On convergence and asymptotic behaviour of semigroups of operators

Abstract

The classical and modern theorems on convergence, approximation and asymptotic stability of semigroups of operators are presented, and their applications to recent biological models are discussed.

This article is part of the theme issue ‘Semigroup applications everywhere’.

1. Introduction

All semigroupies know that neither the original Einar Hille’s ‘Functional Analysis and Semi-groups’ nor its second edition [1] coauthored by Ralph Phillips contain the, now celebrated, convergence theorem for semigroups. But did you know that the latter book does contain that theorem in a germ form, or at least its foretaste? Also, do you know that today convergence theorems can be proved in a much simpler fashion than that suggested by Trotter, Neveu and Kato? Do you know that the classical theorems by nature cannot be applied to situations like singular perturbations, abounding in mathematical biology and other applied fields, and that today there are efficient semigroup-theoretic tools for handling such perturbations? Finally, do you know how nowadays theorems on asymptotic behaviour of semigroups are related and stem from the classical Perron–Frobenius theorem? Do you know of the associated, rich theory of asymptotic behaviour of stochastic semigroups and its recent applications in biology? If the answer to at least one of these questions is negative, this text is for you.

2. Prehistory of convergence theorems

(a). The approximation theorem

Let’s start from the basics. Suppose we are given a family of semigroups {T_r(t), t ≥ 0}, r ∈ (0, 1) in a Banach space $\mathbb{X}$, satisfying the all-important stability condition

$$||T_r(t)||\le M{\text{e}}^{\omega t},$$ 2.1

for some given constants M ≥ 1 and $\omega \in \mathbb{R}.$ The Trotter–Neveu–Kato Approximation Theorem states these semigroups converge to another semigroup, say {T(t), t ≥ 0}, iff their resolvents converge to the resolvent of this semigroup. More precisely

$$\underset{r\to 1-}{lim}{T}_r(t)x=T(t)x,t\ge 0,\hspace{0.17em}x\in \mathbb{X},$$ 2.2

iff

$$\underset{r\to 1-}{lim}{(\lambda -A_r)}^{-1}x={(\lambda -A)}^{-1}x,\lambda >\omega ,\hspace{0.17em}x\in \mathbb{X};$$ 2.3

where A_r is the generator of {T_r (t), t ≥ 0} and A is the generator of {T(t), t ≥ 0}. Recall that the resolvent of the generator is the Laplace transform of the semigroup; because of (2.1) we have

$${(\lambda -A_r)}^{-1}x={\int }_0^{\mathrm{\infty }}{\text{e}}^{-\lambda t}{T}_r(t)x\hspace{0.17em}\text{d}t,\lambda >\omega ,x\in \mathbb{X}.$$

In particular, the implication (2.2) ↠ (2.3) is immediate by the Lebesgue Dominated Convergence Theorem and the real value of the approximation result lies in the fact that it allows deducing convergence of semigroups from the convergence of their Laplace transforms. This is particularly important because the operators forming the semigroups are rarely given in a manageable form but in many cases something substantial can be said about the resolvents. (By contrast, unless extra conditions are met, convergence of Laplace transforms of a sequence of vector-valued functions does not imply point-wise convergence of these functions, see e.g. [2] p. 42, [3] pp. 319–320, or [4] p. 13 and pp. 47–48.)

Three remarks are here in order. The first is that to prove (2.2) it suffices to show (2.3) for just one λ > ω; this is handy, as we seem to remember curious situations where proving (2.3) for all λ > ω was next to impossible because one λ > ω was particularly uncooperative, despite the fact that, as we already said, convergence of the resolvents for just one λ > ω is known to imply this convergence for all λ > ω.

The second remark is that in fact each semigroup {T_r (t), t ≥ 0} may be defined in a different Banach space, say ${\mathbb{X}}_r$; the theorem remains in force provided that ${\mathbb{X}}_r$ in a sense converge, as r → 1, to the space $\mathbb{X}$ where {T(t), t ≥ 0} is defined, with technical changes related to the way (2.2) and (2.3) are then understood. This modification is important from the viewpoint of stochastic processes, and has been devised so that approximation theorems may cover the case of diffusion approximations (H.F. Trotter’s paper [5] was based on his PhD dissertation, and his PhD advisor was William Feller). For reasons that will become clearer in the following sections, we will, however, restrict ourselves to the case where all semigroups act in a single space.

The third and final remark is that convergence in (2.2) is in fact, by nature, uniform for t in bounded subintervals of ${\mathbb{R}}^{+}$: you may prove it for all t ≥ 0 separately or, as is usually easier, prove (2.3) for just one λ > ω, and the convergence in (2.2) will turn out to be uniform anyway. This result initially seems to be a bonus, and to some extent it is, but later this will be shown to be the biggest shortcoming of the discussed theory: the Trottter–Neveu–Kato Theorem cannot describe a convergence that is not uniform for t in bounded subintervals of ${\mathbb{R}}^{+}$!

(b). The convergence theorem

The proposition discussed in the preceding section has a version applicable to the case where the limit semigroup is not known, and even its existence cannot be a priori granted. The Trotter–Neveu–Kato Convergence Theorem says that if

• (a)the semigroups {T_r(t), t ≥ 0}, r ∈ (0, 1) satisfy the stability condition (2.1),
• (b)
  the limit

  $$R_{\lambda }x:=\underset{r\to 1-}{lim}{(\lambda -A_r)}^{-1}x,x\in \mathbb{X},$$ 2.4

  exists for one (and hence for all) λ > ω, and
• (c)the operator R_λ has dense range,

then there is a semigroup {T(t), t ≥ 0} in $\mathbb{X}$ such that (2.2) holds; the limit is uniform, as before, and R_λ, λ > ω is the resolvent of the limit semigroup generator.

Both propositions have been proved independently by H.F. Trotter [5] and J. Neveu [6] (in a special case). The name of T. Kato is inseparably linked to these propositions because of his two-page note [7]: H.F. Trotter in his proof of the convergence theorem is apparently a bit too quick to conclude that the limit pseudo-resolvent R_λ, λ > ω is in fact the resolvent of a certain operator (to be shown to be the generator of the limit semigroup). Corollary 1 of Kato’s [7] may be used to justify Trotter’s proof.

(c). Kato’s Theorem: a foretaste of the glory to come

Both Trotter–Neveu–Kato Theorems seem natural today; we have become accustomed to the fact that statements on semigroups can be translated into statements on resolvents and vice versa. But this awareness has been growing with the theory of semigroups, and simply was not there in 1957 when [1] was published (or at least was not there in full), for if it were, what we call now the Trotter–Neveu–Kato Theorems would had been easily proved much earlier by luminaries such as Hille, Phillips or Yosida. We venture to believe that before 1958 it was not clear whether a path to convergence theorems leading through resolvents is a natural one; it was even less clear how to prove that (2.3) implies (2.2).

To see this, let us come back to T. Kato’s 1954 paper [8], reproduced nearly in extenso in [1]. Its starting point is an intensity matrix indexed by natural numbers, i.e. a matrix (q_i,j)_i,j≥1 such that $\sum _{j\ge 1}{q}_{i,j}=0,\hspace{0.17em}i\ge 1$, and q_i,j ≥ 0 for i ≠ j, and the aim is to prove a generation theorem in the space ℓ¹ of absolutely summable sequences (ξ_i)_i≥1 (with standard norm $||{({\xi }_i)}_{i\ge 1}||=\sum _{i\ge 1}|{\xi }_i|$). Because of the previous work of W. Feller and in light of certain probabilistic intuitions, Kato knows that the maximal operator Q related to (q_i,j)_i,j≥1, given by

$$Q{({\xi }_i)}_{i\ge 1}:={\left(\sum _{j\ge 1}{\xi }_j{q}_{j,i}\right)}_{i\ge 1},$$

on the maximal domain where all the series $\sum _{j\ge 1}{\xi }_j{q}_{j,i}$ converge absolutely and the sequence ${\left(\sum _{j\ge 1}{\xi }_j{q}_{j,i}\right)}_{i\ge 1}$ is a member of ℓ¹, in general is ‘too large’ to generate a semigroup in ℓ¹. He also believes that a restriction of Q should be a generator.

