1 Ergodic branching diffusions with immigration: our setting
This section introduces branching diffusions with immigration (BDI) and their ergodicity properties. In a first subsection, we introduce BDI as strong Markov processes with life time, close to , , ,  but more general in that we allow for quite arbitrary spatial scattering of the descendants generated at a branching event (as in ). Our method is a construction by killing and repasting of strong Markov processes as in , ,  or . In a second subsection we state a ‘spatial subcriticality’ condition which grants positive Harris recurrence with finite invariant occupation measure. A third subsection sketches proofs as far as their techniques are relevant for the rest of the paper.
1.1 BDI processes as strong Markov processes with life time
For , we write , and call single particle space. We call the space of (ordered) particle configurations configuration space; denoting the void configuration, we have . We write for the Borel--field on : is a Polish space. Lebesgue measure on is defined layer-wise (for , its restriction to equals Lebesgue measure on ). The length of a configuration is denoted by , i.e. iff . Sometimes we write a configuration as a point measure , (which equals if ). To measurable functions we associate functions by , i.e.
With these notations, BDI will be a -valued càdlàg strong Markov process with the following properties (A1)–(A4):
(A1) For , on some random time interval which is specified through (A2) and (A3) below, -particle configurations travel in as (strong) solutions to
Here are independent -dimensional Brownian motions. Drift and volatility are assumed to be Lipschitz.
(A2) i) Independently of each other, particles living at the same time are killed at position-dependent rate ; we assume that is measurable and locally bounded on .
ii) We have a transition probability from to such that gives the probability for a particle killed in position to produce offspring, .
iii) We have a transition probability from to with the property
which scatters offspring generated at a branching event relative to the parent’s position: -particle offspring of a particle killed in position will be located in positions
and in case we put .
An important special case contained in (A2) iii) is given by product kernels
for some fixed transition probability on .
if for some probability measure on , independently of , the distribution of newborn particles relative to their parent’s position is spatially homogeneous.
Finally, is the commonly considered case that particles are born exactly at the death position of their parent; we will refer to this special case as local branching.
In this paper, we shall work under (1) and shall not even assume (2), i.e. we allow for arbitrary non-local branching mechanisms.
(A3) For some probability measure on and some constant , single immigrants arrive at constant rate and are located in according to , independently of everything else.
Write for the canonical path space of càdlàg functions with life time , and for the canonical process on . Then (A1)–(A3) above determine uniquely a probability measure on under which is a jump diffusion with life time. As long as , jumps (finitely many on finite time intervals) arrive at rate
By convention, the rate is when . Note that by the Lipschitz assumptions on drift and diffusion coefficient in (A1), and by the local boundedness of in (A2), the life time of the process is the first accumulation point of the sequence of successive jumps times : we have as long as is finite, and . On events , representing the configuration immediately before the jump by , , the new configuration at time is obtained from as follows:
With exception of chosen equal to , jumps change the length of the configuration.
In case , is the void configuration , thus a one-particle configuration with selected by .
When we deal with trajectories of individual particles in the BDI process, we write
for the single-particle motion on . Assumption (A1) on drift and diffusion coefficient grants that the diffusion has infinite life time.
(A4) i) We have almost surely, for every choice of a starting point for .
ii) Reproduction means are (finite and) locally bounded on .
Condition (A4) i) grants that all defined above are almost surely finite stopping times.
Assumptions (A1)–(A4) and all notations of the present subsection will hold throughout the paper. So far, our construction of the BDI process is the canonical one: is the canonical path space of càdlàg functions with life time , is the canonical process on , and we have a unique probability law on such that is strongly Markov with the above properties: a jump diffusion with successive jump times which are finite stopping times and increase towards .
In this subsection, we state a set of sufficient conditions which ensure that the BDI process
is positive Harris recurrent, admitting the void configuration as a recurrent atom (thus in particular, will have infinite life time );
admits a finite invariant occupation measure on .
Up to the general form of our kernel in (A2) iii), we follow the same approach as Löcherbach , , ,
Höpfner and Löcherbach  section 1.4; see also Hammer  section 3
where the same general form of non-local branching was allowed.
Introducing the necessary notation we state the relevant results.
The functions and are bounded on .
