Exponential ergodicity for general continuous-state nonlinear branching processes

By using the coupling technique, we present sufficient conditions for the exponential ergodicity of general continuous-state nonlinear branching processes in both the $L^1$-Wasserstein distance and the total variation norm, where the drift term is dissipative only for large distance, and either diffusion noise or jump noise is allowed to be vanished. Sufficient conditions for the corresponding strong ergodicity are also established.

Intuitively, such process can be identified as a continuous-state branching process with population-size-dependent branching rates and with competition. If γ 0 (x) = a + bx for some a ≥ 0 and b ∈ R and γ i (x) = c i x (i = 1, 2) for some c 1 , c 2 ≥ 0, then the solution to (1.1) is reduced into the classical continuous-state branching process, see [1,7,8,10,12] and references therein. We should mention that, only and if only in this particular case, the solution satisfies the so-called branching property, which means that different individuals act independently with each other. If γ i (x) = c i x (i = 1, 2) for some c i ≥ 0 and γ 0 (x) = b 1 x−b 2 x 2 with some b 1 , b 2 > 0, then the solution to (1.1) is called the logistic branching process in the literature, which can be used to model the population dynamics with competition, see [4,9] for more details. The quadratic regulatory term in the coefficient γ 0 (x) has an ecological interpretation, as it describes negative interactions between each pair of individuals in the population. Similar equations with general coefficients γ 0 (x) to model more general competitions were considered in [20].
Throughout this paper we always assume that (1.1) has the unique non-explosive strong solution, which is denoted by (X t ) t≥0 ; see Subsection 2.1 for related discussions. Let P t (x, ·) and (P t ) t≥0 be the transition function and the transition semigroup of the process (X t ) t≥0 , respectively. We are going to study the asymptotic behavior of the L 1 -Wasserstein distance and the total variation distance between P t (x, ·) and P t (y, ·) for any x, y ∈ R + . As a direct consequence, we will establish sufficient conditions for the exponential ergodicity and the strong ergodicity of the process (X t ) t≥0 .
To the best of our knowledge, there are few known results on this topic. For the classical branching process (i.e. γ 0 (x) = a − bx and γ i (x) = c i x (i = 1, 2) for some b > 0 and a, c i ≥ 0), by the branching property, [13,Theorem 2.4] proved that the total variation distance between P t (x, ·) and P t (y, ·) decays exponentially fast. Recently, under uniformly dissipative condition on γ 0 (x) (see Remark 3.4(1) below) and finite second moment condition on the measure ν (i.e. R + z 2 ν(dz) < ∞), [5,Theorem 4.2] established the exponential decay between P t (x, ·) and P t (y, ·) with respect to the L 1 -Wasserstein distance.
To illustrate our main contributions, we present the following statement for the exponential ergodicity and the strong ergodocity of the process (X t ) t≥0 . The reader can refer to Section 3 for general results. Theorem 1.1. Let (X t ) t≥0 be the unique strong solution to the SDE (1.1) such that assumptions below (1.1) on the coefficients are satisfied. Suppose that there are constants l 0 , k 1 ≥ 0 and k 2 > 0 such that If one of the following three assumptions holds: (1) the function γ 1 (x) is continuous and strictly positive on (0, ∞) such that (2) there are constants α ∈ (0, 2) and c 0 > 0 such that and the function γ 2 (x) is continuous and strictly positive on (0, ∞) such that (3) there are constants α ∈ (1, 2) and c 0 > 0 such that is a non-decreasing function on R + ; then the process (X t ) t≥0 is exponentially ergodic both in the W 1 -distance and the total variation norm.
In the following, we will remark that conditions (1.3) and (1.5) are sharp in some concrete examples.