To prove his claim he devises the following ingenious approximation procedure. He considers the operator D (‘D’ for ‘diagonal’) with domain $\mathcal{D}(D)$ composed of (ξ_i)_i≥1 such that (q_i,iξ_i)_i≥1 ∈ ℓ¹, given by D (ξ_i)_i≥1 = (q_i,iξ_i)_i≥1, and the related off-diagonal operator $O{({\xi }_i)}_{i\ge 1}={\left(\sum _{j\ne i}{\xi }_j{q}_{j,i}\right)}_{i\ge 1}$ on the same domain. He also notes that, because of the assumption that (q_i,j)_i,j≥1 is an intensity matrix, $||Ox||\le ||Dx||$ for $x\in \mathcal{D}(D)$ with equality for non-negative x ∈ ℓ¹. These properties allow proving, with the help of the Hille–Yosida Theorem, that all the operators

$$A_r:=D+rO\text{with}\hspace{0.17em}\mathcal{D}(A_r)=\mathcal{D}(D),$$ 2.5

indexed by r ∈ [0, 1), are generators of (positive) contraction semigroups in ℓ¹ (i.e. that the semigroups generated by A_rs satisfy (2.1) with M = 1 and ω = 0). Moreover, there are handy formulae for the resolvents of A_rs which make it clear that (λ − A_r)⁻¹s increase with r; because of the special lattice structure of ℓ¹, these facts imply that lim_r→1− (λ − A_r)⁻¹ exists.

Perhaps surprisingly to twenty-first-century readers, Kato does not deduce convergence of the semigroups from the established convergence of resolvents (which is easy, if the Trotter–Neveu–Kato Convergence Theorem is available, see [9] Section 8.4.16, [4] Ch. 13, or [10] Section 3.1.4). Instead, he argues that since the resolvents increase with r, so do the semigroups, and establishes convergence of the latter somewhat independently. Consequently, he needs to worry about additional issues which today are easily resolved with the help of the Trotter–Neveu–Kato Theorem. He might have been aware of the perils awaiting those who want to deduce convergence of semigroups from convergence of resolvents. Is this the reason why he so quickly spotted the gap in Trotter’s reasoning?

To complete the story of [8]: Kato successfully completes his reasoning by proving that the generator A of the limiting semigroup is a restriction of Q (and an extension of D + O), and that in fact the semigroup generated by A is in a sense minimal. Notably, even if the implication: convergence of resolvents ↠ convergence of the semigroups is not used here directly, we are in fact facing one of the predecessors of the Trotter–Neveu–Kato Theorem.

3. Kisyński’s proof; Sova and Kurtz’s approach

For the presentation of the results in this section, it will be convenient, instead of considering families of semigroups of operators, to consider their sequences: we start with the semigroups {T_n(t), t ≥ 0}, n ≥ 1 in a Banach space $\mathbb{X}$, such that, as in (2.1), $||T_n(t)||\le M{\text{e}}^{\omega t}$; the generator of the nth semigroup is A_n. Clearly, both Trotter–Neveu–Kato Theorems are still valid in this case, with minor, obvious changes, such as replacing lim_r→1− by lim_n→∞.

Trotter’s proof of the convergence theorem is long and involved; in 1967 a truly beautiful argument by Jan Kisyński was published [11]. The idea is to work in the space $c(\mathbb{X})$ of convergent sequences with values in $\mathbb{X}$, and to deduce the convergence theorem from the Hille–Yosida Theorem. To this end, Kisyński considers the following operator $\mathcal{A}$ in $c(\mathbb{X})$

$$\mathcal{A}{(x_n)}_{n\ge 1}:={(A_n{x}_n)}_{n\ge 1},$$ 3.1

on the domain where each x_n belongs to $\mathcal{D}(A_n)$ and (A_nx_n)_n≥1 is a member of $c(\mathbb{X}).$ Assumption (b) of the convergence theorem allows introducing operators ${\mathcal{R}}_{\lambda },$ λ > ω in $c(\mathbb{X})$ as follows: ${\mathcal{R}}_{\lambda }{(x_n)}_{n\ge 1}:={({(\lambda -A_n)}^{-1}{x}_n)}_{n\ge 1}$, and condition (c) allows concluding that the (common) range of these operators is a dense subspace of $c(\mathbb{X}).$ It is then almost immediate that ${\mathcal{R}}_{\lambda }$ is the left and right inverse of $\lambda -\mathcal{A},$ λ > ω (and thus in particular, that $\mathcal{A}$ is densely defined) and that the (uniform) estimates for the resolvents of A_ns translate to the estimates for ${\mathcal{R}}_{\lambda }.$ This shows that $\mathcal{A}$ is the generator of a semigroup, say $\{\mathcal{T}(t),t\ge 0\}$ in $c(\mathbb{X})$. Then, uniqueness of the Laplace transform leads to the conclusion that

$$\mathcal{T}(t){(x_n)}_{n\ge 1}={(T_n(t)x_n)}_{n\ge 1},t\ge 0,\hspace{0.17em}{(x_n)}_{n\ge 1}\in c(\mathbb{X}).$$

In particular, since the operators $\mathcal{T}(t)$ map $c(\mathbb{X})$ into itself, taking a constant sequence (x)_n≥1, $x\in \mathbb{X}$, we obtain the Trotter–Neveu–Kato Theorem.

Kisyński’s method is much more than a mere trick. First of all it shows that convergence theorems lie in the very centre of the theory of semigroups of operators: on the one hand, the Hille–Yosida Theorem can be obtained from the Trotter–Neveu–Kato Theorem via the Yosida Approximation; on the other, the Trotter–Neveu–Kato Theorem is just the Hille–Yosida Theorem in disguise. In many cases (see for instance Theorem (2.1) in [12] or Corollary 8.4 in [4]) the approximation theorem plays the role of a generation theorem; in a very real sense it is a generation theorem.

Secondly, Kisyński’s paper reveals that the operator $\mathcal{A}$ is a central object in the analysis of convergence theorems (see also below). Interestingly, $\mathcal{A}$ has its counterpart A in the space $\mathbb{X}$; an $x\in \mathbb{X}$ belongs to the domain $\mathcal{D}(A)$ of A if there are $x_n\in \mathcal{D}(A_n)$ such that lim_n→∞ x_n = x and lim_n→∞ A_n x_n exists, and then we agree that Ax = lim_n→∞ A_n x_n (if A is densely defined this definition does not depend on the choice of (x_n)_n≥1). Since properties of A can be translated into properties of $\mathcal{A}$ and vice versa, the operator A, termed the extended limit of A_ns, can be used to deduce convergence theorems.

The latter idea is due to Miroslav Sova [13] (who published it in 1967, without knowing Kisyński’s paper or working in $c(\mathbb{X})$) and Thomas G. Kurtz [12,14], who re-discovered and substantially extended Kisyński’s approach via the operator $\mathcal{A}.$ The Sova–Kurtz version of the convergence theorem is expressed in terms of the extended limit A and says that if

• (a)the semigroups generated by A_ns satisfy the stability condition,
• (b)for some, and hence all λ > ω, the range of λ − A is dense in $\mathbb{X}$ (this gives convergence of resolvents), and
• (c)$\mathcal{D}(A)$ is dense in $\mathbb{X}$ (this gives density of the range of the limit pseudo-resolvent),

then the semigroups converge and the limit semigroup is generated by A.