This assumption guarantees in particular non-explosion of the process : by 1.2.1, has infinite life time almost surely (and from here on, we will take as the usual Skorohod path space of càdlàg functions ). Using the kernel from assumption A2 iii), we define a transition probability on the single-particle space as follows: for and measurable, let
if is such that we put , for some fixed probability measure on .
In the following we write for the Markov generator of the single-particle-motion on
where . Let us introduce a jump diffusion on by defining a generator
here, with from (A2) iii), the integral contribution equals
These generators should be understood in the sense of the corresponding martingale problems.
The jump diffusion can be defined probabilistically in the sense of killing and repasting of strong Markov processes,
cf. Ikeda, Nagasawa and Watanabe , , ,  and Nagasawa :
a diffusive motion according to is killed at position-dependent rate and restarted in a position selected by , independently of everything else.
Since has infinite life time and since is bounded in virtue of assumption 1.2.1, also the jump diffusion has infinite life time.
If we think of as a rate of annihilation/creation of mass, (6) or (7) deal with the total mass of the -resolvent kernel of the jump diffusion . Assumption 1.2.2 generalizes condition (6) in  to kernels satisfying (A2) iii). It implies, see lemma 1.2.4 below, ‘spatial subcriticality’ in the sense of almost certain extinction of families starting from one ancestor located in .
We shall write for the branching diffusion without immigration arising as subprocess of all direct descendants of one or several ancestors at some initial time . When there is need to specify positions for one or for several ancestors, we write or .
Recalling notation for with convention , let
denote the expected occupation measure (finite or not) for starting from one ancestor in .
1.3 Sketching the proofs, and some further notation
This subsection will sketch proofs for lemmata 1.2.4 and 1.2.5 –assertions which generalize results from  to kernels according to (A2) iii)– as far as the techniques which appear are of importance for the rest of the paper.
Proof of lemma 1.2.4: Consider the process starting from one ancestor in . By (A4), the time of the first branching event in is a.s. finite, thus (-valued) and (
-valued) are well-defined random variables. The strong Markov property yields
where we combine definition (8) of with the branching property (i.e. the fact that particles evolve independently). Writing the second contribution on the right hand side conditionally on as
which by definition of in (3) equals
equation (10) takes the form
Note that conditionally on , and up to time of position-dependent killing at rate , is a single-particle diffusion . Introducing the -resolvent kernel of the diffusion
(, measurable), the law of starting from is given by , and we can rewrite (11) as
which allows for iteration. By (11) and (13), the expected occupation measure (8) has the following interpretation: At rate , we erase unit mass travelling along the trajectory of , replace it by mass (generating particles with probability , and then merging these particles), then shift the location of the merged mass to a random position selected according to . The underlying strongly Markovian system (again defined probabilistically by killing and repasting since the corresponding semigroup is i.g. not contractive) has the generator
(notations from (4), (5), (3)) and is thus identified as the jump diffusion on ‘killed’ at position-dependent rate (of course, since we do not assume , speaking of ‘killing’ is abuse of language). Iteration of (13) combined with (14) then provides us with the following explicit solution to (8):
Compare the last proof to , lemma 1.4 and its proof, and to , Prop. 3.2.21 and Cor. 3.2.30, and note the role of the kernel from (A2) iii) which scatters offspring produced at branching events:
Indeed, the kernel defined in , (3.2.23) corresponds exactly to our definition of in (3).
Following definition 4.10 in Ikeda, Nagasawa and Watanabe , the semigroup on the single-particle space
is called expectation semigroup for the branching diffusion without immigration . In case , is the expected number of particles visiting at time which descend from a single ancestor in at time . This semigroup was implicit in definition 1.2.3, via . In analogy with the derivation (10)-(15), one can use the strong Markov property and the branching property to obtain by iteration the representation
which identifies the expectation semigroup as the Feynman-Kac semigroup corresponding to the jump diffusion with generator (5) and the ‘potential’ . We refer to , Thm. 3.2.28 for a full proof under our present assumptions.
Identities of the form (16)+(17) expressing the expected number of particles in terms of the dynamics of a single particle have a long history, going back to (at least) Watanabe .
They are now commonly called ‘many-to-one’-formulas (see e.g. )
and have been generalized in various ways; in particular, in (16) the function may be replaced by a functional depending on the whole path of the process up to time .