Define µ(dx) = x −2 e −x dx. One can verify that for any f ∈ C 2 b (R + ), µ(Lf ) = 0, which implies that µ(dx) is an invariant measure for the operator L. However, On the other hand, according to [11, the case (i)-(ib) after Example 2.18, p. 14], we know that P x (τ 0 = ∞) = 1 for all x > 0, where τ 0 = inf{t > 0 : X t = 0}. This is, the point 0 can be seen as the reflection boundary for the diffusion process (X t ) t≥0 associated with the operator L on [0, ∞). Therefore, the process (X t ) t≥0 is not ergodic, see e.g. [2,  ( with c > 0 and γ 2 (x) = 0. Then, the solution to (1.1) is reduced into the famous Cox-Ingersoll-Ross (CIR) model. In this case, one can easily see that (1.2) and (1) in Theorem 1.1 hold. Therefore, the CIR model is exponentially ergodic in both the W 1 -distance and the total variation distance. On the other hand, denote by τ 1 = inf{t ≥ 0 : X t = 1}. According to [3,Corollary 9], for any x > 1, By letting x → ∞ in the above equality, we can conclude that sup x>1 E x [τ 1 ] = ∞. This together with [19,Lemma 2.1] yields that the CIR model is not strongly ergodic. In particular, this implies that (1.5) with δ > 1 for the strong ergodicity in some sense is sharp.
The approach of our paper is based on recent developments of the couplings for SDEs with Lévy noises via coupling operators, see [14,15,17,18,21] for more details. However, there are a few essential differences between continuous-state nonlinear branching processes and the settings of [14,15,17,18,21]. For example, in the present setting, the diffusion term and the jump noise are allowed to appear together in the SDE (1.1), and moreover both coefficients γ 1 (x) and γ 2 (x) are degenerate on R + (since γ 1 (0) = γ 2 (0) = 0). The differences require much more effort than those as in [14,15,17,18,21] to efficiently apply the coupling technique. In particular, we need consider the coupling operator that contains both local part and non-local part of the associated generator (2.1), which makes the coupling function (e.g., see (4.2) and (4.9)) in the applications of coupling process more complex and delicate than that in [14,15,17,18,21].
The remainder of this paper is arranged as follows. In Section 2, we recall some results from [6] on the strong solution to the SDE (1.1), and then present a Markovian coupling of the solution through the construction of a new coupling operator. General results on the exponential ergodiciy and the strong ergodicity for the SDE (1.1) are stated in Section 3. The proofs of all main results in Section 3 and Theorem 1.1 are given in the last section.

Unique strong solution and its coupling process
This section consists of two parts. We first recall results from [6] on the existence and the uniqueness of the strong solution to the SDE (1.1), and then construct a new Markovian coupling of the solution. (1) there is a constant K > 0 so that (2) there exists a non-decreasing function L(x) on R + such that (3) the function γ 2 (x) is nonnegative and non-decreasing on R + ; (4) γ 0 (x) = γ 0,1 (x) − γ 0,2 (x), where γ 0,1 (x) is continuous on R + , and γ 0,2 (x) is continuous and non-decreasing on R + . For each integer m ≥ 1 there is a non-decreasing concave function r m (x) on R + such that Then, for any initial value X 0 = x ≥ 0, there exists a unique strong solution to the SDE (1.1), and the solution is a strong Markov process (X t ) t≥0 with the generator given by for any f ∈ C 2 b (R + ). To investigate the exponential ergodicity of the process (X t ) t≥0 , we will assume that the drift term γ 0 (x) is dissipative for large distance, see (3.1) below. One can see that condition (1) in Theorem 2.1 holds with K = sup 0≤r≤l 0 Φ 1 (r) under (3.1). On the other hand, we suppose that the function γ 1 (x) is continuous on R + such that γ 1 (0) = 0. Hence, condition (2) in Theorem 2.1 holds with L(x) := sup 0≤y≤x γ 1 (y). We have already supposed that condition (3) is satisfied, see assumptions below the SDE (1.1). Therefore, in the setting of our paper, the SDE (1.1) has a unique strong solution under assumptions of Theorem 3.1 (or Theorem 3.2), and some locally continuous assumptions on the coefficients γ i (x) for all i = 0, 1, 2 (e.g. conditions (4) and (5) in Theorem 2.1).

2.2.
Markovian coupling for continuous-state nonlinear branching process. To study the coupling of the process (X t ) t≥0 determined by (1.1), we begin with the construction of a new coupling operator for its generator L given by (2.1).