This version of the convergence theorem has a clear advantage over the original one. It does not require knowing the resolvents (λ − A_n)⁻¹ in a manageable form, and the latter is often not available (think, for example, of A_ns as being proven to be generators by an intricate perturbation argument). This advantage cannot be overestimated (for a wide spectrum of applications, see e.g. [4,10,15]). In fact, in the light of this version, the approach via resolvents turns out to be not so natural, and indeed much less useful than we have become accustomed to think. In particular, the approach via resolvents hides a number of curious phenomena visible only when looked upon from the perspective of the extended limit. For instance, see [10] Section 3.4, even though all the operators A_r of (2.5) are defined on the common domain $\mathcal{D}(D)$, the domain of their extended limit, i.e. the domain of the generator of the minimal semigroup of Kato’s Theorem, may be larger than $\mathcal{D}(D)$ or even larger than the domain of the closure of D + O.

The following example, which we owe to Jochen Glück, exhibits a similar effect, and may be appreciated by the readers who find the notion of extended limit counterintuitive or unnatural. Let (e_i)_i≥1 be an orthonormal basis in a Hilbert space $\mathbb{H}$, so that any $x\in \mathbb{H}$ may be represented as $x=\sum _{i=1}^{\mathrm{\infty }}{\xi }_i{e}_i$ where ξ_i = (x, e_i), i ≥ 1. For each n ≥ 1, the formula

$$T_n(t)\left(\sum _{i=1}^{\mathrm{\infty }}{\xi }_i{e}_i\right)=\sum _{i=1}^n{\xi }_i{e}_i+{\text{e}}^{-nt}\sum _{i=n+1}^{\mathrm{\infty }}{\xi }_i{e}_i,t\ge 0,$$

defines a strongly continuous semigroup {T_n(t), t ≥ 0} in $\mathbb{H}$, generated by the bounded operator A_n given by

$$A_n\left(\sum _{i=1}^{\mathrm{\infty }}{\xi }_i{e}_i\right)=-n\sum _{i=n+1}^{\mathrm{\infty }}{\xi }_i{e}_i.$$

Clearly, $\underset{n\to \mathrm{\infty }}{lim}{T}_n(t)x=x,\hspace{0.17em}x\in \mathbb{H}$, i.e. the semigroups converge to the semigroup generated by the zero operator (and the limit is even uniform in t ≥ 0). However, the operators A_n, n ≥ 1 cannot converge strongly, since $||A_n||=n$, and in particular do not converge to the zero operator. On the other hand, for any x ∈ H, defining $x_n=\sum _{i=1}^n{\xi }_i{e}_i,n\ge 1,$ we have lim_n→∞ x_n = x and lim_n→∞ A_nx_n = 0. This confirms the fact that the extended limit, and not the ‘normal’ limit, of A_ns is the limit semigroup generator.

4. A version with the limit semigroup defined on a subspace

In this section, for simplicity of notations and exposition, we assume that the semigroups generated by A_ns are equibounded: $||T_n(t)||\le M$ for some M ≥ 1. The general case may be easily reconstructed by standard techniques.

As often as not, the range condition (c) of the Trotter–Never–Kato Theorem (or its Sova–Kurtz version) fails to be satisfied. What happens in this case is clearly seen from the perspective of the operator $\mathcal{A}$ of (3.1): the entire Kisyński’s analysis works except for the argument that $\mathcal{A}$ is densely defined. Thus, $\mathcal{A}$ is, in today’s terminology, a Hille–Yosida operator in that it satisfies all the assumptions of the Hille–Yosida Theorem except for being densely defined. The theory originating from the seminal paper by Wolfgang Arendt [16] and culminating in the treatise [2] reveals that such operators are generators of so-called Lipschitz continuous integrated semigroups, i.e. families {U(t), t ≥ 0} satisfying a certain functional equation and such that $||U(t)-U(s)||\le M|t-s|$ (with the constant M coming from equiboundedness condition). The integrated semigroup generated by $\mathcal{A}$ has the following explicit form:

$$\mathcal{U}(t){(x_n)}_{n\ge 1}={({\int }_0^t{T}_n(s)x_n\hspace{0.17em}\text{d}s)}_{n\ge 1},t\ge 0,{(x_n)}_{n\ge 1}\in c(\mathbb{X}).$$

It follows that for each $x\in \mathbb{X}$ and t ≥ 0, the limit $\underset{n\to \mathrm{\infty }}{lim}{\int }_0^t{T}_n(s)x\hspace{0.17em}\text{d}s$ exists; this was observed by Kurtz [14], long before the rise of the theory of integrated semigroups. As noted by Kisyński [17], this result may be extended in the following way: from the resolvent of the generator of a Lipschitz continuous integrated semigroup one can build a representation of the convolution algebra $L^1({\mathbb{R}}^{+})$ by means of operators in the space where this semigroup is defined, and therefore, there exist the limits

$$H(\varphi )x:=\underset{n\to \mathrm{\infty }}{lim}{\int }_0^{\mathrm{\infty }}\varphi (t)T_n(t)x\hspace{0.17em}\text{d}s,\varphi \in L^1({\mathbb{R}}^{+}),x\in (\mathbb{X}).$$

(Note that for ϕ(s) = e^−λs, s ≥ 0 this formula recovers existence of lim_n→∞ (λ − A_n)⁻¹x.)

An example as simple as A_n : = n A₁, n ≥ 1 where A₁ is the generator of the right shift semigroup in the space $\mathbb{X}:=C_0(0,\mathrm{\infty })$ of continuous functions on (0, ∞) which vanish at both 0 and ∞, shows that these results, as close as they seem to be to asserting convergence of semigroups, cannot be much improved. In this example, the resolvents converge to zero, but the semigroups do not except at x = 0.

It turns out that, in the general case, the subspace, say ${\mathbb{X}}_0$, of $x\in \mathbb{X}$ such that lim_n→∞ T_n(t) x exists and is uniform for t in compact subsets of ${\mathbb{R}}^{+}$ can be characterized by means of the limit pseudo-resolvent R_λ, λ > 0 and/or the extended limit operator A:

$${\mathbb{X}}_0=\overline{\text{Range}{R}_{\lambda }}=\overline{\mathcal{D}(A)},$$ 4.1

where the bar denotes the closure; in ${\mathbb{X}}_0$, the formula T(t)x = lim_n→∞ T_n(t) x defines a limit semigroup. This is a reflection of the fact that each Hille–Yosida operator ‘produces’ also a semigroup, but this semigroup is defined only on the subspace equal to the closure of this operator’s domain. In the case of $\mathcal{A}$, this is the subspace of those sequences which converge to a member of ${\mathbb{X}}_0$.

5. Singular perturbations; Kurtz’s Theorem

In this section, for the sake of applications we will present in §6, we come back to the notations of §2: the operators A_r are generators of semigroups satisfying the stability condition (2.1).

We have established that if (c) in the convergence theorem fails to hold (but the remaining two are satisfied), the limit lim_r→1− T_r (t) x exists and is uniform for t in compact subsets of ${\mathbb{R}}^{+}$ only for x in the subspace ${\mathbb{X}}_0$ of the original Banach space. Outside of ${\mathbb{X}}_0$, the limit usually does not exist.

In the theory of singular perturbations it often happens, though, that the limit exists even if $x\notin {\mathbb{X}}_0$; of course, this limit must then not be uniform as described above. It may be shown that in such a case it is uniform for t in compact subsets of the open right half-axis (0, ∞). For singular perturbation theorists, the reason for that is clear: there are initial (and possibly other) layers involved here, and the convergence simply cannot be uniform around t = 0.

In fact, quite often singular perturbation theorems have the form

$$\underset{r\to 1-}{lim}{T}_r(t)x=T(t)Px,t>0,x\in \mathbb{X},$$ 5.1

where {T(t), t ≥ 0} is a strongly continuous semigroup on a subspace ${\mathbb{X}}_0$ of $\mathbb{X}$ and P is a projection onto ${\mathbb{X}}_0$ (in the sense that P² = P and $Px=x,x\in {\mathbb{X}}_0$). To repeat, in this case, the classical theory does not work and, in particular, condition

$$\underset{r\to 1-}{lim}{(\lambda -A_r)}^{-1}x={(\lambda -B)}^{-1}Px,x\in \mathbb{X},$$ 5.2

(where B is the generator of {T(t), t ≥ 0}) for all (some) λ > 0 is necessary but not sufficient for (5.1).

