ell(t, {{x}})$
(Figure 2),
\begin{equation}
\left\{
\begin{aligned}
y_1&=(r({{x}})+c_0\tau)\cos(\alpha({{x}})+c_1(t-\tau)), \\
y_2&=(r({{x}})+c_0\tau)\sin(\alpha({{x}})+c_1(t-\tau)),
\end{aligned}
\right.\qquad\tau\in[0,\;
t].
\label{eqn24}
\end{equation}
The support of the distribution of
${{\mathbb X}}(t)$
: the Archimedean spiral
$\ell({{x}}, t)$
defined by (3.5) with
${{x}}=(10, 0)$
,
$c_0=2$
,
$c_1=3$
, and time
$t=10$
.
Let
$\xi=\xi({{x}}, {{y}})=\dfrac{|{{y}}|-|{{x}}|}{c_0}$
,
${{y}}\in\ell(t, {{x}})$
, be the total time of radial motion,
$0\leq\xi\leq t$
, such that the remaining time,
$t-\xi$
, is the total time of circular motion.
From Theorem 2.1, the density functions
$p_i({{y}}, t;\, n~|~
\!{\mid}\!{{x}})$
of
${{\mathbb X}}^{{x}}(t)$
are given by
\begin{equation}
p_i({{y}}, t;\, n~|~
\!{\mid}\!{{x}})\,{{\rm d}}{{y}}=\frac{|{{y}}|}{\sqrt{c_0^2+c_1^2}}
q_i(\xi, t-\xi;\, n)\theta({{x}}, {{y}})\delta_\ell({{\rm d}} {{y}}),
\exp\left({-}\,{{\lambda}}_0\xi-{{\lambda}}_1(t-\xi)\right),
\qquad i\in\{0, 1\},\quad
n\geq1 , .
\label{eqn25}
\end{equation}
where
$$\theta({{x}},{{y}})=\dfrac{|{{y}}|}{\sqrt{c_0^2+c_1^2}\,}\exp({-}\,{{\lambda}}_0\xi-{{\lambda}}_1(t-\xi)),$$
and
$q_i(\xi, \eta;\, n)$
are defined by (1.3). If there are no switchings, we have
$$\begin{aligned}
p_0({{y}}, t;\, 0~|~
\!{\mid}\!{{x}})&={{\rm e}}^{-{{\lambda}}_0t}\delta( {{y}}-\widehat r_{c_0t}({{x}})),\\
p_1({{y}}, t;\, 0~|~
\!{\mid}\!{{x}})&={{\rm e}}^{-{{\lambda}}_1t}\delta( {{y}}-\widehat\omega_{c_1t}({{x}})).
\end{aligned}$$
Here,
$$\widehat r_{c_0t}({{x}})={{x}}\Big(1+c_0t\dfrac{{{x}}}{|{{x}}|}\Big)$$
is the radial displacement and
$\widehat \omega_\alpha({{x}})$
denotes the rotation of
${{x}}$
.
The density functions
$p_i({{y}}, t~|~
\!{\mid}\!{{x}})$
,
$i\in\{0, 1\}$
,
${{y}}\in\ell({{x}}, t)$
, can be obtained by summing up in (3.6) similarly to (2.14) and (2.15); see Figure 3.
The regular part of the density function
$p_0(\,{\cdot}\,, 10 ~|~
\!{\mid}\!{{x}})$
with
$c_0=c_1=1$
,
${{\lambda}}_0={{\lambda}}_1=2$
, and the initial point
${{x}}=(1, 1)$
.
4. Self-similarity
The process
${{\mathbb X}}^x={{\mathbb X}}^x(t)\in {{\mathbb R}}_+^1$
is called positive self-similar if there exists a constant
$\gamma > 0$
such that, for any
$x > 0$
and
$R > 0$
,
\begin{equation}
R\,{\cdot}\,\mathbb X^{x}(R^{-\gamma}t)
~\text{is equal in law to}~{{\mathbb X}}^{Rx}(t),\qquad t\geq0 ; ,
\label{eqn26}
\end{equation}
see the definition in [Reference Kyprianou17, Chapter 13].
The following theorem characterizes piecewise deterministic positive (1D) self-similar processes.
Theorem 4.1. Let
${{\mathbb X}}^x={{\mathbb X}}^x(t)\in{{\mathbb R}}^1_+$
,
$x>0$
, be the positive piecewise deterministic time-homogeneous process with two alternating patterns
$\phi_0$
,
$\phi_1$
based on a common rectifying diffeomorphism
$\Phi$
, (2.13), such that
$\phi_0=\Phi^{-1}(\Phi(x)+c_0t)$
and
$\phi_1=\Phi^{-1}(\Phi(x)+c_1t)$
with
$c_0, c_1>0$
.