However, most of this literature tends to focus on the case of local branching mechanisms where particles reproduce exactly at their death position. See  and  for versions admitting non-local branching, also employing an auxiliary process as our jump diffusion , but still under stronger conditions on the offspring mechanism than our assumption (A2) iii)
( assumes in addition constant rates).
Now we can prove lemma 1.2.5.
Proof of lemma 1.2.5: 1) By lemma 1.2.4 (where for ) and in virtue of (7) in assumption 1.2.2, the expected time to extinction of a subprocess of defined by all descendents of the -th immigrant is finite, the -th immigrant choosing its location according to by (A3). Since the process of immigration instants is a Poisson random measure with constant intensity on , the BDI process will a.s. in the long run return infinitely often to the void configuration .
2) By 1), the BDI process can be rewritten in the form of a sum of i.i.d. excursions away from the void configuration . Write for the times of successive returns to , and define a measure on the configuration space by
Sets of positive -measure are visited infinitely often in the long run, a.s. for every choice of a starting point in . Thus is a Harris process (we refer to  and , ,  for Harris recurrence). A Harris process has a unique (up to constant multiples) invariant measure which for the moment we may call . We know that is equivalent to , and we have ratio limit theorems: for pairs of measurable functions , with such that , the limits
exist almost surely, for every choice of a starting point . The structure of as a sum of i.i.d. excursions away from then allows to identify the limits with
3) Associate to the invariant measure on the configuration space defined by (18) an invariant occupation measure on the single particle space via
Let denote the sequence of successive immigration times, write for the time of extinction of the subprocess of all direct descendants of the ancestor who immigrated at time , then
coincides –with from (8) and from (A3)– with
in application of definition 1.2.3. This shows that in (19) equals , up to some multiplicative constant. Combining (15) with (7) in assumption 1.2.2 we see that is finite. This implies , and we have the assertion of the lemma up to choice of norming constants: on is a finite measure, thus we have positive Harris recurrence of the BDI process with finite invariant occupation measure.
4) It remains to determine the constants. Define with notations of 3). As a consequence of (A3) we have almost surely as
for every choice of a starting point for the process (where shows that the right hand side is necessarily larger than ), together with
almost surely as . This establishes
when invariant measure is defined by (18) and invariant occupation measure by (19).
Now, dividing the right hand sides of (18)+(7) and both sides of (20) by and changing notations correspondingly, we get the assertion of the lemma with respect to the invariant probability.
2 Some properties of the invariant probability and the invariant occupation measure
We state and prove two theorems on the invariant measure and the invariant occupation measure. Both will be key tools in the statistical context of sections 3 and 4. Theorem 2.1.3 deals with finite ‘moments’ of the invariant probability of the BDI process of the same order as the reproduction law in (A2) ii). Theorem 2.1.6 gives conditions which grant existence of a continuous Lebesgue density of the invariant occupation measure . The proofs are given in sections 2.2 and 2.3.
For the special case of local branching where particles reproduce exactly at their death position, the existence of a continuous invariant occupation density has been considered by Höpfner and Löcherbach ;
with different methods, Löcherbach  and Hammer  allow for interactions between particles (see remark 2.1.7 below).
In our setting, due to the general form of the kernel in (A2) iii) which scatters offspring generated at a branching event relative to the parent’s position, we take a different approach.
2.1 Two theorems
We introduce further assumptions (not all of these will be in force at the same time) and strengthen preceding ones.
From now on, 1.2.1 and 1.2.2 are always assumed, is the invariant probability of the BDI process on the configuration space , and the invariant occupation measure on the single particle space as specified by (9) in lemma 1.2.5.
There is some natural number such that is bounded on , where
denotes -th moments of the position-dependent reproduction laws at in (A2) ii).
Our next assumption strengthens heavily (7) of assumption 1.2.2. Recall the expectation semigroup for the branching process without immigration from (16),
associated to the expected occupation measure (8), and its
representation as a Feynman-Kac semigroup in the ‘many-to-one’-formula (17) in remark 1.3.1.
