PERTURBATION OF TRANSITION FUNCTIONS AND A FEYNMAN

PERTURBATION OF TRANSITION FUNCTIONS AND A FEYNMAN-KAC FORMULA FOR THE INCORPORATION OF MORTALITY TIMOTHY LANT† AND HORST R. THIEME¦ Abstract. Markov transition kernels are perturbed by output kernels with a special emphasis on building mortality into structured population models. A Feynman-Kac formula is derived which illustrates the interplay of mortality with a Markov process associated with the unperturbed kernel. Mathematics Subject Classification (2000). 47D06, 60J35, 92D25 Keywords. (Markov) transition functions, stochastic continuity, output kernels, Hille-Yosida operators, forward equation, Markov processes, Feynman-Kac formula

1. Introduction Markov transition functions play an important role as a modeling ingredient in structured population models [5, 11] and as an analytic link between the theory of Markov processes and operator semigroups [7, 8, 15]. While we have concentrated on the second aspect in [13], we will here illustrate a perturbation method which builds mortality and reproduction into a model which merely describes structural state transitions. As in [5], the perturbation method is based on output kernels, though our perturbation procedure is immediately applied to the transition functions rather than to the output kernels first. An operator-theoretic version using cumulative output families has been used in [11] and a mixture of both approaches in [12]. Complementary to [5, 11] where mortality is already part of the state transition and birth processes are built in by perturbation, we elaborate here on the incorporation of mortality. To be more specific, let Ω be the state space and x a state point in Ω. Further let B be a σ-algebra of subsets of Ω which represent the state sets. In the most basic scenario, a transition function K(t, r, x, D) describes the probability of the state being in the Date: September 6, 2006. † partially supported by NSF grant DMS-0314529 and SES-0345945. ¦ partially supported by NSF grants DMS-9706787 and DMS-0314529. 1

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T. Lant, H.R. Thieme

set D ∈ B at time t if x has been the state at time r, r < t, with the possibility of dying between r and t being ignored. Let K∞ (r, x, t, D) be the same probability as before but now under a mortality regime which is described by a function ϑ(t, x) which gives the mortality rate at state x at time t. The two transition functions are related by the integral equation K∞ (t, r, t, D) =K(t, r, x, , D) Z t Z (1.1) − ds K(s, r, x, dy)ϑ(s, y)K∞ (t, s, y, D), r

Ω

and we will find K∞ as a solution of this equation. That K∞ describes state transition with mortality can be seen from a Feynman-Kac type formula which can be derived if we assume that the transition function K is associated with a Markov process Xt on a probability space with probability measure P , K(t, r, Xr , D) = P Xr (Xt ∈ D),

(1.2)

t > r, D ∈ B.

Here P Xr is the conditional probability given Xr . If f is a bounded measurable function on Ω, K∞ is related to Xt by the Feynman-Kac type formula · Z ³ Z t ´¸ Xr (1.3) K∞ (t, r, Xr , dy)f (y) = E f (Xt ) exp − ϑ(s, Xs )ds , Ω Xr

r

where E is the conditional expectation given Xr . Typically, Feynman-Kac formulas appear in the literature if Xt is the solution of a stochastic differential equation [14] or K satisfies the Feller-property, R ˜ ˜ i.e., the operators S(t) defined by S(t)f = Ω K(t, r, ·, dy)f (y) map an appropriate space of continuous functions f on Ω into itself [4, 10, 17]. In these cases, (1.3) is used to define K∞ rather than finding K∞ from (1.1). In [13] we have spelt out the restrictions that the Feller-property imposes on transition functions in measure-theoretic terms and have taken this as a motivation to develop a theory without assuming it. More seriously, the Feller property is too restrictive for transition functions in general structured population models. The Feynman-Kac formula (1.3) suggests, but does not prove, that K∞ is non-negative in spite of the negative sign in (1.1). This is the case, indeed, if K is non-degenerate in a sense explained later. The proof is related to the fact that a Markov transition function K is characterized by a (Kolmogorov) forward equation if it is time-autonomous. More precisely there exists a closed linear operator (actually a HilleYosida operator) on the space of signed measures with bounded variation, M(Ω), such that K is uniquely determined as the solution of the

Perturbation of transition functions

equation (1.4)

Z

3

t

K(t, x, ·) = δx + A

K(s, x, ·)ds,

x ∈ Ω, t ≥ 0.