Recall that an operatorL acting on where h(x, y) = f (x) + g(x) for x, y ∈ R + . In this paper, we will use the coupling by reflection of the local part and the refined basic coupling of the non-local part for the operator L. Note that the function γ 2 (x) is non-decreasing on R + . For a given parameter κ > 0, set x κ = x ∧ κ for x > 0. Roughly speaking, when x > y ≥ 0, the refined basic coupling of the non-local part for the operator L is given by for all x ∈ R. Similarly, we can define the case that 0 ≤ x < y. See [18,Section 2] for more details on the refined basic coupling for SDEs with Lévy jumps. Then, for any f ∈ C 2 (R 2 + ) and x > y ≥ 0, we definẽ Here and in what follows, ∂x∂y , and so on. Similarly, we can defineLf (x, y) for the case that 0 ≤ x < y. By using the fact that µ x = δ x * µ −x for any x ∈ R (see [18,Corollary A.2]), one can check that the generatorL constructed above is a coupling operator of L given by (2.1); see [18, Subsection 2.1].
Next, we will construct the SDE on R 2 + associated with the coupling operatorL defined above, and prove the existence of the strong solution to the corresponding SDE. The idea below is partly motivated by [ for all x ∈ R. Recalling µ x = ν ∧(δ x * ν)(dz), we define the following control function with ρ(0, z) = 1 by convention. Consider the following SDE: zÑ(ds, dz, du), Proposition 2.2. For any (x, y) ∈ R 2 + , the system of equations (2.5) is well defined, and has a unique strong solution Proof. Recall that in the setting of our paper, we always assume that (1.1) has a nonexplosive and pathwise unique strong solution (X t ) t≥0 . We are going to show that the sample paths of (Y t ) t≥0 given in (2.5) can be obtained by repeatedly modifying those of the solution of the following equation: By the definition of (B * t ) t≥0 , we can verify that (B * t ) t≥0 is still an F t -Brownian motion. Since the driving Poisson random measure for (1.1) and (2.6) is same, the existence of the strong solution (Z t ) t≥0 to the equation (2.6) is guaranteed by the pathwise unique strong solution to (1.1).
We first claim that the process (Y t ) t≥0 given in (2.5) is the same as N(ds, dz, du). (2.7) Indeed, this immediately follows from the fact that for all z > 0, µ z (R + ) = µ −z (R + ) < ∞, and the identity that for any x, y ∈ R + with x = y, Hence, we next turn to construct the sample paths of (Y t ) t≥0 given in (2.7).
Define the stopping times We consider two cases: t ]. In the following, we will restrict on the event {T 1 > σ 1 } and consider the following SDE: Denote its solution by (Z 2,t := for all t > σ 1 . We further define T 2 = inf{t > 0 : Z In the same way, we can define Y t for t ≤ σ 2 . We then repeat this procedure. Note that is uniformly bounded (thanks to (2.4)) for any t < τ m with m = 1, 2, . . . , where Then only finite many modifications have to be made in the interval (0, t ∧ τ m ).
Finally, by letting m → ∞, we can determine the unique strong solution (Y t ) t≥0 to the SDE (2.7) globally.
With the construction of (Y t ) t≥0 above, we can apply the Itô formula to the SDE (2.5) to obtain the assertion (1). The assertion (2) immediately follows from the SDE (2.5) and the assumption that (1.1) has a non-explosive and pathwise unique strong solution (X t ) t≥0 .
In the following, we call (X t , Y t ) t≥0 determined by (2.5) a (Markovian) coupling process of (X t ) t≥0 . To conclude this part, we will give the preserving order property of the coupling process (X t , Y t ) t≥0 .
be the coupling process determined by (2.5) and with the starting point (x, y). If x > y, then X t ≥ Y t for all t > 0 a.s.
Proof. Denote by P (x,y) and E (x,y) the probability and the expectation of the process (X t , Y t ) t≥0 starting from (x, y), respectively. Let Then, for any x > y and t > 0, where in the last equality we used the fact thatLf n (x, y) = 0 for all x ≥ y, thanks to the definition of the coupling operatorL given by (2.3) and the assumption that the function γ 2 (x) is non-decreasing on R + . Then, by the Fatou lemma, Therefore, for any x > y, P (x,y) (T = ∞) = 1. That is, for any x > y, the coupling process (X t , Y t ) t≥0 associated with the coupling operatorL satisfies that X t ≥ Y t for all t > 0 a.s. Remark 2.4. One can apply the synchronous coupling (instead of the refined basic coupling) of the non-local part to construct another coupling operator for the operator L. For any x > y ≥ 0, the synchronous coupling of the non-local part for the operator L given by (2.1) is given by Then, for any f ∈ C 2 (R 2 + ) and x > y ≥ 0, the corresponding coupling operator L * is defined by The difference betweenL and L * is that the coupling operator L * do not involve the measures µ (x−y)κ and µ −(x−y)κ . The coupling process associated with the coupling operator L * above can be construed directly. Actually, putting (1.1) and (2.6) together, we can check by Itô's formula that the generator of the Markov process (X t , Y t ) t≥0 defined by (1.1) and (2.6) on R 2 + is just the coupling operator L * ; Similarly, we can see that this coupling process (X t , Y t ) t≥0 also enjoys the preserving order property as in Corollary 2.3.