The process
${{\mathbb X}}^x$
is positive self-similar with index
$\gamma>0$
if and only if the underlying patterns are given by
$\Phi(x)=x^{\gamma}$
,
$x\in{{\mathbb R}}^1_+$
.
Proof. Let
${{\mathbb X}}^x$
be the piecewise deterministic time-homogeneous process based on the two patterns
\begin{equation}
\phi_i(t;\, x)=(x^{\gamma}+c_it)^{1/\gamma},\qquad t\geq0,\quad
x>0, \qquad
i\in\{0, 1\} , .
\label{eqn27}
\end{equation}
with
$c_0, c_1>0$
.
Note that the flows
$\phi_i(t;\, x)$
,
$i\in\{0,1\}$
, defined by (4.2) satisfy the scaling relation
\begin{equation}
\phi_i(R^{-\gamma}t;\, R^{-1}x)=R^{-1}\phi_i(t;\, x),\qquad x>0,\ \;
t\geq0, \qquad
i\in\{0, 1\}.
\label{eqn28}
\end{equation}
Moreover, under the time scaling
$t\to R^{-\gamma}t$
the switching intensities are transformed as
\begin{equation}
\lambda_0\to R^\gamma\lambda_0,\qquad \lambda_1\to R^\gamma\lambda_1.
\label{eqn29}
\end{equation}
Therefore, the piecewise deterministic process
$\mathbb X^x(t)$
,
$t\geq 0$
, which follows the patterns (4.2), switching from one to another with alternating intensities
${{\lambda}}_0$
,
${{\lambda}}_1$
, is the positive self-similar continuous process with index
$\gamma$
, (4.1).
Note that this can also be verified by using explicit formulae for the distribution. Let
$\Phi(x)=x^\gamma$
,
$x>0$
. Under the space–time scaling
$x\to R^{-1}x$
,
$t\to R^{-\gamma}t$
the variable
$\xi=\varphi(\Phi(y)-\Phi(x),t)$
, (2.9), used in Theorem 2.1, is transformed as
$\xi\to R^{-\gamma}\xi$
. Hence, by Theorem 2.1 and Equations (4.4) and (4.3), the transition densities
$p_i^{{\mathbb X}}(\,{\cdot}\,, t;\, n~|~
\!{\mid}\! x)$
satisfy the relation
$$\label{eq:ssn-gamma}
R^{-1}p_i^{\mathbb X}(R^{-1}y, R^{-\gamma}t, n~|~
\!{\mid}\! R^{-1}x)
|_{\lambda_0\to R^\gamma\lambda_0,\,\lambda_1\to R^\gamma\lambda_1}
\equiv p_i^{\mathbb X}(y, t, n~|~
\!{\mid}\! x),\qquad n\geq0,\ t>0.$$
The same is fulfilled for
$p_i^{{\mathbb X}}(y, t~|~
\!{\mid}\! x)$
:
$$\label{eq:ssinfty}
R^{-1}p_i^{\mathbb X}(R^{-1}y, R^{-\gamma}t~|~
\!{\mid}\! R^{-1}x)
|_{\lambda_0\to R^\gamma\lambda_0,\,\lambda_1\to R^\gamma\lambda_1}
\equiv p_i^{\mathbb X}(y, t~|~
\!{\mid}\! x),$$
$i\in\{0, 1\}$
, which corresponds to (4.1).
To prove the inverse assertion note that by definition (4.1) (with
$x\downarrow0$
) one can see that the underlying patterns satisfy
$$
\phi_i(t;\, 0)=(c_it)^{1/\gamma},$$
where
$c_i=\phi_i(1;\, 0)^\gamma>0$
.
Due to the semi-group property
$\phi_i(t-s;\, \phi_i(0;\,s))=\phi_i(t;\, 0)$
, we have
$$
\phi_i( t-s;\, (c_is)^{1/\gamma})=(c_it)^{1/\gamma},\qquad 0<s<t.$$
Hence,
$$
\phi_i(t-x^\gamma/c_i;\,\;
x)=(c_it)^{1/\gamma}.$$
Therefore (under the shift
$t\to t+x^\gamma/c_i$
) we have
$$
\phi_i(t;\,\;
x)=(x^\gamma+c_it)^{1/\gamma}.\tag*{$$}$$
Remark 4.1. If the ‘velocities’
$c_0, c_1$
are positive, the process
${{\mathbb X}}^x$
is a subordinator (defined for all
$t\geq0$
).
