Supercritical branching diffusions in random environment

Supercritical branching processes in constant environment conditioned on eventual extinction are known to be subcritical branching processes. The case of random environment is more subtle. A supercritical branching diffusion in random environment (BDRE) conditioned on eventual extinction of the population is not a BDRE. However the quenched law of the population size of a supercritical BDRE conditioned on eventual extinction is equal to the quenched law of the population size of a subcritical BDRE. As a consequence, supercritical BDREs have a phase transition which is similar to a well-known phase transition of subcritical branching processes in random environment.


Introduction and main results
Branching processes in random environment (BPREs) have attracted considerable interest in recent years, see e.g. [2,3,8] and the references therein. On the one hand this is due to the more realistic model compared with classical branching processes. On the other hand this is due to interesting properties such as a phase transition in the subcritical regime. Let us recall this phase transition. In the strongly subcritical regime, the survival probability of a BPRE (Z (1) t ) t≥0 scales like its expectation, that is, P Z where const is some constant in (0, ∞). In the weakly subcritical regime, the survival probability decreases at a different exponential rate. The intermediate subcritical regime is in between the other two cases. Understanding the differences of these three regimes is one motivation of the literature cited above. The main observation of this article is a similar phase transition in the supercritical regime.
Let us introduce the model. We consider a diffusion approximation of BPREs as this is mathematically more convenient. The diffusion approximation of BPREs is due to Kurtz (1978) and had been conjectured (slightly inaccurately) bei Keiding (1975). We follow Böinghoff and Hutzenthaler (2011) and denote this diffusion approximation as branching diffusion in random environment (BDRE). For every n ∈ AE := {1, 2, . . .} let (Z (n) k ) k∈AE 0 be a branching process in the random environment Q , k ∈ AE 0 denotes the associated random walk where n ∈ AE. Set ⌊t⌋ := max{m ∈ AE 0 : m ≤ t} for t ≥ 0. Let the environment be such that S (n) ⌊tn⌋ / √ n t≥0 converges to a Brownian motion (S t ) t≥0 with infinitesimal drift α ∈ Ê and infinitesimal standard deviation σ e ∈ [0, ∞) as n → ∞.
Furthermore assume the mean offspring standard deviation to stabilize at as n → ∞ and if a third moment condition holds, then in the Skorohod topology (see e.g. [7]) where the limiting diffusion is the unique solution of the stochastic differential equations (SDEs) for t ≥ 0 where Z 0 = z and S 0 = 0. The processes (W t ) t≥0 are independent standard Brownian motions. Throughout the paper the notations P z and E z refer to Z 0 = z and S 0 = 0 for z ∈ [0, ∞). The diffusion approximation (1) is due to Kurtz (1978) (see also [5]). Note that the random environment affects the limiting diffusion only through the mean branching variance σ 2 b and through the associated random walk. We denote the process (S t ) t≥0 as associated Brownian motion. This process plays a central role. For example it determines the conditional expectation of Z t for every z ∈ [0, ∞) and t ≥ 0. Moreover the infinitesimal drift α of the associated Brownian motion determines the type of criticality. The BDRE (2) is supercritical (i.e. positive survival probability) if α > 0, critical if α = 0 and subcritical if α < 0, see Theorem 4 of Böinghoff and Hutzenthaler (2011). We will refer to α as criticality parameter. Afanasyev (1979) was the first to discover different regimes for the survival probability of a BPRE in the subcritical regime (see [2][3][4]13] for recent articles). The following characterisation for the BDRE (2) is due to Böinghoff and Hutzenthaler (2011). The survival probability of (Z t ) t≥0 decays like the expectation, that is, P(Z t > 0) ∼ const · E(Z t ) = const·exp (α+ σ 2 e 2 )t as t → ∞, if and only if α < −σ 2 e (strongly subcritical regime). In the 2σ 2 e t as t → ∞ in the weakly subcritical regime α ∈ (−σ 2 e , 0). This article concentrates on the supercritical regime α > 0. Our main observation is that there is a phase transition which is similar to the subcritical regime. Such a phase transition has not been reported for BPREs yet. We condition on the event {Z ∞ = 0} = {lim t→∞ Z t = 0} of eventual extinction and propose the following notation. If P(Z t > 0|Z ∞ = 0) ∼ const·E(Z t |Z ∞ = 0) as t → ∞, then we say that the BDRE (Z t , S t ) t≥0 is strongly supercritical. If the probability of survival up to time t ≥ 0 conditioned on extinction decays at a different exponential rate as t → ∞, then we refer to (Z t , S t ) t≥0 as weakly supercritical. The intermediate regime is referred to as intermediate supercritical regime. Our first theorem provides the following characterisation. The BDRE is strongly for every z ∈ (0, ∞) where G ν is gamma-distributed with shape parameter ν ∈ (0, ∞) and scale parameter 1, that is, where ψ : where β := 2α for every a ∈ (0, ∞).
The proof is deferred to Section 2.
Let us recall the behavior of Feller's branching diffusion, that is, (2) with σ e = 0, which is a branching diffusion in a constant environment. The supercritical Feller diffusion conditioned on eventual extinction agrees in distribution with a subcritical Feller diffusion. This is a general property of branching processes in constant environment, see e.g. Jagers and Lagerås (2008) for the case of general branching processes (Crump-Mode-Jagers processes). Knowing this, Theorem 1 might not be surprising. However, the case of random environment is different. It turns out that the supercritical BDRE for t ≥ 0.
The proof is deferred to Section 2.
It is rather intuitive that the conditioned process is not a subcritical BDRE if σ b > 0. The supercritical BDRE has a positive probability of extinction. Thus extinction does not require the associated Brownian motion to have negative drift. As long as the BDRE stays small, extinction is possible despite the positive drift of the associated Brownian motion. Note that ifŽ t is small for some t ≥ 0, then the drift term ofŠ t is close to α. Being doomed to extinction, the conditioned process (Ž t ) t≥0 is not allowed to grow to infinity. IfŽ t is large for some t ≥ 0, then the drift term ofŠ t is close to −α which leads a decrease of (Ž t ) t≥0 . The situation is rather different in the case σ b = 0. Then the extinction probability of the BDRE is zero. So the drift of (Š t ) t≥0 needs to be negative in order to guaranteeŽ t → 0 as t → ∞. It turns out that if σ b = 0, then the drift of (Š t ) t≥0 is −α and (Ž t ,Š t ) t≥0 is a subcritical BDRE with criticality parameter −α.
We have seen that conditioning a supercritical BDRE on extinction does -in general -not result in a subcritical BDRE. However, if we condition (Z t , S t ) t≥0 on {S ∞ = −∞}, then the conditioned process turns out to be a subcritical BDRE with criticality parameter −α.
Now we come to a somewhat surprising observation. We will show that the quenched law of (Z t ) t≥0 conditioned on extinction agrees in law with the quenched law of a subcritical BDRE. More formally, inserting the second equation of (11) into the equation for dŽ t we see that for t ≥ 0. This is the SDE for the population growth of a subcritical BDRE with criticality parameter −α. As the solution of (13) is unique, this proves the following corollary of Theorem 2.
On the event of non-extinction, the population size Z t of a supercritical BDRE grows like its expectation E(Z t |S t ) as t → ∞.
be the solution of (2) with criticality parameter α ∈ Ê. Then Z t /e St t≥0 is a nonnegative martingale. Consequently for every initial value The limiting variable is zero if and only if the BDRE goes to extinction, that is, P z (Y = 0) = P z (Z ∞ = 0). In the supercritical case α > 0, the distribution of the limiting variable Y satisfies that for all z, λ ∈ [0, ∞) where G ν is gamma-distributed with shape parameter ν ∈ (0, ∞) and scale parameter 1, that is, G ν has distribution (7).
The proof is deferred to Section 2. In particular, Theorem 6 implies that Z ∞ := lim t→∞ Z t exists almost surely and that Z ∞ ∈ {0, ∞} almost surely.