3. Exponential convergence in the L 1 -Wasserstein distance and the total variation distance In this section, we shall give general results about the exponential ergodicity of the process (X t ) t≥0 determined by the SDE (1.1), in terms of both the L 1 -Wasserstein distance and the total variation norm. To present our main result, we first introduce some notation. For a strictly increasing function ψ on R + and two probability measures µ 1 and µ 2 on R + , define where C (µ 1 , µ 2 ) is the collection of measures on R 2 + with marginals µ 1 and µ 2 . When ψ is concave, the above definition gives rise to a Wasserstein distance W ψ in the space of probability measures µ on R + such that R + ψ(z) µ(dz) < ∞. If ψ(r) = r for all r 0, then W ψ is the standard L 1 -Wasserstein distance, which will be denoted by W 1 throughout this paper. Another well known example for W ψ is given by ψ(r) = 1 (0,∞) (r), which leads to the total variation distance The following two results give us the exponential convergence in the L 1 -Wasserstein distance and the total variation norm for the SDE (1.1), respectively. Theorem 3.1. Suppose that there are constants l 0 ≥ 0, k 2 > 0 and a nonnegative function x − y > l 0 .
If one of the following two assumptions holds: (A1) there exist constants β ∈ [1, 2) and k 3 > 0 such that (A2) there exist constants α ∈ (0, 2), β ∈ [α − 1, α) ∩ (0, ∞) and C * , k 3 > 0 such that where µ z is given by (2.2); then there exist positive constants C and λ so that for all t > 0 and x, y ≥ 0, W 1 (P t (x, ·), P t (y, ·)) ≤ Ce −λt |x − y|. when Assumption (A2) holds, then there exist positive constants C and λ so that for all t > 0 and x, y ≥ 0, We make some comments on the assumptions of Theorems 3.1 and 3.2. First, (3.1) is the so-called dissipative condition for large distance on the drift term γ 0 (x). In applications there are a lot of choices for the function Φ 1 ; for example, Φ 1 (r) = Cr corresponds to the standard one-sided locally Lipschitz continuous condition, and Φ 1 (r) = Cr log(4l 0 /r) is the typical one-sided non-Lipschitz continuous condition. Both functions satisfy assumptions in Theorems 3.1 and 3.2. Secondly, since we assume that the function γ 1 (x) is continuous on R + such that γ 1 (0) = 0, (3.2) is satisfied when the function γ 1 (x) is strictly positive on (0, ∞) such that lim inf x→0 This assumption is concerned on the concentration of the Lévy measure ν around zero (small jump activity), and it implies that the measure ν has a component that is absolutely continuous with respect to the Lebesgue measure, see [18,Proposition A.5]. Then (3.4) is equivalently saying that γ 2 (x) ≥ k 3 x β for all 0 ≤ x ≤ l 0 , which is also equivalent that γ 2 (x) is strictly positive on (0, ∞) such that lim inf x→0 γ 2 (x) x β > 0. On the other hand, when γ 2 (x) − γ 2 (y) ≥ k 3 (x − y) β for all 0 < x − y ≤ l 0 (this in particular indicates that the function γ 2 is strictly increasing on R + ), we only require (3.4), which can be fulfilled even for singular measures ν, see the remarks below Theorem 1.1.
As direct consequences of Theorems 3.1 and 3.2, we have the following statement for the exponential ergodicity of the process (X t ) t≥0 in term of the W 1 -distance and the total variation norm. Let P 1 be the space of probability measures having the first finite moment. (1) Under assumptions of Theorem 3.1, there exist a unique invariant probability measure µ ∈ P 1 and a constant λ > 0 such that for all t > 0 and µ 0 ∈ P 1 , where C µ 0 is a positive constant depending on µ 0 .