In the case of a negative velocity the process
${{\mathbb X}}^x$
is defined until hitting zero at time
$\zeta^x=\inf\{t>0~|~
\!{\mid}\! X^x(t)=0\}=\inf\{t>0~|~
\!{\mid}\! T(t)=-x^\gamma\}$
. The distribution of
$\zeta^x$
is known explicitly; see, e.g., [Reference Bogachev and Ratanov2].
Remark 4.2. Consider the time-homogeneous process
${{\mathbb X}}^x$
determined by the alternating patterns
$\phi_0$
,
$\phi_1$
with common diffeomorphism
$\Phi(x)={{\rm e}}^x$
,
$x\in{{\mathbb R}}^{1}$
:
$$
\phi_i(t;\, x)=\log({{\rm e}}^{x}+c_it),\qquad t\geq0,\quad
{{\rm e}}^x+c_it>0, \qquad
i\in\{0, 1\}.$$
If
$c_0, c_1\geq0$
, the process
${{\mathbb X}}^x(t)$
is defined for all
$t\geq0$
. In the case of negative
$c_i$
the process is killed and sent to the cemetery state
$-\infty$
at time
$t_*=\inf\{t>0~|~
\!{\mid}\!{{\mathbb T}}(t)=-{{\rm e}}^x\}$
, where
${{\mathbb T}}(t)$
is the respective telegraph process.
The process
${{\mathbb X}}^x(t)$
possesses the property of additive self-similarity: under time scaling the process takes a spatial shift,
$${{\mathbb X}}^{x-R}({{\rm e}}^{-R}t)~\text{is equal in law to}~{{\mathbb X}}^x(t)-R.$$
Indeed, under transformations
$t\to {{\rm e}}^{-R}t$
and
$x\to x-R$
the switching intensities are transformed as
${{\lambda}}_0\to{{\rm e}}^{R}{{\lambda}}_0$
,
${{\lambda}}_1\to{{\rm e}}^{R}{{\lambda}}_1$
, and
$\xi\to{{\rm e}}^{-R}\xi$
. By Theorem 2.1, the distributions of
${{\mathbb X}}^{x-R}({{\rm e}}^{-R}t)$
and
${{\mathbb X}}^x(t)-R$
coincide.
Appendix A. The auxiliary result
Lemma A.1. Let
$z\in I_t$
be fixed, and
$\xi=\varphi(z, t)$
,
$0\leq\xi\leq t$
, be the (unique) solution of the equation
$z-tc_1=\xi(c_0-c_1)$
, (2.9). Then
$z-c_0\tau\in I_{t-\tau}$
and
$z-c_1\tau\in
I_{t-\tau}$
for sufficiently small
$\tau$
,
$\tau>0$
.
Further, for all
$z\in I_t$
the solution
$\xi=\varphi(z, t)$
of (2.9) satisfies the following identities:
\begin{equation}
\begin{aligned}
\varphi(z-c_0\tau, t-\tau) & \equiv \xi-\tau \quad
& & \text{if }\tau\in[0,\;
\xi], \\
\quad
\varphi(z-c_1\tau, t-\tau) & \equiv \xi \quad
& & \text{if } \tau\in[0,\;
t-\xi].\\
\end{aligned}
\label{eqn30}
\end{equation}
Proof. By substitution of
$z-c_0\tau$
and
$z-c_1\tau$
with z and
$t-\tau$
with t into (2.9) one can obtain
\begin{equation}
z-c_0\tau=\widetilde\xi c_0+(t-\tau-\widetilde\xi)c_1,
\qquad\widetilde\xi=\varphi(z-c_0\tau,\;
t-\tau),
\label{eqn31}
\end{equation}
and
\begin{equation}
z-c_1\tau=\tilde\xi c_0+(t-\tau-\tilde\xi)c_1,\qquad\tilde\xi=\varphi(z-c_1\tau,\;
t-\tau).
\label{eqn32}
\end{equation}
Equation (A.2) is satisfied by
$\widetilde\xi=\xi-\tau$
if
$\tau\leq\xi$
, and (A.3) is satisfied by
$\widetilde\xi=\xi$
if
$\tau\leq t-\xi$
.
Further, note that, by definition,
$z-c_0\tau\notin I_{t-\tau}$
if
$\tau>\xi$
and
$z-c_1\tau\notin I_{t-\tau}$
if
$\tau>t-\xi$
. Hence, the lemma is proved.