Proofs
If σ b = 0 and Z 0 > 0, then the process (Z t ) t≥0 does not hit 0 in finite time almost surely. So the interval (0, ∞) is a state space for (Z t ) t≥0 if σ b = 0. The following analysis works with the state space [0, ∞) for the case σ b > 0 and with the state space (0, ∞) for the case σ b = 0. To avoid case-by-case analysis we assume σ b > 0 for the rest of this section. One can check that our proofs also work in the case σ b = 0 if the state space [0, ∞) is replaced by (0, ∞).
Inserting the associated Brownian motion (S t ) t≥0 into the diffusion equation of (Z t ) t≥0 , we see that (Z t ) t≥0 solves the SDE for t ∈ [0, ∞). One-dimensional diffusions are well-understood. In particular the scale functions are known. For the reason of completeness we derive a scale function for (23) in the following lemma. The generator of (Z t , S t ) t≥0 is the closure of the mapping G : for all z ∈ [0, ∞), s ∈ Ê and every f ∈ C 2 0 [0, ∞) × Ê . for z ∈ [0, ∞) and s ∈ Ê. Then U is a scale function for (Z t ) t≥0 and V is a scale function for (S t ) t≥0 , that is, U(Z t ) t≥0 and V (S t ) t≥0 are martingales. Moreover we have that GU ≡ 0 and GV ≡ 0.
Proof. Note that U is twice continuously differentiable. Thus we get that for all z ∈ [0, ∞). Moreover V is twice continuously differentiable and we obtain that for all s ∈ Ê. This shows GU ≡ 0 ≡ GV . Now Itô's formula implies that for all t ≥ 0. This proves that U(Z t ) t≥0 and V (S t ) t≥0 are martingales.
Proof. Define the first hitting time T x (η) := inf{t ≥ 0 : η t = x} of x ∈ Ê for every continuous path η ∈ C [0, ∞), Ê) . As V is a scale function for (S t ) t≥0 , the optional sampling theorem implies that for all s ∈ Ê and N ∈ AE, see Section 6 in [10] for more details. Thus we get that for all z ∈ [0, ∞), s ∈ Ê and t ≥ 0. The proof of the assertions (29) and (31) is analogous.
Proof of Theorm 5. Analogous to the proof of Theorem 2, we identify the generatorĜ of the BDRE conditioned on {Z ∞ > 0}. Note that for all z ∈ (0, ∞). Similar arguments as in (35) and GU ≡ 0 result in