Condition (5.2) may imply (5.1) provided that the semigroups involved possess additional regularity properties (like uniform analyticity; see e.g. [4], Ch. 31 and 41 for details; the latter condition can be for example efficiently checked if we are working in a Hilbert space and the semigroups are related to forms, see [18]). A different set of conditions guaranteeing that (5.2) implies (5.1) has been devised by Kurtz [15,19,20]. Kurtz’s singular convergence theorem is usually expressed in terms of the extended limit of generators; the following (simplified) resolvent-version may be easily deduced, e.g. from combined Lemma 7.1 and Theorem 42.2 in [4].

Theorem 5.1. —

Suppose that for some λ > 0

$$\underset{r\to 1-}{lim}{(\lambda -(1-r)A_r)}^{-1}x={(\lambda -A_0)}^{-1}x,t\ge 0,x\in \mathbb{X},$$

where A₀ is the generator of a strongly continuous semigroup {S(t), t ≥ 0} such that

$$Px:=\underset{t\to \mathrm{\infty }}{lim}S(t)x,x\in \mathbb{X},$$

exists. Then condition (5.2) (for some λ > 0, with the same P) implies (5.1), and the limit is uniform for t in compact subsets of (0, ∞); for $x\in {\mathbb{X}}_0$ the limit is uniform for t in compact subsets of [0, ∞).

6. Thin layer approximation in modelling of signalling pathways

Both the Trotter and Neveu papers were written with applications to stochastic processes in mind. But the full impact of the approximation theory of semigroups of operators on Markov processes, and its intrinsic beauty, could only be measured after the monograph [15] by Ethier and Kurtz was published; this monograph was to a large degree based on T. G. Kurtz’s extensions and refinements of the existing results discussed partly above.

Applications of the theory to other branches of mathematics, including mathematical biology, are also numerous, featuring scenarios as apparently distant as modelling fish populations and studying mutations and recombinations on and between DNA strands. A glimpse of this vast field is offered by e.g. [4,21]. Here we would like to discuss briefly one yet more recent development (see [22] for a discussion of the relation of the results presented below to the seminal paper by Hale & Raugel [23] and the entire resulting, growing literature on thin layer approximation).

To give a bit of biological background: the so-called B cells (or B lymphocytes) are key cells of adaptive immune response which, when activated, secrete highly specific antibodies, which in turn bind viruses and restrict their spread. B cells’ activation is initiated by B cell receptors aggregation that follows antigen engagement. Antigen-bound B cell receptors can activate specific signal-transmitting protein molecules in the B cell, called kinases. Kinases, activated on the membrane, diffuse in cytoplasm of the B cell, where they are inactivated by other enzymes called phosphatases. Characteristics of this diffusion process change with geometry, and the effect is particularly visible in the case of B cells, which are roughly spherical and their distinguishing feature is that their nucleus is of a radius more than 0.9 of the cell radius. As a result, cytoplasmic kinases remain in the close vicinity of the membrane.

In Tomasz Lipniacki group’s paper [24], B cell cytoplasm is modelled as a shell between two concentric spheres, one representing cell membrane with the unit radius and the other representing nuclear membrane with radius r ∈ (0, 1). All kinases, whether active or inactive, are diffusing in this shell with the same diffusion coefficient, which implies that the total concentration (i.e. the ‘population density’) of active and inactive kinases remains constant over space and time, and may be assumed equal to 1. A somewhat simplified reaction–diffusion equation for active kinase concentration K has the following form:

$$\frac{\mathrm{\partial }K}{\mathrm{\partial }t}=\text{d}\mathrm{\Delta }K+f(K)t\ge 0,$$ 6.1

where f(K) : = K(1 − K) − (cK/(K + M)) for some constant c > 0, describes kinase auto-activation and their inactivation by uniformly distributed phosphatases. This equation is supplemented by Robin type boundary condition on the outer sphere (i.e. membrane, where kinases are activated) and by Neumann type condition on the inner sphere (i.e. nuclear membrane):

$$aR(1-K_{\text{out}})=\text{d}\hspace{0.17em}n{(\mathrm{\nabla }K)}_{\text{out}},n{(\mathrm{\nabla }K)}_{\text{inn}}=0.$$ 6.2

Here, a > 0 is a kinase activation coefficient, R is surface concentration of active receptors, K_out is the value of K at the outer boundary, and thus (1 − K_out) is the concentration of inactive kinases at the boundary. Furthermore, $n{(\mathrm{\nabla }K)}_{\text{out}}$ is the normal derivative at the outer boundary of radius r₀ = 1, and $n{(\mathrm{\nabla }K)}_{\text{inn}}$ is the normal derivative at the inner boundary of radius r.

Since the shell is thin, it seemed tempting to conjecture that these equations should, as r → 1, be approximated by a reaction–diffusion equation on the two-dimensional unit sphere. An issue to be resolved was, however, what happens with boundary conditions which, in the limit, with disappearance of the boundary, cease to have meaning. It was first confirmed by numerical simulations that the boundary conditions should become part of the reaction term in the limit reaction–diffusion, and then, in [22], proved formally that solutions of (6.1) with boundary conditions (6.2) (rescaled by replacing aR by (1 − r)aR so that the flux through the outer boundary remains constant in the approximating procedure) converge as r → 1 to solutions of

$$\frac{\mathrm{\partial }K}{\mathrm{\partial }t}=d{\mathrm{\Delta }}_{\text{LB}}K+f(K)+aR(1-K),$$ 6.3

which is a 2D equation on the unit sphere with Δ_LB denoting the Laplace–Beltrami operator. The intriguing part of this result is that, in agreement with biological intuitions, the left-hand part of the boundary condition (6.2) ‘jumps’ into the limit master equation (comp. (6.1) to (6.3)).

There are three obstacles one encounters in proving this theorem within the theory of semigroups of operators: (a) nonlinearity in the boundary condition, (b) nonlinearity in the equation, and (c) explaining in what sense (solutions to) equations in shells should converge to (solutions to) an equation on the unit sphere. Problem (a) is taken care of by switching to equations for inactive kinases $K^{\diamond }:=1-K$. As to (b), the fact that f(0) ≥ 0 and f(1) ≤ 0 allows reducing the problem to the case where f is globally Lipschitz, and then the main result of [25] says that convergence of solutions of semi-linear equations can be deduced from convergence of semigroups. As to (c), the idea, accomplished by a family of isomorphism, is to look at the thinner and thinner shells through a finer and finer magnifying glass. From this perspective, all the semigroups of interest act in a single space of continuous functions on a shell of unit thickness, and describe diffusion processes with faster and faster radial component. As a result, because of averaging properties of diffusion, solutions to the studied equations become less and less dependent on the radius and in the limit become identical to functions on the sphere. (In other words, ${\mathbb{X}}_0$ of (4.1) is the subspace of functions on the unit shell that do not depend on the radius, and we have an example of relation (5.1)) Moreover, the Robin boundary conditions describe disappearance of diffusing inactive kinases at the cell’s membrane (i.e. activation of these kinases); in the limit, this becomes a description of kinases’ activation at a point of the unit sphere, which is no longer a boundary but the state-space of the underlying diffusion process.

The papers [26,27] extend the result just described to the case where there are two thin layers lying on two sides of a semi-permeable membrane. The role of boundary conditions is then played by various sorts of transmission conditions, but the effect is the same: transmission conditions become integral parts of the limit master equation, and describe jumps of a limit process from one side of the membrane to the other. The fact that the same biological phenomenon, depending on ‘geometry’, may be described by a boundary/transmission condition or a term in the master equation is thoroughly discussed, and connections to the underlying, so-called Lévy local time of stochastic process are drawn.

