Ergodic Behavior of Non-conservative Semigroups via Generalized Doeblin’s Conditions

Abstract

We provide quantitative estimates in total variation distance for positive semigroups, which can be non-conservative and non-homogeneous. The techniques relies on a family of conservative semigroups that describes a typical particle and Doeblin’s type conditions inherited from Champagnat and Villemonais (Probab. Theory Relat. Fields 164(1–2):243–283, 2016) for coupling the associated process. Our aim is to provide quantitative estimates for linear partial differential equations and we develop several applications for population dynamics in varying environment. We start with the asymptotic profile for a growth diffusion model with time and space non-homogeneity. Moreover we provide general estimates for semigroups which become asymptotically homogeneous, which are applied to an age-structured population model. Finally, we obtain a speed of convergence for periodic semigroups and new bounds in the homogeneous setting. They are illustrated on the renewal equation.

Appendices

Appendix A: Branching Models, Absorbed Markov Process and Semigroups

The techniques by coupling used in this paper have been extensively developed in probability, in particular for the study of branching processes and killed process, see the introduction for references. Let us present here informally the probabilistic objects and the interpretation of the auxiliary semigroup.

For that purpose we consider a population of individuals with a trait belonging to the space $\mathcal{X}$. This population can die or give birth to some offsprings with a rate which depends on their trait and independently one from each other (branching property). Moreover the trait may vary in an homogeneous way and without memory (Markov property). Let us also assume that some subspace $\mathcal{S}$ of $\mathcal{X}$ is absorbing, meaning that each individual whose trait reaches this set stop dividing and keeps a constant trait. Writing $V_{t}$ the set of individuals at time $t$ and $(X_{t}^{i} : i \in V _{t})$ the set of their traits, the branching and Markov properties and the absorbing property of $\mathcal{S}$ ensure that

$$\delta_{x} M_{s,t}(f)=\mathbb{E} \biggl( \sum_{i\in V_{t}} f(X_{t}^{i})1_{X _{t}^{i} \notin\mathcal{S}} \Bigm| X_{s}=\delta_{x} \biggr) $$

is a semigroup. In general, it is not conservative, since its mass

$$m_{s,t}(x)=\delta_{x} M_{s,t}{\mathbf{1}}=\mathbb{E}\bigl(\#\bigl\{ i \in V_{t} : X_{t}^{i} \notin\mathcal{S} \bigm| X_{s}=\delta_{x}\bigr\} \bigr) $$

can decrease by absorption in $\mathcal{S}$ or death of individual or created by births. The trait of a typical non-absorbed individual is then given by the auxiliary conservative inhomogeneous semigroup

$$\delta_{x}P^{(t)}_{u,v}f= \frac{\delta_{x} M_{u,v}(fm_{v,t})}{m_{u,t}(x)}=\frac{\mathbb{E} ( \sum_{i\in V_{t}} f(X_{v}^{i})1_{X_{v}^{i} \notin\mathcal{S}} \mid X_{u}=\delta_{x} )}{\mathbb{E}(\#\{ i \in V_{t} : X _{t}^{i} \notin\mathcal{S}\} \mid X_{u}=\delta_{x} )}= \mathbb{E}\bigl(f\bigl(Y_{v}^{(t)}\bigr) \bigm| Y_{u}^{(t)}=x\bigr), $$

where $X_{v}^{i}$ is the trait of the ancestor of $i$ at time $v$ and $Y^{(t)}$ is the inhomogenous Markov process associated to $P^{(t)}$. Thus, $Y^{(t)}$ is the process describing the dynamics of the trait of a typical individual, which is alive at time $t$ and non-absorbed. Proving that it is ergodic ensures the ergodicity of $\delta_{x} M _{s,t}{\mathbf{1}}/m_{s,t}(x)$ as $t$ goes to infinity. In this paper, we make a coupling for that, with Doeblin conditions which ensure exponential uniform ergodicity. Thanks to [12], this Doeblin condition can be rewritten in terms of coupling constants on the original semigroup $M$.

In homogeneous-time setting, two particular classes of processes have attracted lots of attention. First, if we make $\mathcal{S}=\varnothing$, then $X$ is a branching process and

$$\delta_{x} M_{s,t}(f)=\mathbb{E} \biggl( \sum_{i\in V_{t}} f(X_{t}^{i}) \Bigm| X_{s}=\delta_{x} \biggr) $$

is its first moment semigroup which provides the mean number of individuals with a given trait. The auxiliary process describes the dynamical of the trait along the ancestral lineage of an individual chosen uniformly at random, when the population is becoming large. More generally, the genealogical tree of the population can be constructed from this typical lineage, which is called spine construction.