(2) Under assumptions of Theorem 3.2, there exist a unique invariant probability measure µ ∈ P 1 and a constant λ > 0 such that for all t > 0 and µ 0 ∈ P 1 , where C µ 0 is a positive constant depending on µ 0 .
and the finite second moment condition for the jump measure ν, as well as some growth conditions on the coefficients γ 1 (x) and γ 2 (x), [ with α ∈ (0, 2) and α 1 > 1.
(2) We mention that, by the remarks below [11, Example 2.18], one can easily give examples such that the assumptions of Theorem 3.1 (or Theorem 3.2) are satisfied, but for any x > 0, P x (τ 0 < ∞) > 0 (or even =1), where τ 0 = inf{t > 0 : X t = 0}. Therefore, under assumptions of Theorem 3.1 (or Theorem 3.2) the invariant probability measure of the process (X t ) t≥0 could be allowed to have an atom at {0}.
The following assertion is furthermore concerned on the strong ergodicity of the process (X t ) t≥0 . Theorem 3.5. Under assumptions of Theorem 3.2, if (3.1) is strengthened into the condition that there are a constant l 0 ≥ 0 and two nonnegative functions Φ 1 and Φ 2 such that where Φ 1 is the same as that in Theorem 3.2, and then the process (X t ) t≥0 is strongly ergodic, i.e., there exist the unique invariant probability measure µ and constants C, λ > 0 such that for all t > 0 and x ≥ 0, (3.5) is stronger than (3.1) (by choosing l 0 > 0 large enough if necessarily). A typical example for the function Φ 2 in Theorem 3.5 is that Φ 2 (r) = c 0 r δ with c 0 > 0 and δ > 1.
We close this section with the following examples on the coefficient γ 0 (x). Example 3.6.
We have the following typical choice of functions g in the definition (4.2) for ψ.
Proof. This lemma follows from the preserving order property for the coupling process (X t , Y t ) t≥0 associated with the coupling operatorL proved in Corollary 2.3, and the arguments in part (2)  (1) We first verify the assertion when (A2) is satisfied. At the beginning we will prove it under the assumption that In this case, (3.4) is reduced into which is equivalent to Throughout the proof, without loss of generality we may and can assume that l 0 ≥ 1 and κ ∈ (0, 1]. According to the definition (2.3) of the coupling operatorL, we know that for any f ∈ C 2 (R + ) and any x > y ≥ 0, In the following, we will take f to be the function ψ defined by (4.2), where the constants c 1 , c 2 > 0 and the function g are determined later. According to Lemma 4.1(1), ψ ′′ (r) ≤ 0 for all r > 0, and so by the mean value theorem, for any x > y ≥ 0 and z > 0, moreover, thanks to Lemma 4.1(2), for 0 < x − y ≤ l 0 and 0 < z ≤ l 0 , we see Furthermore, taking and recalling g ′ (r) = (α − β)r α−β−1 + c 3 Φ 1 (r)r α−β−2 , we arrive at that for any x, y ∈ R + with 0 < x − y ≤ l 0 , On the other hand, for any x − y > l 0 , according to all the estimates above for the function ψ and (3.1), where in the last inequality we used the fact that ψ ′′ ≤ 0 and the definition of ψ. According to (4.5), (4.6) and Lemma 4.1(3), we know that for any 0 < y < x, This along with Lemma 4.3 yields that for any t > 0 and x, y ∈ R + , W ψ (P t (x, ·), P t (y, ·)) ≤ ψ(|x − y|)e −λt .
Hence, the required assertion follows from the inequality above and Lemma 4.1 (3). When one can follow the arguments above to obtain the desired assertion. Indeed, in this case we can get rid of the term involved µ (x−y)κ (R + ) in estimates forLψ(x − y) for any x, y ∈ R + with 0 < x − y ≤ l 0 , since this term is non-positive. We also note that under (4.7) we can also directly apply the coupling operator L * and the associated coupling process mentioned in Remark 2.4; however, such coupling can not deal with the case that (4.4) is satisfied. This explains the reason why we adopt the refined basic coupling for the non-local part of the operator L, rather than simply apply the synchronous coupling.