0

Here δx is the Dirac measure concentrated at x. If ϑ is bounded, K∞ satisfies an equation of the same type with A being replaced by A∞ where Z (A∞ µ)(D) = (Aµ)(D)− µ(dx)ϑ(x), µ ∈ D(A) ⊆ M(D), D ∈ B. D

This is consistent with the backwards equations in [14, Thm.7.13] and [10, Exc.9.21, 24(iii)]. More generally, every perturbation of K by an autonomous output kernel is related to a perturbation of the operator A. The time-autonomy of the transition function is no real restriction as every non-autonomous transition function can be represented by an autonomous transition function on a larger state-space which incorporates time [2, 2.2.9]. This also holds for output kernels as we show in the appendix. 2. Autonomous Transition kernels ˜ be measurable spaces with σ-algebras B and B˜ of subsets Let Ω and Ω ˜ of Ω and Ω respectively. ˜ × B → R is called a Definition 2.1. [3, 10.3.1] A function L : Ω (measure) kernel if ˜ (a) L(x, ·) is a non-negative measure on B for all x ∈ Ω. ˜ (b) L(·, D) is a B-measurable function for all t ≥ 0, D ∈ B. A measure kernel L is called bounded if supx∈Ω˜ L(x, Ω) < ∞. Definition 2.2. A function K : R+ × Ω × B → R is called a transition function if (a) K(t, x, ·) is a non-negative measure on B for all t ≥ 0, x ∈ Ω. (b) K(t, ·, D) is a B-measurable function for all t ≥ 0, D ∈ B. (c) K(0, x, D) = 1 if x ∈ D and K(0, x, D) = 0 if x ∈ Ω \ D. (d) There exist δ, c > 0 such that K(t, x, Ω) ≤ c for all t ∈ [0, δ], x ∈ Ω. A transition function K is called a Markov transition function if it satisfies the Chapman-Kolmogorov equations, Z K(t + r, x, D) = K(t, y, D)K(r, x, dy), (2.1) Ω x ∈ X, t, r ≥ 0, D ∈ B.

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T. Lant, H.R. Thieme

A transition function is called a transition kernel if K(·, D) is BR+ × B measurable where BR+ is the σ-algebra of Borel sets on R+ and BR+ ×B is the product σ-algebra, in other words if K is a measure kernel. If K(t, x, ·) is a signed measure rather than a non-negative measure, we speak about signed transition functions (kernels). In this case, (d) needs to be replaced by (d’) There exist δ, c > 0 such that |K(t, x, D)| ≤ c for all t ∈ [0, δ], x ∈ Ω, D ∈ B. Notice that, if K is a transition function, K(t, ·) is a measure kernel ˜ = Ω, but K itself may not be a kernel. with Ω Remark 2.1. Sometimes the term ‘transition function’ is used such that the Chapman-Kolmogorov equations are included [7]. We follow the use in [15, Sec.3.2] and [9, Sec.2.1] though it may not be clear whether they use ‘Markov’ to highlight the Chapman-Kolmogorov equations or the assumption that K(t, x, Ω) ≤ 1 (or = 1) which we do not make (cf. [3, 10.3.1]). We use ‘Markov’ in order to emphasize the connection of the Chapman-Kolmogorov equations to Markov processes [2, Sec.2.2] [7, Ch.4 (1.9)]. The Chapman-Kolmogorov equations imply that Markov transition functions are exponentially bounded. Lemma 2.3. [13] Let K be a Markov transition function. There exist ω ∈ R, M ≥ 1 such that K(t, x, Ω) ≤ M eωt for all t ≥ 0, x ∈ Ω. Definition 2.4. A transition function K is called non-degenerate if a measureRµ ∈ M(Ω) is necessarily the zero measure whenever, for any D ∈ B, Ω K(t, x, D)µ(dx) = 0 for almost all t > 0. Sufficient conditions for non-degeneracy can be formulated if Ω is a topological space. Recall that the Baire σ-algebra is the smallest σ-algebra such that all bounded continuous functions on Ω are measurable. If Ω is a normal space, the Baire σ-algebra is generated by the open Fσ sets in Ω. An Fσ -set is the countable union of closed sets. A Kσ -set is the countable union of compact sets [3] . Proposition 2.5. [13] A transition function K is a non-degenerate transition kernel if one of the following two assumptions hold: (a) Ω is a σ-compact space (i.e. a locally compact space and a Kσ set) and K is weakly stochastically continuous, i.e., K(t, x, U ) → 1 as t → 0 whenever U is an open Kσ -set in Ω with compact closure and x ∈ U .

Perturbation of transition functions

5

(b) Ω is a normal space and K is stochastically continuous, i.e., K(t, x, U ) → 1 as t → 0 whenever U is an open Baire set in Ω and x ∈ U . That the assumption for K in (a) is weaker than in (b) is not surprising as every σ-compact space is normal. (Every σ-compact space is paracompact [6, XI.7] and every paracompact space is normal [6, VIII.2].) 3. Perturbation by output kernels A measure kernel L : Ω × (BR+ × B) → R+ is called an output kernel for the transition function K if there exists some r > 0 such that (3.1)

sup L(x, [0, r] × Ω) < ∞ x∈Ω

and (3.2)

Z

¡ ¢ K(r, x, dy)L(y, [0, t] × D) =L x, [r, r + t] × D Ω

∀x ∈ Ω, r, t ≥ 0, D ∈ B. The name output kernel and the relation (3.2) are motivated as follows. Let β(y, D) be the output into the state set D by an individual with state y, e.g., the amount of offspring with a state in D produced by an individual with state in x. If K is a Markov transition kernel, Z ³Z ´ (3.3) L(x, Γ × D) = K(s, x, dy)β(y, D) ds Γ