With notation , we have
Assumption 2.1.2 implies in particular that the function in (6) is bounded, thus (7) of assumption 1.2.2 holds for any choice of an immigration measure . Property (21) is known in the general theory of semigroups as uniform exponential stability. We refer to , Ch. V, Sec. 1 for a number of equivalent characterizations which can be used to check our assumption 2.1.2 whenever the semigroup is strongly continuous on the Banach space of continous functions vanishing at infinity.
of the single-particle motion killed at rate .
The semigroup (22) was already implicit in the proof of lemma 1.2.4, see (12).
For this semigroup,
we shall now require existence of heat kernel bounds (which Hammer 
used to investigate the invariant measure on , see remark 2.1.7 below).
For sufficient conditions implying such bounds, we refer to Dynkin  theorem 0.5 p. 229 appendix paragraph 6, or Friedman  theorem 4.5 p. 141.
The semigroup in (22) admits densities with respect to Lebesgue measure which are continuous in for fixed and admit bounds
for some fixed and some positive constant .
Heat kernel bounds 2.1.4 will be a key tool in our proof for the existence of a continuous invariant occupation density, as well as for the results in section 3 below. We stress that 2.1.4 is a strong assumption: even with and constant killing rate it does not hold for Ornstein-Uhlenbeck one-particle motion when the OU parameter is different from . On the other hand, by Dynkin  p. 229, assumption 2.1.4 does hold for all choices of a Hölder continuous and bounded killing rate whenever the single-particle motion is such that uniform ellipticity holds on and all , in (A1) are bounded.
Our final assumption requires that the transition probability of (3) admits bounds of convolution type.
There exists some finite measure on the single-particle space such that
Note that (24) is essentially a condition on the transition probability of (A2) iii). Clearly assumption 2.1.5 covers the case of a product structure (2) where for some probability measure on : here we take and have equality in (24). It also covers the case of absolutely continuous product structures
for -finite measures on when -densities depend on and but are uniformly dominated by
where ; then (24) holds for . Note that we do not require the -finite measure to be Lebesgue-absolutely continuous. Beyond (25) and (26), we see from (3) that assumption 2.1.5 controls in some sense the distance of a ‘typical’ child from its parent’s position.
The following is the second main probabilistic result: heat kernel bounds 2.1.4 for particle motion killed at rate and convolution bounds 2.1.5 on the scattering of offspring at branching events allow to obtain a continuous Lebesgue density for the invariant occupation measure.
Theorem 2.1.6 will be proved in section 2.3.
Assume 1.2.1, 2.1.4, 2.1.5, and suppose that the immigration measure is such that condition (7) of 1.2.2 is satisfied. If , suppose in addition that is absolutely continuous with Lebesgue density for some . Then the invariant occupation measure on (a finite measure by lemma 1.2.5) admits a continuous Lebesgue density .
Höpfner and Löcherbach  proved existence of a continuous Lebesgue density for in the special case of local branching,
i.e. when of (A2) iii) is of product type (2) with .
Their approach, using stochastic flows of diffeomorphisms, is not directly applicable in our case of non-local branching where we allow for jumps in the distribution of newborn particles, reflected in the jump diffusion with generator (5).
However, it can be adapted to our setting by using duality theory for (Feller) semigroups. This approach, which will be taken up in another paper, leads to a continuous invariant occupation density under an alternative set of conditions on the single particle motion and the branching and reproduction mechanism.
In the present work however, we restrict to the setting of assumptions 2.1.4 and 2.1.5, since the heat kernel bounds (23) will also be used (independently) in the proofs of our results in section 3 below.
ii) For the case of local and binary branching, Löcherbach  considered a generalization of the model where coexisting particles move as interacting diffusions. In a -setting, assuming uniform ellipticity, Malliavin calculus establishes the existence of a continuous invariant occupation density (theorem 4.2 in ).
iii) Assuming existence of Lebesgue densities for and of transition densities for as in (25)-(26
) such that the Fourier transforms ofand are integrable, Hammer  used Fourier methods to deduce existence of a continuous Lebesgue density of the invariant measure on the configuration space from the heat kernel bound assumption 2.1.4 (see assumptions 2.2.1, 2.2.5 and theorem 2.2.8 in ), where continuity on is understood layer-wise, i.e. for every the restiction of to admits a Lebesgue density which belongs to . We shall not make use of this result in the present paper.