Second, if the individuals neither die nor give birth, we get a Markov process in the space trait $\mathcal{X}$ and

$$\delta_{x} M_{s,t}(f)=\mathbb{E}\bigl(f(X_{t})1_{X_{t}\notin\mathcal{S}} \bigm| X_{s}=x\bigr) $$

Assume that $X_{t}$ is eventually absorbed as $t$ goes to infinity a.s. and consider the distribution of the process conditioned on non-absorption:

$$\mathbb{P}_{x}(X_{t} \in. \mid X_{t} \notin\mathcal{S})=\frac{ \delta_{x} M_{0,t}}{m_{0,t}(x)}=\delta_{x}P^{(t)}_{0,t}. $$

The ergodic behavior of $P^{(t)}$ and its convergence to a distribution $\nu$ yields the convergence of the conditioned distribution (Yaglom limit) to the quasistationary distribution. At fixed time $t$, $P^{(t)}$ describes the dynamic of the trait for trajectories non-absorbed at time $t$.

Appendix B: Measure Solutions to the Renewal Equation

We give here the details about the construction of the homogeneous renewal semigroup. It is based on the dual renewal equation

$$ \partial_{t} f_{t}(a)- \partial_{a} f_{t}(a)+b(a)f_{t}(a)=2b(a)f _{t}(0), \quad t,a\geq0. $$

(B.1)

Integrating this equation along the characteristics, we obtain the mild formulation (also called Duhamel formula)

$$ f_{t}(a)=f_{0}(a+t) \mathrm{e}^{-\int_{0}^{t}b(a+\tau)\,d\tau}+2 \int_{0}^{t}\mathrm{e}^{-\int_{0}^{\tau}b(a+\tau')\,d\tau'}b(a+ \tau)f _{t-\tau}(0)\,d\tau. $$

(B.2)

The first result is about the well-posedness of this equation in $\mathcal{B}_{b}(\mathbb{R}_{+})$.

Lemma B.1

For all$f_{0}\in\mathcal{B}_{b}(\mathbb{R}_{+})$there exists a unique family$(f_{t})_{t\geq0}\subset\mathcal{B}_{b}(\mathbb{R}_{+})$solution to (B.2). Additionally if$f_{0}\geq0$then$f_{t}\geq0$for all$t\geq0$.

Proof

First we use the Banach fixed point theorem on a truncated problem. For $T>0$ and $f_{0}\in\mathcal{B}_{b}(\mathbb{R}_{+})$ we define the operator $\varGamma:\mathcal{B}_{b}([0,T])\to\mathcal{B}_{b}([0,T])$ by

$$\varGamma g(t)= f_{0}(t)\mathrm{e}^{-\int_{0}^{t}b(\tau)\,d\tau}+2\int _{0}^{t} \varPhi(\tau)g(t-\tau)\,d\tau. $$

We easily have

$$\|\varGamma g_{1}-\varGamma g_{2}\|_{\infty}\leq2\int_{0}^{T}\varPhi (\tau) \,d\tau\,\|g_{1}-g_{2}\|_{\infty}, $$

so $\varGamma$ is a contraction if $2\int_{0}^{T}\varPhi<1$ and there is a unique fixed point in $\mathcal{B}_{b}([0,T])$. Additionally since $\varGamma$ preserves non-negativity when $f_{0}\geq0$, we get that the fixed point $g$ is non-negative when $f_{0}$ is non-negative. Since the contraction constant $2\int_{0}^{T}\varPhi$ is independent of $f_{0}$, we can iterate to obtain a unique function $g\in\mathcal{B} _{b}(\mathbb{R}_{+})$ which satisfies

$$g(t)= f_{0}(t)\mathrm{e}^{-\int_{0}^{t}b(\tau)\,d\tau}+2\int_{0}^{t} \varPhi(\tau)g(t-\tau)\,d\tau $$

for all $t\geq0$. Now we set for all $t,a\geq0$

$$f_{t}(a)=f_{0}(a+t)\mathrm{e}^{-\int_{0}^{t}b(a+\tau)\,d\tau}+2\int _{0}^{t}\mathrm{e}^{-\int_{0}^{\tau}b(a+\tau')\,d\tau'}b(a+\tau)g(t- \tau)\,d\tau, $$

which is a solution to (3.11) since by definition $f_{t}(0)=\varGamma g(t)=g(t)$. The uniqueness is a direct consequence of the uniqueness of $g$. The non-negativity follows from the non-negativity of $g$ when $f_{0}\geq0$, and the boundedness is given by the inequality

$$\|f_{t}\|_{\infty}\leq\|f_{0}\|_{\infty}+2\sup_{0\leq s\leq t}\bigl|g(s)\bigr|. $$