(2) We next verify the assertion when (A1) is satisfied. Let ψ be the function defined by (4.2). Then, we get from estimates for ψ, (3.2) and (3.1) that, for any x > y with 0 < x − y ≤ l 0 , where in the first inequality we used the fact that thanks to (3.2). Furthermore, we choose Similarly, with possible choice of constants c 1 , c 2 and c 3 in the definition of ψ, one can follow the argument in part (1) to verify the desired assertion. The details are omitted here.
Proof of Theorem 3.2. For simplicity, we only verify the case that (A2) is satisfied and that µ z (R + ) ≥ C * z −α for all z ∈ (0, κ] (i.e., (4.4) holds), since one can prove the desired assertion similarly (and even easier) for other cases. Without loss of generality, we assume that l 0 ≥ 1 and κ ∈ (0, 1]. For any n ≥ 1, define f n ∈ C 2 (R + ) such that where b > 0 chosen later, θ = (α − β)/2 ∈ (0, 1), and ψ is defined by (4.2) (which is the one in part (1) of the proof of Theorem 3.1 with some modification on the associated constant c 3 ). We will verify that there exists a constant λ > 0 such that for any n ≥ 1 and x − y > 1/n, Here and in what follows, for any nonnegative functions f, g, f ≍ g means that there is a constant c ≥ 1 such that In the following, let ψ 0 (r) = b(r/(1 + r)) θ , and l * 0 ∈ (0, κ] determined later. Then, by (3.1), for n ≥ 1 and 1/n ≤ l * 0 ≤ x − y ≤ l 0 , where in the first inequality we used the fact that ψ ′′ 0 ≤ 0 and the definition of the coupling operatorL given by (2.3). On the other hand, with the same function ψ (with the same function g and the constants c 0 , c 1 , c 2 ) in the proof of Theorem 3.1, we find that for n ≥ 1 and 1/n ≤ l * 0 ≤ x − y ≤ l 0 , g(x−y) AND JIAN WANG 0, we can find l * 0 ∈ (0, 1] small enough such that for all n ≥ 1 and 1/n < x − y ≤ l * 0 ≤ 1,L Hence, for any n ≥ 1 and 1/n < x − y ≤ l * 0 , Finally, according to (4.6) and the facts that ψ ′ 0 ≥ 0 and ψ ′′ 0 ≤ 0, we find that for any x − y > l 0 , Combining all the estimates above forLf n (x − y), we can obtain (4.8), thanks to the fact that f n (r) ≍ (1 + r) for all r = 0. The proof is complete.
Proof of Corollary 3.3. By [6, Proposition 2.3], we see that under assumptions of Theorem 3.1 (or Theorem 3.2), there exist constants C 1 , K > 0 such that for all x ∈ R + and t > 0, It then follows that δ x P t ∈ P 1 and hence µP t ∈ P 1 for each µ ∈ P 1 , where P 1 is the space of all probability measures on (R + , B(R + )) with the first finite moment. With this at hand, the proof of Corollary 3.3 essentially follows from that of [16,Corollary 1.8]. We omit the details here.
On the other hand, for any l 0 < x − y ≤ 2l 0 , Combining with all the estimates above, we obtain that there is a constant λ > 0 such that for all x, y ∈ R + with x > y and n ≥ 1, Lf n (x, y) ≤ −λ.
This along with the fact that f n (r) ≍ 1 (0,∞) for all n ≥ 1 and Lemma 4.3 in turn yields that there exists a positive constant C so that for all t > 0 and x, y ∈ R + , P t (x, ·) − P t (x, ·) Var ≤ Ce −λt .
Hence, the desired assertion follows from the proof of Corollary 3.3.
(3) We consider the function x → e cx δ with c, δ > 0 on R + . For any x, y ∈ R + with x − y ≥ l 0 and some l 0 > 0 large enough, where c 0 and θ are positive constants. Here, in the first inequality we used the fact that the function x → e cx δ x is increasing for x > 0 large enough, and the second inequality follows from the fact that e cx δ x ≥ c 0 x θ for x > 0 large enough, where θ > 0 can be chosen to be any positive constant. With aid of this inequality, we can prove the desired assertion by following the argument in (2).