4.1. A Dynamical System Driven by an Inhomogeneous Point Process
Here, we follow the theory in Section 7.4 of [
28] (the original results are presented in [
29]). Let
be the probability space, equipped with a filtration
,
, such that for
, the corresponding sigma-algebras are nested as
. Let
be a bounded, strictly positive random variable adapted to
. Let
be a
-adapted inhomogeneous point process with the conditional intensity
, and the compensator
, given via
By choosing the random variable appropriately, we can regulate the “temporal density” of jump points of ; indeed, depending on the magnitude of , the jump points of can either arrive in a statistically rapid succession, or, to the contrary, disperse farther away from each other. For what is to follow, it is a necessary requirement that the intensity is a random variable—both and are, however, adapted to the same filtration , so that if the sequence of values of h up to t is given, so is the sequence of values of m.
We now denote the corresponding jump process of
via
:
Above, the notation “
” denotes the left-limit at
t. Let
be the corresponding random measure of
:
Let us now define a stochastic process
on a Euclidean space
as follows:
where
are suitable (for the purpose of stochastic integration above) random variables, adapted to
. Clearly,
is also
-adapted by construction.
Our task here is to compute the infinitesimal generator of
, that is, for a test function
, we would like to compute
To compute the expectation above, we need to adapt the integral form of
in Equation (
50) to the Itô formula for Lévy-type stochastic integrals (see Chapter 4 of [
25]). However, the problem is that the random measure in the right-hand side of Equation (
50) is not that of the standard Poisson point process with constant intensity, but that of the point process with random intensity
, defined above.
Our first step here is, therefore, the transformation of the stochastic integral in the right-hand side of Equation (
50) to an integral against the random measure of a standard Poisson point process. According to Theorem 7.4.I of [
28] (also see [
29]), the random point process
, defined via
is the standard Poisson point process with intensity 1, where
is the random, albeit
-adapted, compensator process of
, defined above in Equation (
47). Subsequently, we can write the stochastic integral in Equation (
50) from
t to
as
with
being the random measure of the standard Poisson point process with intensity 1; in particular, its intensity measure
[
25] is given via
Substituting the above integral into Equation (
50), we write
Next, recalling the Itô formula for Lévy-type stochastic integrals in Chapter 4 of [
25], we write
Applying the conditional expectation on both sides, we obtain, for the left-hand side and the first term in the right-hand side,
where in the second expression the conditional expectation disappears in the leading order term because the
and
are both
-adapted.
The expectation of the stochastic integral is somewhat more complicated. Observe that
where in the right-hand side both leading order terms are
-adapted, and the last term is given via
At this point, we further assume that the probability of
having a discontinuity for
is
, which implies that
The stochastic integral above can then be split into two parts as
where the conditional expectation of the last integral is
.
The expectation of the first integral in the right-hand side above can be written as
where we make use of the fact that the limits of integration are
-adapted, and the integrand is non-anticipative, so that the expectation can be carried into the integral and split into the product of the corresponding expectations of the integrand and the Poisson random measure.
For the expectation of the integrand, we use the same argument as above for the limits of integration—namely, we assume that the probability of a discontinuity appearing in the integrand for
is
. For the expectation of the Poisson random measure, we recall Equation (
54). This further leads to
Assembling the terms together, we obtain the infinitesimal generator in the form
where in the last identity we denote
, for brevity.
It is worth noting that the form of the infinitesimal generator above extends naturally onto the free-flight configuration of the dynamics—it suffices to set the variable intensity
above. In this case,
in Equation (
50) is driven solely by the integral over the vector field
alone.
4.2. Random Dynamics of Two Spheres
To adapt the general stochastic process in Equation (
50) to the dynamics of spheres, we need to relate
,
,
and
to the variables of the dynamics. Obviously,
is the state vector of the system, and thus it incorporates the coordinates and velocities of both spheres:
Subsequently,
is related to the deterministic component of the dynamics, which is the evolution of the coordinates
and
for given velocities
and
according to Equation (
1):
To specify
, we observe that the instantaneous change of velocities in Equation (
2) can be written, with help of the jump process
, as
where
,
. Therefore, we can define
via
Then, we write the process in Equations (
1) and (
67) as the following stochastic differential equation [
25]:
with
,
and
given via Equations (
65), (
66) and (
68), respectively.
It remains to specify the variable intensity
, which should activate the point process
when both Equations (
43) and (
44) hold concurrently, and be zero in the free-flight configuration. Here, we define
as
Above,
,
are constant parameters, and
is the standard mollifier of the delta-function
, given via
where the constant parameter
c ensures the proper normalization. For a function
, which is continuous at zero, we thus have
that is,
can serve as the delta-function in the limit
, while remaining smooth for finitely small
. We denote the anti-derivative of
as
:
Clearly, as , becomes the usual Heaviside step-function.
Observe that, in the collision configuration in Equations (
43) and (
44), the variable intensity of the point process in Equation (
70), as required in [
28,
29], is indeed
-adapted, strictly positive and bounded, since, first,
is
-adapted by construction, and, second, whenever Equations (
43) and (
44) hold, we have
where
E is the constant energy of the system of two spheres. As the intensity of jumps is globally bounded, and the discontinuities in the solution of Equation (
69) may occur solely as a result of these jumps, the assumption that the probability of discontinuity occurring on the time scale of
is
, used in the computation of the expectation of the stochastic integral in Equations (
58)–(
63), is indeed valid.
