PRELIMINARIES

State-space:

Transition functions: A set of real-valued functions , defined on is called a process (or transition function) if

,

, and

.

It is called standard

and honest if

,

for some (and then for all) t>0.

THE Q-MATRIX

For a standard process P the right-hand derivative exists and defines a q-matrix whose entries satisfy

, , and

.

We set , .

Suppose that Q is given. Assume that Q is stable, that is for all i in S. A standard process P will then be called a Q-process if its q-matrix is Q.

THE KOLMOGOROV DIFFERENTIAL EQUATIONS

For simplicity, we shall assume that Q is conservative, that is

for all . Under this condition, every Q-process P satisfies the backward equations,

but might not satisfy the forward equations,

STATIONARY DISTRIBUTIONS

A collection of positive numbers is a stationary distribution if and

Recipe: Find a collection of strictly positive numbers such that

Such an m is called an invariant measure for Q. If , then we set

and hope that satisfies (2).

BIRTH-DEATH PROCESSES

Transition rates:

( - birth rates)

( - death rates) ( )

Solve: , , that is,

and, for ,

Solution: and

MILLER'S EXAMPLE

Rates: fix r>0 and set

,

.

Solution: and

So,

, where ,

and hence if r>1,

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WHAT IS GOING WRONG?

Rates:

,

.

The relative proportion of births to deaths is r and so, if r>1, the ``process'' is clearly transient.

In fact, the ``process'' is explosive: Q is not regular. R.G. Miller showed that Q needs to be regular for the recipe to work.

THE MOTIVATING QUESTION

If Q is regular, then there exists uniquely a Q-process, namely the minimal process: the minimal solution , to BE.

If Q is not regular there are infinitely many Q-processes, infinitely many of which are honest.

Question: Suppose that there exists a collection of strictly positive numbers such that

Does  admit an interpretation as a stationary distribution for any of these processes?

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AN INVARIANCE RESULT

Let be a collection of strictly positive numbers. We call m a subinvariant measure for Q if

and an invariant measure for Q if

It is called an invariant measure for P if

Theorem: Let P be an arbitrary Q-process. If m is invariant for P, then m is subinvariant for Q, and invariant for Q if and only if P satisfies the forward equations FE over S.

Corollary: If m is invariant for the minimal process F, then m is invariant for Q.

A CONSTRUCTION PROBLEM

Suppose that Q is a stable and conservative q-matrix, and that m is subinvariant for Q.

Problem 1: Does there exist a Q-process for which m is invariant?

Problem 2: Does there exist an honest Q-process for which m is invariant?

Problem 3: When such a Q-process exists, is it unique?

Problem 4: In the case of non-uniqueness, can one identify all Q-processes (or perhaps all honest Q-processes) for which m is invariant?

THE RESOLVENT

Let P be a transition function. If we write

for the Laplace transform of , then enjoys the following (analogous) properties:

,

, and

.

is called the resolvent of P. Indeed, if is a given resolvent, in that it satisfies these properties, then there exists a standard (!) process P with  as its resolvent.

IDENTIFYING Q-PROCESSES USING RESOLVENTS

Now, if one is given a stable and conservative q-matrix Q, together with a resolvent which satisfies the backward equations,

then determines a standard Q-process. In particular, as ,

, and

.

One can also use the resolvent to determine whether or not the Q-process is honest, for this happens when and only when

IDENTIFYING INVARIANT MEASURES USING RESOLVENTS

Theorem: Let P be an arbitrary process and let be its resolvent. Then, m is invariant for P if and only if it is invariant for  , that is,

if and only if

THE EXISTENCE OF A Q-PROCESS WITH A GIVEN INVARIANT MEASURE

Theorem: Let Q be a stable and conservative q-matrix, and suppose that m is a subinvariant measure for Q. Let be the resolvent of the minimal Q-process and define and by

,

and

.

Then, if d=0, m is invariant for the minimal Q-process. Otherwise, if

,

for all , there exists a Q-process P for which m is invariant.

THE EXISTENCE OF A Q-PROCESS WITH A GIVEN INVARIANT MEASURE

Theorem continued: The resolvent of one such process is given by

and this process is honest if and only if

for all . A sufficient condition for there to exist an honest Q-process for which m is invariant is that m satisfies

for all .

Corollary: If m is a subinvariant probability distribution for Q, then there exists an honest Q-process with stationary distribution m. The resolvent of one such process is given by (4).

THE SINGLE-EXIT CASE

Suppose that Q is a single-exit q-matrix, that is, the space of bounded, non-negative vectors which satisfy

has dimension 1. (The minimal process has only one available ``escape route'' to infinity). Then, the condition

is necessary for the existence of a Q-process for which the specified measure is invariant; the Q-process is then determined uniquely by

NON-UNIQUENESS

Consider a pure-birth process with strictly positive birth rates , but imagine that we have two distinct sets of birth rates, and , which satisfy

Let and define by

for r=0,1 and . The measure , given by

is subinvariant for Q.

NON-UNIQUENESS

The resolvents of two distinct Q-processes for which m is invariant are given by

and

Interpretation: The first process chooses between (0,0) and (1,0) with equal probability as the starting point following an explosion, no matter which was the most recently traversed path, and the second process traverses alternate paths following successive explosions.

THE REVERSIBLE CASE

Suppose that Q is symmetrically reversible with respect to m, that is, , . Then, , and so we arrive at the following corollary due to Hou and Chen.

Corollary: If Q is reversible with respect to m, then there exists uniquely a Q-function P for which m is invariant if and only if , for all . It is honest and its resolvent is given by

Moreover, P is reversible with respect to m in that (or, equivalently, ).

BIRTH-DEATH PROCESSES

Suppose that the birth rates and death rates are strictly positive. The q-matrix Q is then regular if and only if

Proposition: Let be the essentially unique invariant measure for Q.

(i) m is invariant for the minimal Q-process if and only if (6) holds.

(ii) When (6) fails, there exists uniquely a Q-process P for which m is invariant if and only if m is finite, in which case P is the unique, honest Q-process which satisfies FE; P is positive recurrent and its stationary distribution is obtained by normalizing m.

-INVARIANCE

Suppose that , where 0 is an absorbing state and C is irreducible (for F). Let . A collection of strictly positive numbers is called a -subinvariant measure for Q if

and a -invariant measure for Q if

It is called a -invariant measure for P, where P is any transition function, if

Proposition: A probability distribution is a -invariant measure for some if and only if it is a quasistationary distribution: for ,

-INVARIANCE FOR F

Theorem: If m is -invariant for P, then m is -subinvariant for Q, and -invariant for Q if and only if P satisfies the forward equations over C. For example, if m is -invariant for the minimal process, then it is -invariant for Q.

Theorem: If m is -invariant for Q, then it is -invariant for F if and only if the equations

have no non-trivial solution for some (and then all) .

Theorem: If m is a finite -invariant measure for Q, then

where , and m is -invariant for F if and only if equality holds in (8).

Q-PROCESSES WITH A GIVEN -INVARIANT MEASURE

Theorem: Suppose that Q is single-exit and that m is a finite -subinvariant measure for Q. Then, there exists a Q-process for which m is -invariant if and only if

The resolvent of any Q-process for which m is -invariant must be of the form

where

and e satisfies .

Q-PROCESSES WITH A GIVEN -INVARIANT MEASURE

Theorem continued: Conversely, if

then all Q-processes for which m is -invariant can be constructed in this way by varying e in the range