7. An introduction to asymptotic stability: the Perron–Frobenius theorem

Let us turn to the question of asymptotic stability: a good starting point is a continuous version of the classical Perron–Frobenius theorem. To recall it, let $\mathbb{X}$ be the space ${\mathbb{R}}^n$ with the scalar product 〈 · , · 〉. If $x={({\xi }_i)}_{i=1,\dots ,n}\in \mathbb{X}$ then we write x > 0 when ξ_i > 0 and x ≥ 0 when ξ_i ≥ 0 for $i=1,2,\dots ,n$. Let Q = (q_i,j)_1≤i,j≤n be an n × n real matrix. Then the formula T(t)x = xe^tQ defines a semigroup {T(t), t ≥ 0} on $\mathbb{X}$. We assume that

• (P)q_ij ≥ 0 for i ≠ j,
• (I)for all i and j, there exists a sequence (i₁, i₂, …, i_m) such that i₁ = i, i_m = j and $q_{i_k,i_{k+1}}>0$ for k = 0, 1, …, m − 1.

Condition (P) guarantees that the semigroup {T(t), t ≥ 0} is positive, i.e. T(t)x ≥ 0 for x ≥ 0. From both conditions (P) and (I), it follows that there are: a constant $r\in \mathbb{R}$ and vectors x* > 0 and y* > 0 such that

$$\underset{t\to \mathrm{\infty }}{lim}{\text{e}}^{-rt}T(t)x=x^{\ast }⟨y^{\ast },x⟩\text{for\hspace{0.17em}}x\in \mathbb{X},$$ 7.1

and the convergence in (7.1) is exponential. The fact that the limit projection has rank 1 is a consequence of the irreducibility assumption (I). A semigroup {T(t), t ≥ 0} satisfying condition (7.1) is said to have asynchronous exponential growth. It should be mentioned that one can find in the literature, e.g. [28], a more general definition of asynchronous exponential growth, where it is only assumed that e^{−r t}T(t)x converges to a non-zero finite rank operator.

This theorem is a consequence of quite non-trivial properties of the matrix Q. First of all, the number r is a dominant eigenvalue of Q, which means that any other eigenvalue λ satisfies Re λ < r. Also, the general eigenspace associated with r is one-dimensional, and x* and y* are left and right eigenvectors corresponding to r, i.e. x*Q = rx* and Qy* = ry* (it follows that T(t)x* = e^rtx*). Finally, the functional α defined by α(x) = 〈y*, x〉 is positive, i.e. α(x) ≥ 0 for x ≥ 0.

If assumption (I) is dropped, the dominant eigenvalue $r\in \mathbb{R}$ still exists, but (7.1) needs not hold as described. More specifically, if the general eigenspace associated with r is one-dimensional then (P) implies (7.1) but the inequalities x* > 0 and y* > 0 need not be fulfilled. If the general eigenspace associated with r is not one-dimensional the behaviour of the semigroup {T(t), t ≥ 0} is more complex. For example, if Q is the 2 × 2 matrix with q₁₁ = q₁₂ = q₂₂ = 1 and q₂₁ = 0, then r = 1 is the dominant eigenvalue and T(t)(ξ₁, ξ₂) = e^t(ξ₁, ξ₂ + ξ₁t). This means that lim_t→∞ e^{−r t}t⁻¹T(t)x = xR, where R is the 2 × 2 matrix determined by xR = (0, ξ₁). Generally, if (P) holds, then the semigroup {T(t), t ≥ 0} has polynomial exponential growth, i.e. there exists a non-negative integer k and a n × n non-zero matrix R with non-negative elements such that, for each x, lim_t→∞ e^{−r t}t^1−kT(t)x = xR. More precisely, k × k is the size of the largest Jordan block related to the eigenvalue r, and xR is a linear combination of certain eigenvectors related to these Jordan blocks.

If (P) is not satisfied, a dominant eigenvalue, defined as an eigenvalue with the largest real part, may be complex, and there may exist a number of eigenvalues with this property. These two new features result in two new types of asymptotic behaviour: using the Jordan–Chevalley decomposition of Q one shows that (a) if imaginary parts of all dominant eigenvalues are commensurable, then the semigroup has a periodic polynomial exponential growth and (b) if they are not, the semigroup has an almost periodic exponential growth. To summarize: asymptotic behaviour of the semigroup {T(t), t ≥ 0} is determined by its behaviour on the invariant subspace $\mathbb{Y}$, spanned by generalized eigenspaces, with the largest dimension, related to dominant eigenvalues.

If we turn to semigroups in infinite-dimensional spaces, new problems appear because spectral properties of infinite-dimensional semigroups are more complex. For example, consider the intensity matrix Q indexed by integers, with elements q_i,i+1 = q_i,i−1 = 1, q_ii = −2 and q_ij = 0 in other cases. Since this Q is related to a bounded linear operator on the space l¹, we may think of the semigroup {T(t), t ≥ 0} defined by T(t) = e^tQ. It can be shown that this is a stochastic semigroup, i.e. $||T(t)x||=||x||$ for x ≥ 0. The matrix Q also satisfies conditions (P) and (I) and thus one could expect that there exists x* > 0 such that T(t)x* = x*, but that is not the case. In other words, for any t, r = 1 is not an eigenvalue of T(t), even though the spectral radius of T(t) is 1, and thus condition (7.1) fails to hold. Non-compactness of trajectories $t\mapsto T(t)x$ is one of the main reasons for this situation.

The second problem in infinite-dimensional spaces is related to the rate of convergence in (7.1). Let {T(t), t ≥ 0} be a semigroup on a Banach space $\mathbb{X}$. We say that this semigroup has asynchronous (or balanced) exponential growth if there exist $r\in \mathbb{C}$, non-zero $x^{\ast }\in \mathbb{X}$, and a non-zero linear functional $\alpha :\mathbb{X}\to \mathbb{C}$ such that

$$\underset{t\to \mathrm{\infty }}{lim}\hspace{0.17em}{\text{e}}^{-rt}T(t)x=x^{\ast }\alpha (x)\text{for\hspace{0.17em}}x\in \mathbb{X}.$$ 7.2

This definition implies that r is a simple eigenvalue of the infinitesimal generator A of the semigroup {T(t), t ≥ 0}. In finite-dimensional cases, convergence in (7.2) is exponential because the spectrum of A is finite and all eigenvalues λ of A, other than r, satisfy the inequality Re λ < Re r. But in the infinite-dimensional case, condition (7.2) does not guarantee that r is an isolated element of the spectrum of A and the rate of convergence in (7.2) need not be exponential.

A final note (in this section): let ${\mathbb{X}}_0=\mathrm{Ker}\hspace{0.17em}\alpha$; it is easy to see that ${\mathbb{X}}_0$ is invariant for the semigroup. By restricting {T(t), t ≥ 0} to this subspace, the problem of the rate of convergence (7.2) is replaced by that of the rate of convergence of $||{\text{e}}^{-rt}T(t)x||\to 0$ for $x\in {\mathbb{X}}_0$.

8. Asymptotic decomposition via spectral properties

There are several results on asymptotic behaviour of strongly continuous semigroups which are in the spirit of the previous section. These results are based on compactness properties of semigroups and are thoroughly discussed in monographs [29,30].

