Boundary States: Absorption, Reflection, And Immigration

What Can Happen at Zero?

In a birth–death process, the state records a nonnegative integer quantity, such as a population size. From a positive state, the process may increase by one through a birth or decrease by one through a death. At zero, a downward jump would leave the state space. The boundary rule therefore determines whether zero traps the process or whether an upward transition can take it away from zero. Immigration is one common mechanism for that upward transition.

Let \((X_t)_{t\ge0}\) be a continuous-time Markov chain on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), with state space \(S=\{0,1,2,\ldots\}\) or a finite interval of nonnegative integers. Its natural filtration is \(\mathcal{F}_t=\sigma(X_s:0\le s\le t)\). We assume the chain is well-defined and non-explosive, so it makes only finitely many jumps on each bounded time interval. Write \(Q=(q_{ij})\) for its generator: for \(j\ne i\), \(q_{ij}\) is the instantaneous rate of a jump from \(i\) to \(j\), and \(q_{ii}=-\sum_{j\ne i}q_{ij}\). Thus each generator row sums to zero.

At zero, there is only one possible outgoing transition in this model: \(0\to1\). If its rate \(\lambda_0\) is zero, the chain cannot leave zero, so zero is absorbing. If \(\lambda_0>0\), the chain leaves zero for state \(1\); this positive boundary birth rate can represent immigration. The term reflecting refers to suppressing a move below zero, not to adding a jump to a different state. A reflecting boundary can still be absorbing if no upward transition is allowed. In many population models, the boundary is called reflecting and has \(\lambda_0>0\), so the process can leave zero.

Generator Rules at the Boundary

Proof. By the birth–death definition, a jump from zero to a negative state is not allowed, and the only possible destination different from zero is state \(1\), with rate \(\lambda_0\). The diagonal generator entry is the negative sum of outgoing rates, so \(q_{00}=-\lambda_0\). If \(\lambda_0=0\), every off-diagonal entry in row zero is zero, and the chain cannot leave zero. If \(\lambda_0>0\), then \(q_{01}>0\), so zero is not absorbing. This proves the boundary statement in both directions.

For \(i\ge1\), the two possible destinations have rates \(\lambda_i\) and \(\mu_i\), so their sum is \(r_i\), and the diagonal entry is \(-r_i\). When \(r_i>0\), the usual competing-exponential-clock construction gives independent waiting times with rates \(\lambda_i\) and \(\mu_i\) for the two types of jump. The probability that neither clock rings by time \(t\) is \(e^{-\lambda_i t}e^{-\mu_i t}=e^{-r_i t}\), which is the survival probability of an exponential holding time with rate \(r_i\). The probability that the birth clock rings first is \(\int_0^\infty \lambda_i e^{-\lambda_i t}e^{-\mu_i t}\,dt=\lambda_i/r_i\); interchanging the two rates gives \(\mu_i/r_i\) for a death first. If \(r_i=0\), both rates are zero, so there is no outgoing jump. \(\square\)

This rule uses the actual outgoing transitions, not a label attached to the boundary. In particular, \(\lambda_0>0\) does not create a downward transition at zero; it only permits the upward transition. At a finite upper boundary, \(\lambda_N=0\) prevents upward jumps from \(N\), but it does not prevent a downward jump if \(\mu_N>0\).

Immigration Rates and Finite Upper Boundaries

In a population model, immigration may be represented by a constant arrival rate \(\alpha>0\) that contributes to births at every population size. If per-individual births also occur at rate \(\beta\), a typical birth rate is \(\lambda_i=\alpha+\beta i\). At zero this gives \(\lambda_0=\alpha\), which is precisely the rate of leaving zero by immigration. This rate convention does not alter the set of possible jumps: from any state the only birth destination remains the next state, and the only death destination remains the previous state when it exists.

Common Boundary Errors

A common mistake is to set the birth rate at a finite upper boundary to zero and then call that state absorbing. For the first example, the incorrect argument would be: “\(\lambda_2=0\), so state \(2\) is absorbing.” But the displayed row is \((0,1,-1)\), whose off-diagonal entry \(q_{2,1}=1\) allows a jump away from \(2\). The correct conclusion is that state \(2\) has no upward births but can die to state \(1\). To make state \(2\) absorbing in that example, one must also set \(\mu_2=0\). Its row then becomes \((0,0,0)\).

Another error is to interpret reflection as a jump from zero to a negative state followed by an automatic return. That is not the birth–death generator on the nonnegative integers: the negative state is outside \(S\), so no such transition is listed. Reflection here means that the downward move is suppressed. Whether zero is absorbing then depends on the upward rate \(\lambda_0\): it is absorbing when that rate is zero and permits departure when that rate is positive.

These distinctions matter whenever a model is translated into a generator. Each off-diagonal entry must correspond to an allowed transition and a nonnegative rate; the diagonal is then fixed as minus the sum of those outgoing rates. Checking the row sums catches arithmetic errors, while checking each boundary row catches modeling errors that row sums alone cannot identify.