Continuous flows driving branching processes and their nonlinear evolution equations

1 Introduction

Let (b_k)_k⩾1 be a sequence of positive numbers such that ∑k⩾1bk⩽1 and consider the following parabolic equation on measures

with the initial condition F₀ = F, where c > 0 and for a function F : M(ℝ^d) → [0, 1] we denoted by F′ its variational derivative.

The first goal of this paper is to show that equation (1.1) has a mild solution having a probabilistic representation by means of the distribution of the d-dimensional Brownian motion and the non-local pure branching process on the finite configurations of M(ℝ^d), induced by the sequence (b_k)_k⩾1. Recall that the measure-valued branching processes intend to be stochastic models for the population dynamics. The time evolution of a system of particles is characterised by a branching mechanism and a spatial motion which describes the movement of the particles between the branching moments. The spatial motion is a right Markov space with state space a Lusin topological space E, while the state space of the branching process is a set of measures on E, namely, either M(E) (:= the set of all finite measures on E) for the superprocesses (see [1, 2, 15, 16, 18, 19, 22, 29, 30]), or Ê (:= the set of all finite configurations of E) for the non-local branching processes; see e.g. [8, 10, 11, 26, 31].

If in (1.1) we replace the Laplace operator with a second order integro-differential operator generating a Markov process with state space ℝ^d, then the obtained equation has a solution with a similar probabilistic representation, and notice that it is based on the same pure branching process. In this way we complete the answer given in [8] to the challenging suggestion of T.E. Harris from [23] , page 50, to investigate branching processes for which “each object is a very complicated entity; e.g., an object may itself be a population”. In fact, in Section 5 we consider equation (1.1) on M(E), with the generator of a right Markov process on the Lusin topological space E instead of the Laplacian on ℝ^d.

Our second goal is to emphasize a class of branching processes which are driven by a right continuous flow Φ on E, in the sense that such a measure-valued process X^ admits a representation by means of a second branching process X0^ and of the flow on measures induced by Φ,

It turns out that the branching process X0^ has the same branching mechanism as X^ , however X0^ has no spatial motion and therefore we call it pure branching process.

The representation (1.2) holds in the case of a superprocess for which the branching mechanism Ψ is independent of the spatial variable, Ψ : [0, ∞) → ℝ is of the form

where b, c ∈ ℝ, c ⩾ 0, and N is a measure on (0, ∞) such that N(u ∧ u²) < ∞, and the spatial motion is a right continuous flow on E, regarded as a deterministic Markov process. Recall that a superprocess X^ provides a stochastic solution for the nonlinear evolution equation written formally as

with the initial condition v₀ = f, where D is the generator of the spatial motion and Ψ is the branching mechanism. The nonlinear evolution equation associated with the pure branching process X0^ is the particular case of (1.3) with no spatial motion term on the right hand side, that is, with D ≡ 0. Actually, we consider the integral version of equation (1.3), or equivalently, we work with the mild solutions of it.

A basic step of our approach is to solve equation (1.3) with D the generator of a right continuous flow Φ on E, to find a solution v_t, t ⩾ 0, having the following two probabilistic representations:

where e_f is the exponential mapping on M(E), e_f (μ):=e^−∫fdμ for all μ ∈ M(E); with the terminology of [18] , page 133, v_t is written as the “log-potential of X t0^ and Φ_t”. We prove analogous results for non-local branching processes, provided that the flow and the distributions of the branching are compatible.

If 𝒧 and 𝒧⁰ are the extended weak generators of the superprocesses X^ and respectively X0^ on M(E), then we have

where D^ is the generator of the continuous flow on M(E) induced by Φ. In this way, the representation (1.2) may be interpreted as a consequence of regarding 𝒧 as a modification of 𝒧⁰ with the first order operator D^ , which is a substitute for a “drift type” operator acting in the considered infinite dimensional frame. This is an exemplification of a general strategy developed in [3] ; for more details see assertion (iv) of Remark 3.5 below.

A key point in obtaining the claimed probabilistic solution to (1.1) is the representation (1.2) applied on M(ℝ^d) as state space for the spatial motion.

We present now the structure of the paper.

The first main result (Theorem 3.2) is stated in Section 3. It is preceded by the description of the weak generator of a superprocess (Proposition 3.1), which completes results from [2, 19, 22, 30]. Results related to equation (1.3) and (1.4) are gathered in assertion (iii) of Theorem 3.2. Corollary 3.4, and Remark 3.5.