Let A be the infinitesimal generator of a strongly continuous semigroup {T(t), t ≥ 0} on a Banach space $\mathbb{X}$ and let $\mu \in \mathbb{C}$ be an isolated point of the spectrum of A. Since λ → R_λ is a holomorphic function it can be expanded as a Laurent series

$$R_{\lambda }=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(\lambda -\mu )}^n{U}_n,$$

for 0 < |λ − μ| < δ and some sufficiently small δ > 0. The coefficients U_n of this series are bounded operators given by the formulae

$$U_n=\frac{1}{2\pi i}{\int }_{\gamma }\frac{R_{\lambda }}{{(\lambda -\mu )}^{n+1}}\hspace{0.17em}\text{d}\lambda ,n\in \mathbb{Z},$$

where γ is the positively oriented boundary of a disc with radius <δ and centre at μ. The operator P = U₋₁ is called the residue of R_λ at μ. We have U_−(n+1) = (A − μ)ⁿP. If there exists a k > 0 such that U_−k ≠ 0 while U_−n = 0 for all n > k, then μ is called a pole of R_λ of order k.

Let $(\mathcal{L}(\mathbb{X}),||\cdot ||)$ be the space of all bounded linear operators on $\mathbb{X}$ endowed with the operator norm. Denote by $\mathcal{K}(\mathbb{X})\subseteq \mathcal{L}(\mathbb{X})$ the set of all compact operators on $\mathbb{X}$. A strongly continuous semigroup {T(t), t ≥ 0} on $\mathbb{X}$ is called eventually compact if there exists t₀ > 0 such that the operator T(t₀) is compact. It can be shown that an eventually compact semigroup is norm continuous for t ≥ t₀. A strongly continuous semigroup {T(t), t ≥ 0} on $\mathbb{X}$ is called quasi-compact if

$$\underset{t\to \mathrm{\infty }}{lim}inf\{||T(t)-K||:K\in \mathcal{K}(\mathbb{X})\}=0.$$

It is clear that an eventually compact semigroup is quasi-compact but not vice versa. The asymptotic behaviour of a quasi-compact semigroup is fully characterized by the following theorem (see [29, p. 333]).

Theorem 8.1. —

Let {T(t), t ≥ 0} be a quasi-compact semigroup on a Banach space $\mathbb{X}$ with generator A. Then the set {λ ∈ σ(A) : Re λ ≥ 0} is finite (or empty) and consists of poles of R_λ of finite algebraic multiplicity. If we denote these poles by λ₁, …, λ_m, and their residues by P₁, …, P_m with poles of order $k_1,\dots ,k_m$, we have $T(t)=T_1(t)+T_2(t)+\cdots +T_m(t)+R(t)$, where

$$T_n(t)={\text{e}}^{{\lambda }_{n}t}\sum _{j=0}^{k_n-1}\frac{t^j}{j!}{(A-{\lambda }_n)}^j{P}_n,\mathit{for\hspace{0.17em}}t\ge 0\mathit{\hspace{0.17em}and\hspace{0.17em}}1\le n\le m,$$ 8.1

and

$$||R(t)||\le M\hspace{0.17em}{\text{e}}^{-\epsilon t}\mathit{\hspace{0.17em}for\; some\hspace{0.17em}}\epsilon >0,\hspace{0.17em}M\ge 1\mathit{\hspace{0.17em}and\; all\hspace{0.17em}}t\ge 0.$$ 8.2

For a particular instance of this theorem, consider the case when there exists a unique dominant eigenvalue $r\in \mathbb{C}$ being a first-order pole. Then r is the spectral bound s(A) of A, i.e.

$$s(A)=sup\{\text{Re}\hspace{0.17em}\lambda :\lambda \in \sigma (A)\},$$

and P = lim_λ→r(λ − r)R_λ. From Theorem 8.1, it follows that

$$||{\text{e}}^{-rt}T(t)-P||\le M\hspace{0.17em}{\text{e}}^{-\epsilon t}\text{for some\hspace{0.17em}}\epsilon >0,\hspace{0.17em}M\ge 1\text{\hspace{0.17em}and all\hspace{0.17em}}t\ge 0.$$ 8.3

Theorem 8.1 is also a proper background for a generalization of the Perron-Frobenius theorem to positive semigroups on Banach lattices. To obtain a version of the Perron–Frobenius theorem for such a semigroup, one typically checks, instead of (I), that the semigroup is irreducible; this means that for every $x\in {\mathbb{X}}_{+}\setminus \{\mathbf{\text{0}}\}$ and for every $x^{\ast }\in {\mathbb{X}}_{+}^{\ast }\setminus \{\mathbf{\text{0}}\}$ there exists a t > 0 such that x*(T(t)x) > 0 (here, ${\mathbb{X}}_{+}=\{x\in \mathbb{X}:x\ge 0\}$). Indeed, the irreducibility rules out the possibility for the range of P to be more than one-dimensional, and together with quasi-compactness implies asynchronous exponential growth of the semigroup. A similar result can be obtained if we replace quasi-compactness by an assumption based on the notion of the measure of non-compactness of A, e.g. [28].

The asynchronous exponential growth (7.2) is a special case of the situation when there exists the limit

$$\underset{t\to \mathrm{\infty }}{lim}\hspace{0.17em}{\text{e}}^{-rt}T(t)=P,$$ 8.4

where $P:\mathbb{X}\to \mathbb{X}$ is an arbitrary non-zero operator, and the convergence can be uniform, strong or weak. The latter property was studied in [31,32], and a comprehensive overview of applications of this theory to population dynamics can be found in [33]. If $\tilde{T}(t):={\text{e}}^{-rt}T(t)$ and $\tilde{\mathbb{X}}:=\mathrm{Ker}\hspace{0.17em}P$, then condition (8.4) in the case of strong convergence is equivalent to strong asymptotic stability of the semigroup $\{\tilde{T}(t),t\ge 0\}$, i.e. to the condition

$$\underset{t\to \mathrm{\infty }}{lim}||\tilde{T}(t)x||=0\text{for\hspace{0.17em}}x\in \tilde{\mathbb{X}}.$$ 8.5

In the case when the semigroup is bounded and the set $\sigma (A)\cap (i\mathbb{R})$ is at its most countable, the strong asymptotic stability holds iff the adjoint operator A* has no pure imaginary eigenvalues [34,35]. Some versions of this result for unbounded semigroups are proved in [36].

The question of deriving the optimal rate of decay in (8.5) is highly non-trivial. This problem is extensively discussed in [37]. To give a taste of this theory, we recall one of the results of [38] (Theorem 2.4). Let {T(t), t ≥ 0} be a strongly continuous semigroup on a Hilbert space $\mathbb{X}$ with generator A such that $\sigma (A)\cap (i\mathbb{R})=\mathrm{\varnothing }$. Then for a fixed α > 0, the following conditions are equivalent:

• (i)R_is = O(|s|^α), s → ∞,
• (ii)$||T(t)A^{-1}||=O(t^{-\alpha })$, t → ∞,
• (iii)$||T(t)A^{-1}x||=o(t^{-\alpha })$, t → ∞, $x\in \mathbb{X}$.

9. Stochastic semigroups

Stochastic semigroups are defined on the space L¹ = L¹(X, Σ, m) with a σ-finite measure m. A linear operator P : L¹ → L¹ is called stochastic if P(D)⊆ D, where D is the set of densities

$$D:=\{f\in \hspace{0.17em}{L}^1:\hspace{0.17em}f\ge 0,\hspace{0.17em}||f||=1\}.$$

A strongly continuous semigroup {P(t), t ≥ 0} of linear operators on L¹ is called a stochastic semigroup if all operators P(t) are stochastic. The iterates of a stochastic operator form a discrete-time semigroup, and by writing P(t) = P^t for these iterates, we may formulate most definitions and results for both types of semigroups in a unified manner.