 □

Lemma B.1 allows to define for all $t\geq0$ the operator $M_{t}$ on $\mathcal{B}_{b}(\mathbb{R}_{+})$ by setting $M_{t}f_{0}:=f_{t}$ for all $f_{0}\in\mathcal{B}_{b}(\mathbb{R}_{+})$. Then for $\mu\in\mathcal{M}_{+}(\mathbb{R}_{+})$ we define the positive measure $\mu M_{t}$ by setting for all Borel set $A\subset \mathbb{R}_{+}$

$$(\mu M_{t})(A):=\mu(M_{t}\mathbf{1}_{A}). $$

The axioms of a positive measure are satisfied. First it is clear that $(\mu M_{t})(\varnothing)=0$ and that $(\mu M_{t})(A\cup B)=(\mu M _{t})(A)+(\mu M_{t})(B)$ when $A$ and $B$ are two disjoint Borel sets. The last axiom deserves a bit more attention. Let $(A_{n})_{n\geq0}$ be an increasing sequence of Borel sets and $A=\bigcup_{n\geq0}A_{n}$. We want to check that $(\mu M_{t})(A)=\lim_{n\to\infty}(\mu M_{t})(A _{n})$. The sequence $(\mathbf{1}_{A_{n}})_{n\geq0}$ is an increasing sequence of Borel functions which converges pointwise to $\mathbf{1} _{A}$. By positivity of the semigroup, $(M_{t}\mathbf{1}_{A_{n}})_{n \geq0}$ is an increasing sequence of Borel functions bounded by $M_{t}\mathbf{1}$. Thus this sequence admits a pointwise limit $f_{t}\in\mathcal{B}_{b}(\mathbb{R}_{+})$ which clearly satisfies the Duhamel formula (3.11) with $f_{0}=\mathbf{1}_{A}$. By uniqueness of the solution to the Duhamel formula we get that $M_{t}\mathbf{1}_{A_{n}}\to M_{t}\mathbf{1}_{A}$ pointwise. Then by dominated or monotone convergence we deduce that $(\mu M_{t})(A)= \mu(M_{t}\mathbf{1}_{A})=\lim_{n\to\infty}\mu(M_{t}\mathbf{1}_{A _{n}})=\lim_{n\to\infty}(\mu M_{t})(A_{n})$. Finally for a signed measure $\mu\in\mathcal{M}(\mathbb{R}_{+})$ we set obviously $\mu M_{t}:=\mu_{+} M_{t}-\mu_{-} M_{t}$.

The family $(M_{t})_{t\geq0}$ such defined is a semigroup which satisfies Assumption 2.1. The semigroup property is a consequence of the uniqueness of the solution to the Duhamel formula (3.11). The positivity has been proved in Lemma B.1. For the strong positivity it follows from the Duhamel formula (3.11) that for all $t,a\geq0$

$$m_{t}(a)\geq\mathrm{e}^{-\int_{0}^{t} b(a+\tau)\,d\tau}>0. $$

The compatibility $(\mu M_{t})(f)=\mu(M_{t}f)$ follows directly from the definition of $\mu M_{t}$ and the definition of Borel functions.

It is claimed in Sect. 3.2.2 that the family $(\mu M_{t})_{t\geq0}$ is a measure solution to the renewal equation. Measure valued solutions to structured population models drew interest in the last few years [9, 10, 24, 28, 29]. They are mainly motivated by biological applications which often require to consider initial distributions which are not densities but measures (Dirac masses for instance). For us it is additionally the suitable framework to apply our ergodic result in Theorem 3.5. We refer to [24] for the proof that the family $(\mu M_{t})_{t \geq0}$ is a measure valued solution to Eq. (3.10) for any $\mu\in\mathcal{M}(\mathbb{R}_{+})$. Here we only give a heuristic argument which consists in differentiating the semigroup property $\mu M_{t} f=\mu M_{s} M_{t-s}f$ with respect to $s\in[0,t]$. The chain rule gives

$$\partial_{s}(\mu M_{s})M_{t-s}f+\mu M_{s}\partial_{s}(M_{t-s}f)=0 $$

and since $M_{t}f$ is a solution to the dual Eq. (B.1) this gives

$$\partial_{s}(\mu M_{s})M_{t-s}f-\mu M_{s}\mathcal{A}(M_{t-s}f)=0 $$

where $\mathcal{A}$ is the unbounded operator defined on $C^{1}( \mathbb{R}_{+})$ by $\mathcal{A}f(a)=f'(a)-b(a)f(a)+2b(a)f(0)$. Taking $s=t$ we get that for all bounded and continuously differentiable function $f$

$$\partial_{t}(\mu M_{t}f)=\mu M_{t}(\mathcal{A}f), $$

which is a weak formulation of Eq. (3.10).