The non-local branching processes on configuration spaces are investigated in Section 4. First, we highlight relevant absorbing sets of finite configurations with respect to a pure branching process on Ê (in Proposition 4.1). Theorem 4.2 is the second main statement of the work. Results similar to (1.4) and Corollary 3.4, but valid on Ê, are collected in Remark 4.3, which also indicates absorbing sets for the pure branching process, when the branching mechanism is the classical one.

The proofs of the two theorems use a unifying argument (Lemma 3.3), which is applicable on both spaces of measures (M(E) and Ê) and we put its proof in the Appendix. Recall that in [9] it was developed a method of proving the path regularity for both superprocesses and non-local branching processes. The Appendix also presents briefly complements on the right continuous flows and the extended weak generator of a Markov process.

The application in solving equation (1.1) is given in Section 5, Corollary 5.2.

In Section 2 we present basic notations and preliminary facts on the right Markov processes, branching processes, and potential theoretical tools we shall use further: transition semigroups and resolvents of kernels, excessive functions, the fine topology, absorbing sets, and right continuous flows on measures.

2 Preliminaries

For a σ-algebra 𝒢 we denote by [𝒢] (resp. p𝒢) the vector space of all real-valued (resp. the set of all positive, numerical) 𝒢-measurable functions on E. Also, for a set of real-valued functions 𝒞 we denote by σ(𝒞) the σ-algebra generated by 𝒞, by [𝒞] the vector space spanned by 𝒞, and by p𝒞 (resp. b𝒞) the set of all positive (resp. bounded) functions from 𝒞.

Excessive functions. Let E be a Lusin topological space (i.e., E is homeomorphic to a Borel subset of a compact metric space) with Borel σ-algebra 𝒝(E). Let further 𝒰 = (U_α)_α>0 be a sub-Markovian resolvent of kernels on (E, 𝒝(E)). We denote by 𝒠(𝒰) the set of all 𝒝(E)-measurable 𝒰-excessive functions, i.e., u ∈ 𝒠(𝒰) if and only if u is a nonnegative, numerical, 𝒝(E)-measurable function, αU_αu ⩽ u for all α > 0, and lim_α→∞ αU_αu(x) = u(x), x∈ E.

If β > 0, we denote by 𝒰_β the sub-Markovian resolvent of kernels 𝒰_β = (U_β+α)_α>0. A 𝒰_β-excessive function is also called β-excessive.

Let X = (Ω, 𝒡, 𝒡_t, X_t, θ_t, 𝕇^x, ζ) be the right Markov process on E having 𝕋 = (T_t)_t⩾0 as transition function, T_tf(x) = 𝔼^x(f(X_t), t < ζ), t ⩾ 0, f ∈ p𝒝(E) (:= the set of all positive 𝒝(E)-measurable functions on E). The resolvent of kernels 𝒰 = (U_α)_α>0 on (E, 𝒝(E)) associated with (T_t)_t⩾0 is defined as Uαf:=∫0∞e−αtTtfdt , f ∈ p𝒝(E).

Fine topology. The fine topology is the coarsest topology on E making continuous all β-excessive functions for some (and equivalently for all) β > 0. Recall that a function f from p𝒝(E) is finely continuous if and only if t ↦ f(X_t) is a.s. right continuous on [0, ζ). Consequently, due to the right continuity of the paths of the process X, each continuous function on E is finely continuous; see [4, 32] for details on the potential theory for Markov processes with a general state space.

Absorbing set. A set A ∈ 𝒝(E) is called absorbing (with respect to the Markov process X) if R βE\A1=0 on A, where R βA1:=inf{u∈𝒠(𝒰β):u⩾1A} for some β > 0. If A is absorbing then it is finely open, U_β1_E\A = 0 on A, and the restriction of X to A is again a right process; see e.g. [32] or [13] . Basic examples of absorbing sets are [v = 0] and [v < ∞] for some v ∈ 𝒠(𝒰_β). Examples of absorbing sets for branching processes will be given in Proposition 4.1 and Remark 4.3 (iii) below.

Trivial Markov process. Let X0=(X t0)t⩾0 be the Markov process on E for which each point is a trap, i.e., 𝕇x(X t0=x)=1 for all t ⩾ 0 and x ∈ E, or equivalently, each kernel from its transition function (T t0)t⩾0 is the identity operator, T t0f=f for every t ⩾ 0 and f ∈ p𝒝(E). Clearly, any Borel subset of E is absorbing with respect to the trivial Markov process on E.

Branching processes. We endow M(E) with the weak topology and let 𝒨(E) denote the Borel σ-algebra on M(E).

A second measure state space we will consider is the set Ê ⊆ M(E) of all finite configurations of E, E^:={ ∑i⩽i0δxi:i0∈𝕅,i0⩾1,xi∈E for all 1⩽i⩽i0 }∪{0} , where 0 denotes the zero measure. The set Ê is endowed with the weak topology on the finite measures on E and the corresponding Borel σ-algebra 𝒝(Ê); see e.g. [14] .