A stochastic semigroup {P(t), t ≥ 0} on L¹ = L¹(X, Σ, m) is called asymptotically stable if there exists a density f* such that

$$\underset{t\to \mathrm{\infty }}{lim}||P(t)f-f^{\ast }||=0\text{for}\hspace{0.17em}f\in D.$$ 9.1

From (9.1), it follows that f* is an invariant density, i.e. P(t)f* = f* for t ≥ 0. If the semigroup {P(t), t ≥ 0} is generated by an evolution equation u′(t) = Au(t), then its asymptotic stability means that the stationary solution u(t) = f* is globally asymptotically stable in the sense of Lyapunov on the set of densities D. Observe that if there is an invariant density f*, then the stochastic semigroup {P(t), t ≥ 0} is asymptotically stable iff $\underset{t\to \mathrm{\infty }}{lim}||P(t)f||=0$ for $f\in L_0^1=\{f\in L^1:{\int }_{X}f(x)\hspace{0.17em}m(\text{d}x)=0\}$.

There are several results concerning asymptotic stability of stochastic semigroups. We start with the lower function theorem of Lasota & Yorke [39]. Let f⁻(x) = max (0, − f(x)). A function h ∈ L¹, such that h ≥ 0 and h ≠ 0, is called a lower function for a stochastic semigroup {P(t)}_t≥0 if

$$\underset{t\to \mathrm{\infty }}{lim}||{(P(t)f-h)}^{-}||=0\text{for every\hspace{0.17em}}f\in D.$$ 9.2

The Lasota–Yorke theorem says that a (continuous or discrete time) stochastic semigroup is asymptotically stable iff it has a lower function h. This theorem was generalized in many directions. The interested reader is referred to the book [40], where one can find versions formulated in the language of von Neumann algebras and to the paper [41] which provides a comprehensive overview of earlier results and presents new theorems for semigroups on Banach lattices including a version with lower functions depending on f.

The second theorem we discuss here shows that convergence of trajectories of a stochastic semigroup to a compact set leads to interesting asymptotic properties. A stochastic semigroup {P(t), t ≥ 0} is called strongly (weakly) constrictive [42] if there exists a strongly (weakly) compact subset F of L¹ such that

$$\underset{t\to \mathrm{\infty }}{lim}inf\{||P(t)f-g||:g\in F\}=0,\text{for\hspace{0.17em}}f\in D.$$

Theorem 9.1. —

If the semigroup {P(t), t ≥ 0} is weakly constrictive then there exist: a t₀ > 0, densities g₁, …, g_r with disjoint supports such that P(t₀)g_i = g_i, and positive periodic functionals α_i(t, f) with period t₀ such that

$$\underset{t\to \mathrm{\infty }}{lim}||P(t)f-\sum _{i=1}^r{\alpha }_i(t,f)g_i||=0\text{for\hspace{0.17em}}f\in L^1.$$ 9.3

(To recall: the support of a measurable function $f:X\to \mathbb{R}$ is defined by the formula $\mathrm{supp}f=\{x\in X:f(x)\ne 0\}$). Historically, Theorem 9.1 was first proved in [42] under assumption of strong constrictivity; 2 years later Komorník [43] showed that weak constrictivity suffices.

A semigroup {P(t), t ≥ 0} satisfying (9.3) is called asymptotically periodic. In particular, for h ∈ L¹, the set F : = {f ∈ D : f ≤ h} is weakly compact. We deduce from Theorem 9.1 that if

$$\underset{t\to \mathrm{\infty }}{lim}||{(P(t)f-h)}^{+}||=0\text{for every\hspace{0.17em}}f\in D,$$ 9.4

then the semigroup {P(t), t ≥ 0} is asymptotically periodic.

Theorem 9.1 can be strengthened considerably in the case when {P(t), t ≥ 0} is a continuous time stochastic semigroup. It is easy to check that in this case densities g_i are invariant and functionals α_i do not depend on t. Moreover, if f ∈ D and $\mathrm{supp}f\subseteq \mathrm{supp}{g}_i$, then P(t)f → g_i as t → ∞. This means that there is a projection H onto the subspace spanned by the densities $g_1,\dots ,g_r$ such that P(t)f → Hf as t → ∞.

Examples of applications of the lower function theorem and Theorem 9.1 are given in [44]. The property of constrictivness was generalized and studied in many papers (see [40] for a comprehensive overview of these results).

10. Asymptotic decomposition via kernel minorants

In this section, we present two theorems on asymptotic behaviour of continuous-time stochastic semigroups (and some of their applications), that stem from the theory of Harris operators [45–47].

A stochastic semigroup {P(t), t ≥ 0} is called partially integral if there exist a t > 0 and a measurable function q(t, · , · ) : X × X → [0, ∞) such that ${\int }_X{\int }_{X}q(t,x,y)\hspace{0.17em}m(\text{d}x)\hspace{0.17em}m(\text{d}y)>0$ and

$$P(t)f(y)\ge {\int }_{X}q(t,x,y)f(x)\hspace{0.17em}m(\text{d}x)\text{for\hspace{0.17em}}f\in D.$$ 10.1

Here is a simple criterion for asymptotic stability of partially integral semigroups (see [48]).

Theorem 10.1. —

Suppose that a partially integral stochastic semigroup {P(t), t ≥ 0} has a unique invariant density f*. If f* > 0 a.e., then the semigroup {P(t), t ≥ 0} is asymptotically stable.

In particular, from Theorem 10.1, it follows that if a partially integral stochastic semigroup is irreducible and has an invariant density then it is asymptotically stable.

The main problem with applying Theorem 10.1 is that often it is not easy to find an invariant density. To get around this difficulty, a method of studying asymptotic properties of stochastic semigroups via their asymptotic decomposition was devised in [49]. To present this method, from now on we assume additionally that (X, ρ) is a separable metric space and $\mathrm{\Sigma }=\mathcal{B}(X)$ is the σ-algebra of Borel subsets of X, and consider stochastic semigroups {P(t), t ≥ 0} which satisfy the following condition:

(K) for every x₀ ∈ X there exist an ε > 0, a t > 0, and a measurable function η ≥ 0 such that $\int \eta (x)\hspace{0.17em}m(\text{d}x)>0$ and

$$q(t,x,y)\ge \eta (y)\text{for\hspace{0.17em}}x\in B(x_0,\epsilon ),y\in X,$$ 10.2

where $B(x_0,\epsilon )=\{x\in X:\hspace{0.17em}\rho (x,x_0)<\epsilon \}$. It is clear that stochastic semigroups satisfying condition (K) are partially integral. Now we present the main theorem of [49].

Theorem 10.2. —

Let {P(t), t ≥ 0} be a stochastic semigroup which satisfies (K). Then there exist an at most countable set J, a family of invariant densities {f*_j}_j∈J with disjoint supports {A_j}_j∈J, and a family {α_j}_j∈J of positive linear functionals defined on L¹ such that

• (i)
  for every j ∈ J and for every f ∈ L¹, we have

  $$\underset{t\to \mathrm{\infty }}{lim}||{\mathbf{1}}_{A_j}P(t)f-{\alpha }_j(f)f_j^{\ast }||=0,$$ 10.3
• (ii)
  if $Y=X\setminus \bigcup _{j\in J}{A}_j$, then for every f ∈ L¹ and for every compact set F, we have

  $$\underset{t\to \mathrm{\infty }}{lim}{\int }_{F\cap Y}P(t)f(x)\hspace{0.17em}m(\text{d}x)=0.$$ 10.4

We note that not only the sets A_j, j ∈ J, are disjoint but also their closures are disjoint, which means that the measures μ_j(dx): = f*_j(x) m(dx) have disjoint topological supports [50].

In order to formulate corollaries to Theorem 10.2, we need an auxiliary notion. A stochastic semigroup {P(t), t ≥ 0} is called sweeping with respect to a set B ∈ Σ if for every f ∈ D

$$\underset{t\to \mathrm{\infty }}{lim}{\int }_{B}P(t)f(x)\hspace{0.17em}m(\text{d}x)=0.$$

Corollary 10.3. —

Assume that a stochastic semigroup {P(t), t ≥ 0} satisfies condition (K) and has no invariant densities. Then {P(t), t ≥ 0} is sweeping from compact sets.

