Introduction

A simple homogeneous degradation, decay (e.g. memory decay), death, or departure process can be described as a directed graphical model (stoichiometric equation in chemics):

$$  N_A \stackrel{\lambda}{\longrightarrow} {\varnothing} \; \; \; \qquad (1)$$

where:

$$ A \text{  :=  is a species of interest} $$

$$ N_A \text{  :=  is the number of particles of species of interest} $$

$$ \lambda \text{  :=  is the rate constant of degradation events} $$

$$ \varnothing \text{  :=  is a species of no special interest} $$

In a homogeneous process the rate is defined as a constant $$\lambda$$ so that $$\lambda \cdot dt $$ gives the probability that a randomly chosen particle of A reacts, degrades, dies, decays, or departs during the time interval $$[t, t + dt) $$ where t are time and dt an (infinitesimally) small time step.

In an in- or nonhomogeneous process the rate is dependent on time or state  $$\lambda(t) \text{ or } \lambda(N_A(t)) $$ so that $$\lambda(t) \cdot dt \text{  or  } \lambda(N_A(t)) \cdot dt $$ gives the probability that a randomly chosen particle of A reacts, degrades, dies, decays, or departs during the time interval $$[t, t + dt) $$ where t are time and dt an (infinitesimally) small time step.

The stoichiometric equations or the DAGs are now:

$$ N_A \stackrel{\lambda(t)}{\longrightarrow} {\varnothing}  \; \; \; \;\text{  with time-dependent rate }  \qquad (2.1)$$

$$ N_A \stackrel{\lambda(N_A)}{\longrightarrow} {\varnothing} \; \; \; \text{  with state-dependent rate }  \qquad (2.2)$$

The Definition of the (In-)Homogeneous Counter Decay Process

The "naive" implementation of the solution of the decay counter process is based not on dt but on a discrete approximation $$\Delta t >> dt$$ so it is wise to reuse and modify the second definition of the Poisson process. The decay process $$N_A(t), t \in [0, \infty)$$ is called a homogeneous counter decay process with rate $$\lambda > 0 $$ if all of the following conditions hold:

$$1. \;\;N_A(0) = N_{A_0} > 0\; \; \text{ ; } $$

$$2. \;\;N_A(t) \text{ has independent and stationary increments} $$

then we have

$$P(N_A(\Delta t) = 0)  \approx  1 - \lambda \Delta t \qquad (3.1.1)$$

$$P(N_A(\Delta t) = -1)  \approx  \lambda \Delta t  \qquad (3.1.2)$$

$$P(N_A(\Delta t) = -2) \approx  0 \qquad (3.1.3)$$

or more precisely

$$P(N_A(\Delta t) = 0) = 1 - \lambda \Delta t + o(\Delta t) \qquad (3.2.1)$$

$$P(N_A(\Delta t) = -1) = \lambda \Delta t + o(\Delta t) \qquad (3.2.2)$$

$$P(N_A(\Delta t) = -2) =  o(\Delta t) \qquad (3.2.3)$$

where "little o"  denotes a function g(.) $$ o(\Delta t) := g(\Delta t) $$ that 'vanishes' faster than $$\Delta t $$ when

$$ \lim_{\Delta t \to 0} \frac{g(\Delta t)}{\Delta t} = 0, \qquad (4) $$

or short

$$ o(\Delta t) \rightarrow 0 \text{ as } \Delta t \rightarrow 0 \; \; .$$