Further (M, 𝒨) will denote either (M(E), 𝒨(E)) or (Ê, 𝒝(Ê)). Recall that a right Markov process with state space M is called branching process provided that for any two independent copies X and X′ of the given process on M, starting respectively from two measures μ and μ′, X + X′ and the process starting from μ + μ′ are equal in distribution. Recall also that if p₁, p₂ are two finite measures on M, then their convolution p₁ * p₂ is the finite measure on M defined for every h ∈ p𝒨 by

A bounded kernel Q on (M, 𝒨) is called branching kernel provided that

where Q_μ denotes the measure on M such that ∫Mh dQμ=Qh(μ) for all h ∈ p𝒨. It turns out that: a right Markov process with state space M is a branching process if and only if its transition function is formed by branching kernels. For a function f ∈ bp𝒝(E) we shall consider the mappings l_f : M → ℝ₊ and e_f : M → [0, 1] defined as

Notice that 𝒨 is generated by {l_f : f ∈ bp𝒝(E)} and also by {e_f : f ∈ bp𝒝(E)}.

Right continuous flows on measures. We present in Appendix (A3) the basic facts on the right continuous flows.

If t ⩾ 0 define the kernel Q t0 on M(E) as

We may consider the right continuous flow on M(E) induced by the right Markov process X. More precisely, let Φ^=(Φ^t)t⩾0 be the family of mappings on M(E) defined as Φ^t(μ):=Xt(𝕇μ) , μ ∈ M(E); recall the notation Xt(𝕇μ)(A):=∫E𝕇x(X t−1(A))μ(dx) for all A ∈ 𝒝(E). Then Φ^ is a right continuous flow on M(E) and its transition function is 𝕈0=(Qt0)t⩾0 . It is easy to check that Q t0 , t ⩾ 0, is a branching kernel on M(E) and consequently, the (deterministic) Markov process induced by Φ^ is actually a branching process on M(E), it is the (X, 0)-superprocess.

Let Φ = (Φ_t)_t⩾0 be a right continuous flow on E. Recall that Φ may be regarded as a deterministic right Markov process on E with infinite lifetime, X = (Ω, 𝒡, 𝒡_t, X_t, θ_t, 𝕇^x): Ω = E, 𝒡 = 𝒡_t = 𝒝(E), X_t(x) := Φ_t(x) =: θ_t(x) for all x ∈ Ω, and 𝕇^x = δ_x. In this particular case, the right continuous flow on M(E) induced by X, as described before, is also denoted by Φ = (Φ_t)_t⩾0: if μ ∈M(E) then Φ_t(μ) is the transport of μ by means of Φ_t, i.e.,

If μ ∈Ê, μ=∑i=1kδxi , then Φ_t(μ) also belongs to Ê, Φt(∑i=1kδxi)=∑i=1kδΦt(xi) , t ⩾ 0. Consequently, the right continuous flow Φ on E also induces a right continuous flow on Ê, we shall denote it by Φ too; here we put Φ_t(0) = 0.

Let A ∈ 𝒝(E) be an absorbing set with respect to the right continuous flow Φ on E, i.e., Φ_t(x) ∈A for all x ∈ A. Then  is an absorbing set with respect to the right continuous flow on Ê induced by Φ.

3 Right continuous flows driving superprocesses

In the sequel the fixed right Markov process X = (Ω, 𝒡, 𝒡_t, X_t, θ_t, 𝕇^x) with state space E and infinite lifetime is called spatial motion, and let 𝒰 be its resolvent. We also fix a branching mechanism, that is, a function Ψ : E × [0, ∞) → ℝ of the form

where c ⩾ 0 and b are bounded 𝒝(E)-measurable functions, and N : p𝒝((0, ∞)) → p𝒝(E) is a kernel such that N(u ∧ u²) ∈bp𝒝(E). Examples of branching mechanisms are Ψ(λ) = −λ^α for 1 < α ⩽ 2 and Ψ(λ) = λ^α for 0 < α < 1.

We present now briefly the construction of the measure-valued branching Markov process associated with X and Ψ, the (X, Ψ)-superprocess; cf. [2, 22, 30].

Equation (1.3) has a unique mild solution, more precisely, for each f ∈ bp𝒝(E) the equation

has a unique solution (t, x) ↦ V_tf(x) jointly measurable in (t, x) such that sup_0⩽s⩽t ||V_sf||_∞ < ∞ for all t > 0 and [0, ∞) ∋ t ↦V_tf(x) is right continuous for all x ∈ E, provided that t ↦T_tf(x) has this property.