Appendix C: The Max-Age Semigroup

As for the homogeneous renewal equation, to build a solution to Eq. (3.17) we use a duality approach. We start with the (backward) dual equation

$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \partial_{s} f_{s,t}(a)+\partial_{a}f_{s,t}(a)+b(a)f_{s,t}(0)=0, & s< t,\ 0\leq a< a_{s}, \\ f_{s,t}(a_{s})=0,& s< t, \\ f_{t,t}(a)=f_{t}(a),& 0\leq a< a_{t}. \end{array}\displaystyle \right . $$

(C.1)

Integrating this equation along the characteristics we get the Duhamel formula

$$ f_{s,t}(a)=f_{t}(a+t-s)+ \int_{s}^{t} b_{\tau}(a+\tau -s)f_{\tau,t}(0) \,d\tau $$

(C.2)

where we have denoted $b_{t}(a):=b(a)\mathbf{1}_{[0,a_{t})}(a)$ and $f_{t}$ has been extended by 0 beyond $a_{t}$.

Lemma C.1

For all$t>0$, $f_{t}\in\mathcal{B}_{b} ([0,a_{t}) )$, and$s\in[0,t]$, there exists a unique$f_{s,t}\in\mathcal{B}_{b} ([0,a _{s}) )$which satisfies (C.2). Additionally if$f_{t}\geq0$then$f_{s,t}\geq0$.

We do not repeat the proof of this result since it follows exactly the strategy of the proof of Lemma B.1. As for the homogeneous renewal equation we define the semigroup $(M_{s,t})_{0 \leq s\leq t}$ on $(\mathcal{X}_{t})_{t\geq0}= ([0,a_{t}) )_{t \geq0}$, first on the right hand side by setting for all $f_{t} \in\mathcal{B}_{b}([0,a_{t}))$

$$M_{s,t}f_{t}:=f_{s,t} $$

where $f_{s,t}$ is the unique solution to Eq. (C.2), and then on the left by setting for all $\mu\in\mathcal{M}([0,a_{s}))$ and all Borel set $A\subset [0,a_{t})$

$$(\mu M_{s,t})(A)=\mu_{+}(M_{s,t}\mathbf{1}_{A})-\mu_{-}(M_{s,t} \mathbf{1}_{A}). $$

For any $\mu\in\mathcal{M} ([0,a_{s}) )$ the family $(\mu M_{s,t})_{s\leq t}$ is a measure solution to Eq. (3.17). As for the homogeneous case a non rigorous justification is obtained by differentiating the semigroup property $\mu M_{s,t}f=\mu M_{s,r}M_{r,t}f$ with respect to $r\in[s,t]$ and using that $M_{r,t}f$ is solution to (C.1).

The semigroup property for the family $(M_{s,t})_{t\geq s\geq0}$ is a consequence of the uniqueness of the solution to the Duhamel formula (C.2). We now verify Assumption 2.1. The positivity has been proved in Lemma C.1. For the strong positivity it suffices to check that $m_{s,t}(0)>0$. Indeed if $m_{s,t}(0)>0$ for all $0\leq s\leq t$ the Duhamel formula ensures that for $a< a_{s}$

$$m_{s,t}(a)=\mathbf{1}_{a+t-s< a_{t}}+\int_{s}^{t} b_{\tau}(a+ \tau-s)m_{\tau,t}(0)\,d\tau\geq\underline{b} \int_{s}^{t} \mathbf{1}_{a+\tau-s< a_{\tau}}\,m_{\tau,t}(0)\,d\tau>0. $$

The positivity of $m_{s,t}(0)$ is clear if $t-s< a_{t}$ since

$$m_{s,t}(0)\geq\mathbf{1}_{t-s< a_{t}}. $$

Consider now the case $t-s\geq a_{t}$. The function $r\mapsto m_{r,t}(0)$ is continuous on $[s,t-a_{t}]$ since for $r\leq t-a_{t}$ we have

$$m_{r,t}(0)=\int_{r}^{t} b_{\tau}(\tau-r)m_{\tau,t}(0)\,d\tau. $$

Assume by contradiction that there exists $r_{0}\in[s,t-a_{t}]$ such that $m_{r_{0},t}(0)=0$ and $m_{r,t}(0)> 0$ for all $r\in(r_{0},t]$. Then the equality above would give for $r=r_{0}$

$$0=\int_{r_{0}}^{t} b_{\tau}(\tau-r_{0})m_{\tau,t}(0)\,d\tau, $$

which is not possible since the integrand on the right hand side is positive for $\tau$ close to $r_{0}$. Finally the compatibility condition readily follows from the definition of $\mu M_{s,t}$.