Appendix B. Proof of Proposition 2.1
System (2.7),
$n=1$
, and (2.8) give the density functions
$p_0^{{\mathbb T}}(z, t;\,\;
1)$
and
$p_1^{{\mathbb T}}(z, t;\,\;
1)$
. Indeed,
$$
\begin{aligned}
p_0^{{\mathbb T}}(z, t;\;1) & = \int_0^t{{\lambda}}_0{{\rm e}}^{-{{\lambda}}_0\tau}{{\rm e}}^{-{{\lambda}}_1(t-\tau)}
\delta(z-\tau c_0-(t-\tau)c_1)\,{{\rm d}}\tau\\
& = \frac{{{\lambda}}_0}{\|c_0-c_1\|} \exp({-}\,{{\lambda}}_0\xi-{{\lambda}}_1(t-\xi))
{\mathbf{1}}_{\{0<\xi<t\}} \\
& = \lambda_0\theta(\xi, t-\xi),
\end{aligned}$$
where
$\xi=\varphi(z, t)$
,
$\xi\in(0, t)$
, is the solution of (2.9). Similarly,
$p_1^{{\mathbb T}}(z, t;\,\;
1)={{\lambda}}_1\theta(\xi, t-\xi)$
. This corresponds to (1.2),
$n=1$
, with
$q_i(\xi, \eta;\, 1)$
defined by (1.3) (
$k=0$
) and
$\theta$
defined by (2.10).
By recalling Lemma A.1 in Appendix A and (2.10),
\begin{equation}
\begin{aligned}
{{\rm e}}^{-{{\lambda}}_0\tau}
\theta(\varphi(z-c_0\tau, t-\tau), t-\tau-\varphi(z-c_0\tau, t-\tau))\\
& = {{\rm e}}^{-{{\lambda}}_0\tau}\theta(\widetilde\xi, t-\tau-\widetilde\xi)
\big|_{\widetilde\xi=\xi-\tau} \\
\equiv\theta(\xi,\;
t-\xi){\mathbf{1}}_{\{\tau<\xi\}} ,
\quad
{{\rm e}}^{-{{\lambda}}_1\tau}
\theta(\varphi(z-c_1\tau, t-\tau), t-\tau-\varphi(z-c_1\tau, t-\tau))\\
& ={{\rm e}}^{-{{\lambda}}_1\tau}\theta(\widetilde\xi, t-\tau-\widetilde\xi)
\big|_{\widetilde\xi=\xi}
\equiv\theta(\xi,\;
t-\xi){\mathbf{1}}_{\{\tau<t-\xi\}}.
\end{aligned}
\label{eqn33}
\end{equation}
Moreover, by applying (A.1) and (B.1) one can obtain the following identities, which are sufficient to finish the proof:
\begin{equation}
\begin{aligned}
\int_0^t{{\rm e}}^{-{{\lambda}}_0\tau}
\varphi(z-c_0\tau, t-\tau)^m(t-\tau-\varphi(z-c_0\tau, t-\tau))^k \\
&
\times\theta(\varphi(z-c_0\tau, t-\tau), t-\tau-\varphi(z-c_0\tau, t-\tau))\,{{\rm d}}\tau
=\theta(\xi, t-\xi)\int_0^\xi(\varphi(z, t)-\tau)^m(t-\varphi(z, t))^k\,{{\rm d}}\tau \\
&
=\theta(\xi, t-\xi)\frac{\xi^{m+1}}{m+1}(t-\xi)^k,\qquad \xi=\varphi(y, t);
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\int_0^t{{\rm e}}^{-{{\lambda}}_1\tau}
\varphi(z-c_1\tau, t-\tau)^m(t-\tau-\varphi(z-c_1\tau, t-\tau))^k \\
&
\times\theta(\varphi(z-c_1\tau, t-\tau), t-\tau-\varphi(z-c_1\tau, t-\tau))\,{{\rm d}}\tau
=\theta(\xi, t-\xi)\int_0^{t-\xi}\varphi(z, t)^m(t-\tau-\varphi(z, t))^k\,{{\rm d}}\tau \\
&
=\theta(\xi, t-\xi)\xi^m\frac{(t-\xi)^{k+1}}{k+1},
\end{aligned}
\end{equation}
$\xi=\varphi(z, t)$
; cf. [Reference Kolesnik and Ratanov16, Chapter 4].
Acknowledgements
We express our thanks to the referees for useful comments that helped to improve the paper. A.D.C. and B.M. are members of the research group GNCS of INdAM.
