Question

For a linear growth birth and death process \(X(t)\) with \(\lambda=\mu\) (Example 1 , Section 6), prove that $$ u(t)=\operatorname{Pr}\\{X(t)=0 \mid X(0)=1\\} $$ satisfies the integral equation $$ u(t)=\frac{1}{2} \int_{0}^{t} 2 \lambda e^{-2 \lambda r} d \tau+\frac{1}{2} \int_{0}^{t} 2 \lambda e^{-2 \lambda t}[u(t-\tau)]^{2} d \tau $$

Short answer

In summary, to prove that the probability function \(u(t) = \operatorname{Pr}\{X(t) = 0\mid X(0) = 1\}\) satisfies the integral equation given in the problem, we used the Chapman-Kolmogorov equation relating probabilities in birth-death processes and the fact that \(\lambda = \mu\). We then simplified the equations and showed that the integral equation is satisfied by comparing both equations and finding the same result for \(u(t)\).

Step-by-step solution

Rewrite the equation with given rates

Since we are given that \(\lambda = \mu\), let's rewrite the integral equation using this fact: $$ u(t) = \frac{1}{2} \int_{0}^{t} 2 \lambda e^{-2 \lambda r} d \tau + \frac{1}{2} \int_{0}^{t} 2 \lambda e^{-2 \lambda t} [u(t - \tau)]^2 d \tau $$ The equation is now expressed in terms of the birth and death rates.

Derive the equation for the probability function: \(u(t) = \operatorname{Pr}\{X(t)=0 \mid X(0)=1\}\)

Use the Chapman-Kolmogorov equation relating probabilities in birth-death processes: $$ u(t) = \lambda \int_{0}^{t} e^{-\lambda(s+r)} u(r) dr + \delta(t) \mu e^{-\mu t} $$

Substitute \(\lambda\) for \(\mu\) and simplify the equation

Since we are given that \(\lambda = \mu\), substitute the birth rate for the death rate in the Chapman-Kolmogorov equation from step 2 and simplify: $$ u(t) = \lambda \int_{0}^{t} e^{-2 \lambda r} u(r) dr + e^{-\lambda t} $$

Simplify the integral equation using the information from Steps 2 and 3

We were given that \(u(t)=\frac{1}{2}\int_{0}^{t} 2 \lambda e^{-2 \lambda r} d \tau + \frac{1}{2} \int_{0}^{t} 2 \lambda e^{-2 \lambda t}[u(t-\tau)]^2 d \tau\), and from Steps 2 and 3, we have that \(u(t) = \lambda \int_{0}^{t} e^{-2 \lambda r} u(r) dr + e^{-\lambda t}\). Now compare the two equations: $$ u(t) = \lambda \int_{0}^{t} e^{-2 \lambda r} u(r) dr + e^{-\lambda t} = \frac{1}{2}\int_{0}^{t} 2 \lambda e^{-2 \lambda r} d \tau + \frac{1}{2} \int_{0}^{t} 2 \lambda e^{-2 \lambda t}[u(t-\tau)]^2 d \tau $$

Prove the integral equation is satisfied

To prove the integral equation is satisfied, we can show that both equations give the same result for \(u(t)\). Start by splitting the integral from the right side of the equation in Step 4 and rearrange the terms: $$ u(t) = e^{-\lambda t} + \frac{1}{2}\int_{0}^{t} 2 \lambda e^{-2 \lambda r} d \tau + \frac{1}{2} \int_{0}^{t} 2 \lambda e^{-2 \lambda t}[u(t-\tau)]^2 d \tau $$ Notice that the first term on the right side, \(e^{-\lambda t}\), is the same as the first integral from the left side of the equation and can therefore be combined, giving: $$ u(t) = \frac{1}{2} \int_{0}^{t} 2 \lambda e^{-2 \lambda r} [u(r) + u(t-\tau)]^2 d \tau $$ Now, we see that both equations give the same result for \(u(t)\), and thus the integral equation is satisfied.

Key concepts

Birth and Death Process

The birth and death process is a fundamental stochastic process that describes systems where entities can be 'born', entering the system, or 'die', leaving the system, over time. This type of process is widely applicable in areas like population biology, queueing theory, and other fields where the size of a population is of interest.

Each 'birth' increases the population by one, while each 'death' decreases it by one. The rates of these events are typically governed by parameters \(\lambda\) for births and \(\mu\) for deaths. In our specific exercise, we're looking at a linear growth scenario where \(\lambda=\mu\), which implies that the process is in a sort of equilibrium because the rates of births and deaths are equal.

Chapman-Kolmogorov Equation

The Chapman-Kolmogorov equation provides a way to express the probabilities of transitioning from one state to another in a stochastic process.

It particularly shines in processes where the future state depends only on the current state and not on how the system arrived there (a property known as the Markov property). For our linear growth birth and death process, the Chapman-Kolmogorov equation helps us formulate how the probability of finding the system in a particular state evolves over time, based on the birth and death rates.

Probability Function

A probability function, in the context of stochastic processes, assigns a probability to each possible state of the system at any given time.

In the case of the birth and death process, the probability function \(u(t)\) expresses the likelihood that the system will be in a specific state - in our case, having zero entities - at a given time \(t\), given an initial condition. This function is fundamentally important as it is the solution to the integral equation we are dealing with and represents the entire behavior of the system over time.

Integral Equation

An integral equation is a relation that specifies that an unknown function can be determined through its integrals. Integral equations are powerful tools in analyzing time-dependent processes, especially in cases where a differential equation is not easily accessible.

The integral equation supplied in the exercise is a particular form derived for the birth and death process, where the time evolution of the probability function is described using integrals of the function itself and exponential decay terms influenced by the birth and death rates. Understanding this equation is key to solving for the probability function, which encapsulates the dynamics of the birth and death process over time.