The mappings f ↦V_tf form a nonlinear semigroup of operators on bp𝒝(E).

For each t ⩾ 0 there exists a unique Markovian kernel T^t on (M(E), 𝒨(E)) such that

Since the family (V_t)_t⩾0 is a (nonlinear) semigroup on bp𝒝(E), it follows that 𝕋^=(𝕋^t)t⩾0 is a transition function (M(E), 𝒨(E)). By Theorem 4.9 from [2] , under a Feller-type regularity condition, there exists a Borel right process X^ , with state space M(E), having the transition function T^ , called (X, Ψ)-superprocess.

Let 𝒰^=(U^α)α>0 be the Markovian resolvent of kernels on (M(E), 𝒨(E)) induced by 𝕋^ .

Let

Then b′ ⩾ 0 and consider the transition function 𝕋b′=(T tb′)t⩾0 of the process obtained by killing X with the multiplicative functional induced by b′,

If u ∈ bp𝒝(E) then by Corollary 4.3 in [2] we have

We also have

The extended weak generator of a superprocess. Let (L, 𝒟(L)) be the extended weak generator of the spatial motion X on E and (𝒧, 𝒟(𝒧)) the extended weak generator of the (X, Ψ)-superprocess on M(E); see Appendix (A4) for the basic notions about the extended weak generator of a Markov process.

Let (D^,𝒟(D^)) be the extended weak generator of the right continuous flow Φ^ on M(E) induced by the right Markov process X, Φ^t(μ):=Xt(𝕇μ) , μ ∈M(E), that is, the extended weak generator of the transition function 𝕈0=(Q t0)t⩾0 , Q t0F(μ):=F(μ∘Tt) for all F ∈ bp𝒨(E), μ ∈M(E), and t ⩾ 0.

4 Right continuous flows driving non-local branching processes on finite configurations

Let (b_k)_k⩾1 be a sequence of functions from bp𝒝(E) such that ∑k⩾1bk⩽1, let m1:=∥∑k⩾1kbk∥∞ and assume that 1 < m₁ < ∞. We fix also a constant c such that 0<c⩽m1m1−1.

Recall that the set Ê of all finite configurations of E is identified with the union of all symmetric k-th powers E^(k) of E, i.e., if k ⩾ 1 then E^(k) is the factorisation of the Cartesian product E^k by the equivalence relation induced by the permutation group σ^k. Hence E^=∪k⩾0E(k), where E⁽⁰⁾ := {0}; see e.g. [11, 26]. For each k ⩾ 1, let B_k be a Markovian kernel from E^(k) to E.

If ϕ ∈ p𝒝(E), 0 ⩽ ϕ ⩽ 1, then by Proposition 4.1 from [8] (see also Theorem 3.1 in [10]), the integral evolution equation

has a unique solution t ↦ H_tϕ, jointly measurable in (t, x) ∈ ℝ₊ × E, such that 0 ⩽ H_tϕ ⩽ 1. Here, for a function h ∈ bp𝒝(E) we have denoted by h^(k), k ⩾ 1, the function on E^(k) defined as h^(k)(x) := h(x₁) · · · h(x_k) for all x = (x₁, . . . , x_k) ∈E^(k). We also denote by ĥ: Ê → ℝ₊ the function defined as ĥ_|E^(k) = h^(k) if k ⩾ 1 and ĥ(0) = 1.

The integral evolution equation (4.1) associated with a non-local branching process on Ê is formally equivalent to the equation

with the initial condition h₀ = ϕ, where L is the generator of the spatial motion; see e.g. Remark 4.2 (ii) from [8].

It turns out that the nonlinear semigroup (H_t)_t⩾0 induces a branching semigroup of kernels (Ĥ_t)_t⩾0 on Ê such that H^tϕ^=Htϕ^ for all ϕ ∈ bp𝒝(E), 0 ⩽ ϕ ⩽ 1. According to Theorem 4.1. from [8], under some additional assumptions, there exists a branching right Markov process X^ with state space Ê (depending on the spatial motion X, (b_k)_k⩾1, the sequence of kernels (B_k)_k⩾1, and c) having the transition function (Ĥ_t)_t⩾0.

Non-local pure branching processes on Ê. We consider now the non-local pure branching process X0^ on Ê for which the spatial motion is the trivial Markov process X⁰. In this case equation (4.1) becomes

Let (H t0ϕ)t⩾0 be the solution to equation (4.3), then the transition function of X0^ is (H t0^)t⩾0.

The next proposition emphasises absorbing sets with respect to a pure branching processes on Ê, it is a version of a result from [6].