The dynamics in Equations (
69) and (
70) function as follows:
The existence of strong solutions to Equations (
69) and (
70) in the collision configuration in Equations (
43) and (
44) is a subject that merits a separate discussion. For the purpose of this work, we assume that bounded strong solutions are sufficiently generic for typical initial conditions, so that the corresponding statistical formulation (the forward Kolmogorov equation) of the dynamics is reliable enough for the description of large ensembles of solutions.
The definition of the jump intensity above in Equation (
70) also indicates that the probability that the jump in the point process does not arrive during the collision window is
, regardless of the values of
or
. Indeed, observe that one can write
which means that if the contact zone is traversed completely (that is, the jump has not arrived), then the compensator
in Equation (
47) is incremented by
. Clearly, to mimic the collisions of hard deterministic spheres in
Section 2, one eventually needs to take
and
, so that, first, the contact zone Equation (
43) becomes infinitely thin, and, second, the jump arrives with probability 1 whenever the spheres are in the collision configuration.
The corresponding infinitesimal generator of Equations (
69) and (
70) is given via
Changing back to the original variables
,
,
and
, we write the infinitesimal generator of Equation (
69) in the form
where
and
are the functions of
,
,
and
given in Equation (
2). The jump portion of the generator above in Equation (
77) is not translationally invariant, and thus the process in Equation (
69) is not a Lévy process. However, it is a Lévy-type Feller process [
25,
30], whose infinitesimal generator can be reformulated in the Courrège form [
31] via an appropriate change of variables.
The next step is to obtain the corresponding forward Kolmogorov equation [
24,
26,
27] for the probability density of states of the system, which is easily achieved via the integration by parts. Let
be the corresponding probability distribution of the random process above. We can then integrate Equation (
77) against
F and obtain, with help of Equation (
40),
where
is the volume element of the coordinate space, and
is the area element of the sphere of zero momentum and constant energy (the subscript denotes the number of spheres in the system).
Above, the terms with spatial derivatives in
and
can be integrated by parts, with the condition that the boundary effects are not present. For the part with
we can write, for fixed
and
,
where we used Equations (
5)–(
7) (note that
and
remain on the same zero momentum—constant energy sphere), and in the last identity renamed
,
and vice versa, since the integral occurs over the same velocity sphere. As a result, we can recombine the terms as
Assuming that
is arbitrary, we can strip the integral over
and obtain the equation for
F alone:
Unlike the Liouville problem in Equations (
8)–(
10), here observe that the effect of collisions is present in the equation itself, and is not contingent upon additional properties imposed on
F.
4.3. Extension to Many Spheres
Here, we extend the previously formulated dynamics onto K spheres, with the corresponding coordinates and velocities , . Observe that we have possible pairs of spheres. To define their random interactions, we introduce independent instances of the point process, each assigned to the pair of ith and jth spheres.
For
, let us define
where the two nonzero entries in the
-vector above are in the
th and
th slots. Let
be the set of
independent inhomogeneous point processes with conditional intensities
, given via
where
is defined in Equation (
70). Let
be the set of corresponding random measures for
. Then, the
K-sphere dynamics is defined via the following system of stochastic differential equations:
The process above in Equation (
85) is also a Lévy-type Feller process [
25,
30], which lives on the sphere of zero momentum and constant energy
As above for two spheres, here we assume that the total momentum of the system is zero without loss of generality. The infinitesimal generator of such process is, apparently, given via
In the
and
variables, this translates into
where the notation
specifies that all velocity arguments in
are set to the corresponding velocities
, except for
ith and
jth, which are set to
and
via Equation (
2).
Observe that the dynamics in Equation (
85) is a direct extension of the dynamics of two spheres in Equation (
69) onto multiple spheres—the evolution of the coordinates is governed by the same equations, and the velocities of each sphere are coupled to all other spheres via the independent point processes
. In particular, there is no provision for a collision of more than two spheres at once (which is often discussed in the literature [
7,
9]); however, given the fact that the collisions in the hard sphere model are instantaneous, we assume that the event of a three-sphere collision is improbable for a “generic” initial condition.
It is interesting to note that the properties of an infinitesimal generator similar to Equation (
88) are studied in [
21] in the same context (that is, a system of
K particles interacting according to Equation (
2)). However, the collision part of the infinitesimal generator in [
21] is scaled differently, as if the intensity parameter
in Equation (
88) is set to
. Thus, as
, the particles described by the generator in [
21] cease colliding upon contact. It is, however, unclear where the form of the infinitesimal generator in [
21] comes from; the generator in [
21] appears to be postulated, rather than derived from an underlying SDE.
To obtain the corresponding forward equation, we follow the same principle as for the two spheres above. First, we integrate Equation (
88) against the probability density
F and obtain
where
is the volume element of the coordinate space of the
K spheres,
is the area element of the corresponding velocity sphere of zero momentum and constant energy, and the term with the spatial derivatives is integrated by parts assuming that the boundary terms vanish. Now, for all terms with
and
, we have, for fixed coordinates,
where we use Equations (
5)–(
7), and observe that, for fixed coordinates, the variables
and
sample the same zero momentum and constant energy sphere as do
and
. Finally, stripping the integral over
, we arrive at the forward Kolmogorov equation in the form
Note that Equation (
91) admits solutions that are symmetric under the reordering of the spheres. These solutions are “physical”, that is, they correspond to real-world scenarios where it is impossible to statistically tell the spheres apart.