Our next aim is to find conditions guaranteeing that a stochastic semigroup {P(t), t ≥ 0} satisfies the Foguel alternative, i.e. it is asymptotically stable or sweeping from all compact sets [44]. The following result is a step in this direction.

Corollary 10.4. —

If a stochastic semigroup {P(t), t ≥ 0} is irreducible and satisfies condition (K) then the Foguel alternative holds.

Sometimes we can replace irreducibility by a weaker condition. We say that a substochastic semigroup {P(t), t ≥ 0} is weakly irreducible if:

(WI) there exists a point x₀ ∈ X such that for each ε > 0 and for each density f we have

$${\int }_{B(x_0,\epsilon )}P(t)f(x)\hspace{0.17em}m(\text{d}x)>0\text{for some\hspace{0.17em}}t>0.$$ 10.5

Corollary 10.5. —

Suppose X is a compact space. Then a stochastic semigroup which satisfies (K) and (WI) is asymptotically stable.

The proof of this corollary can be found in [50] (see remarks after corollary 5.3). A brief argument is the following. Conditions (K), (WI) and the property that the measures μ_j have disjoint topological supports imply that the set J in Theorem 10.2 is empty or is a singleton. Since X is a compact space the set Y in Theorem 10.2 is empty. Thus J is a singleton and consequently the semigroup is asymptotically stable.

New results concerning positive operators on Banach lattices similar in spirit to Theorems 10.1 and 10.2 may be found in [51].

11. Applications

Stochastic semigroups play a special role in application in biology, chemistry and physics [44,52,53]. In this section, we present two examples exemplifying usefulness of the results of §10 in biology. The first example is a general model of gene regulatory networks built on the ideas presented in papers [54–56]. The second model describes the evolution of immune status considered in [57,58].

(a). Gene expression models

Gene expression is a process of production and decay of various biomolecules dependent on activity of a number of genes. Changes in concentration of these biomolecules can be basically described by a system of differential equations, but this system depends on random parameters related to the genes’ activity. Let us assume that we have k different configurations related to gene activity or inactivity. For each fixed configuration i ∈ I = {1, …, k} concentration of d different biomolecules changes according to a system of differential equations of the form x′ = bⁱ(x), $x\in {\mathbb{R}}^d$. Since the speed of production of biomolecules is bounded but their decay is proportional to their concentration, x is a vector from the set C = [0, M₁] × … × [0, M_d], where M_j is the maximum concentration of j-type of biomolecules. Thus, for a fixed configuration i the evolution of concentration x is defined by a semiflow $({\pi }_t^i)$ on C, where ${\pi }_t^i(x_0)$ is the solution x(t) of the system x′ = bⁱ(x) with initial condition x(0) = x₀. Then the stochastic semigroup {P_i(t), t ≥ 0} on the space L¹(C) with the generator A_if(x) = −div(bⁱf) on a suitable domain describes evolution of densities of concentration. Assume that the configuration i changes according to a k × k dimensional intensity matrix Q(x) such that $C\ni x\mapsto Q(x)$ is continuous. Finally, combining both ingredients: semiflows $({\pi }_t^i)$ and jumps between them, we obtain a stochastic semigroup {P(t), t ≥ 0} on the space $\mathbb{X}=L^1(C\times I,\mathcal{B}(C\times I),m)$, where m is the product measure given by m(B × {i}) = |B| for each Borel subset B of C, 1 ≤ i ≤ k, and | · | is the Lebesgue measure on C. Let f(x, i) = f_i(x) and let f = (f₁, …, f_k), Af = (A₁f₁, …, A_k f_k) be vertical vectors. Then the generator $\tilde{A}$ of the semigroup {P(t), t ≥ 0} is given by $\tilde{A}f=Af+f\cdot Q$ (on the same domain).

We can study asymptotic stability of the semigroup {P(t), t ≥ 0} by applying Corollary 10.5 (see [50] for details). Condition (WI) holds if there exists a point x = (x₀, i₀) ∈ C × I such that a cumulative flow of the form ${\pi }_{t_n}^{i_n}\circ \cdots \circ {\pi }_{t_1}^{i_1}(x)$ joins each point (x, i) ∈ C × I with a point arbitrarily close to x. Condition (K) can be verified by checking that the vectors

$$b^2(x_0)-b^1(x_0),\dots ,b^k(x_0)-b^1(x_0),\hspace{0.17em}[b^i,b^j]{(x_0)}_{1\le i,j\le k},\hspace{0.17em}[b^i,[b^j,b^l]]{(x_0)}_{1\le i,j,l\le k},\dots ,$$

span the space ${\mathbb{R}}^d$ and q_ij(x₀) > 0 for i ≠ j. The symbol [α, β] is the Lie bracket of two vector fields α and β, i.e. a new vector field given by

$${[\alpha ,\beta ]}_j(x)=\sum _{k=1}^d({\alpha }_k\frac{\mathrm{\partial }{\beta }_j}{\mathrm{\partial }{x}_k}(x)-{\beta }_k\frac{\mathrm{\partial }{\alpha }_j}{\mathrm{\partial }{x}_k}(x)).$$

(b). Dynamics of immune status

The immune status is the concentration of specific antibodies, which appear after infection with a pathogen and remain in serum, providing protection against future attacks of that same pathogen. Over time the number of antibodies decreases until the next infection. While fighting the invader the immunity is boosted and then it is gradually waning, until it is boosted by a new encounter. Concentration of antibodies between subsequent infections is a decreasing function x(t), which satisfies a differential equation of the form x′(t) = g(x(t)), where g(x) < 0 for x > 0. It is assumed that the time it takes the immune system to clear infection is negligible and that if x is the concentration of antibodies at the moment of infection, then ψ(x) > x is the concentration of antibodies just after clearance of infection. The moments of infections are distributed according to a Poisson process with intensity c > 0. Thus the immune status is a stochastic process (ξ_t)_t≥0: a flow on the interval [0, ∞), with jumps at random moments [57].

We assume that ψ is a C¹-function and there exists an at most countable family of disjoint intervals (a_i, b_i), i ∈ I, such that $[0,\mathrm{\infty })=\bigcup _{i\in I}[a_i,b_i]$ and ψ′(x) ≠ 0 for x ∈ (a_i, b_i) and i ∈ I. Let φ_i be the inverse function of $\psi {|}_{(a_i,b_i)}$, and let I_x = {i : φ_i(x) ∈ (a_i, b_i)}. The process of jumps from x to ψ(x) induces the stochastic operator

$$Pf(x)=\sum _{i\in I_x}f({\phi }_i(x))|{\phi }_i^{\prime }(x)|.$$

The evolution of densities of the process (ξ_t)_t≥0 is described by the stochastic semigroup {P(t), t ≥ 0} on the space L¹[0, ∞) with the generator Af = −(gf)′ + c Pf − c f.

The long-time behaviour of this semigroup is described in [58], as a consequence of Corollary 10.4 above. It is quite simple to check that the semigroup is irreducible, but condition (K) needs more attention. From the Dyson–Phillips expansion, it follows that (K) is fulfilled if there exists an x > 0 such that g(ψ(x)) ≠ ψ′(x)g(x). This requirement is satisfied for all sufficiently small x > 0 because g(0) = 0 and g(ψ(0)) > 0. Therefore, by Corollary 10.4, the semigroup under consideration is either asymptotically stable or sweeping from compact subsets of [0, ∞). In [58], sufficient conditions for asymptotic stability and for sweeping are given. Notably, the latter property can be interpreted as asymptotic permanent immunity of the population.

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Authors' contributions

A.B. wrote the part on convergence of semigroups. R.R. wrote the part on asymptotic behaviour of semigroups.

Competing interests

We declare we have no competing interests.

Funding

A.B.’s research was supported by National Science Center (Poland)grant no. 2017/25/B/ST1/01804 and R.R.’s research was supported by National Science Center (Poland) grant no. 2017/27/B/ST1/00100.