Ω

is the cumulative output into the set D which an individual which has state x at time 0 produces during the time set Γ. It easily follows from the Chapman-Kolmogorov equations that L defined by (3.3) is an output kernel. Lemma 3.1. Let L be an output kernel for the transition function K and g : R+ × Ω → R+ be measurable from BR+ × B to BR+ . Then Z Z K(t, x, dz) L(z, d(s, y))g(s, y) Ω R+ ×Ω Z = L(x, d(s, y))g(s − t, y) ∀t > 0, x ∈ Ω. [t,∞)×Ω

Proof. For g = χ[0,r]×D this follows from the definition of an output kernel. By standard measure theoretic arguments, the statement is extended first to g = χΓ×D with Γ ∈ BR+ , then to g = χD˜ with ˜ ∈ BR+ × B, further to linear combinations of such functions with D positive scalars and then to general non-negative g. ¤

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Lemma 3.2. Let L be an output kernel for the Markov transition function K. Then L is exponentially bounded, i.e. there exist Λ > 0, ω ∈ R such that L(x, [0, t] × Ω) ≤ Λeωt for all t ≥ 0, x ∈ Ω. More precisely, if K has a positive exponential growth bound, L has the same growth bound; if K has a negative growth bound, L is bounded, and if K is bounded, L grows linearly. Proof. By (3.1) there exist r, M > 0 such that L(x, [0, r] × Ω) ≤ M ˜ ≥ 1, ω ∈ R such that for all x ∈ Ω. By Lemma 2.3, there exist M ωt ˜ K(t, x, Ω) ≤ M e for all t ≥ 0. Let t > 0. Choose n ∈ N such that (n − 1)r < t ≤ nr. By (3.2), n X L(x, [0, t] × Ω) ≤ L(x, [0, r] × Ω) + L(x, [(j − 1)r, jr] × Ω) ≤M + ≤M +

j=2

n Z X

K((j − 1)r, x, dy)L(y, [0, r] × Ω) Ω

j=2 n X

K((j − 1)r, x, Ω)M ≤ M

n ³X

j=2

˜ =M M

˜ e(j−1)ωr M

´

j=1

n−1 X

(eωr )j

j=0

If ω > 0, ωrn ˜ e − 1 ≤ MM ˜ eωr(n−1) ≤ M M ˜ eωt . L(x, [0, t] × Ω) ≤ M M eωr − 1 If ω < 0, ˜ 1 − eωrn MM ˜ L(x, [0, t] × Ω) ≤ M M ≤ . 1 − eωr 1 − eωr If ω = 0, ˜ n ≤ MM ˜ (1 + (t/r)). L(x, [0, t] × Ω) ≤ M M

¤ We consider the integral equation Z (3.4) ψ(t, x, D) = φ(t, x, D) +

L(x, d(s, y))ψ(t − s, y, D),

[0,t)×Ω

where φ and ψ are signed transition functions with φ being given and ψ being wanted. Let Z be the ordered Banach space of signed transition functions ψ : [0, σ] × Ω → R with norm n o kψkλ = sup e−λt |ψ(t, x, D)|; t ∈ [0, σ], x ∈ Ω, D ∈ B .

Perturbation of transition functions

7

The integral equation (3.4) can be rewritten in operator form as (I − Ψ)ψ = φ with the linear bounded operator Ψ on Z given by Z (3.5) [Ψψ](t, x, D) = L(x, d(s, y))ψ(t − s, y, D). [0,t)×Ω

The operator norm of Ψ satisfies Z (3.6) kΨkλ ≤ sup x∈Ω

e−λs L(x, d(s, y)).

[0,σ]×Ω

I − Ψ has a positive bounded inverse given by the Neumann series −1

(3.7)

(I − Ψ)

=

∞ X

Ψj ,

j=0

if the spectral radius of Ψ is strictly smaller than 1. While this condition may look too abstract, requiring kΨkλ < 1 for large enough λ is too strong for some applications. A good compromise may be to require kΨ2 kλ < 1. Ψ2 is given by Z 2 ˜ d(s, y))ψ(t − s, y, D), [Ψ ψ](t, x, D) = L(x, [0,t)×Ω

with

Z

Z

˜ D) ˜ = L(x,

L(x, d(t, y)) R+ ×Ω

R+ ×Ω

L(y, d(s, z))χD˜ (t + s, z).

Let r ∈ (0, σ). Then Z ˜ d(s, y)) ≤ L(x, ˜ [0, r] × Ω) + e−λr L(x, ˜ [r, σ] × Ω) e−λs L(x, [0,σ]×Ω Z ≤ L(x, [0, r], dy)L(y, [0, r] × Ω) Ω Z −λr +e L(x, [0, σ], dy)L(y, [0, σ] × Ω). Ω

We see that kΨ2 kλ < 1 for sufficiently large λ > 0 if the assumption in the next theorem is satisfied. Theorem 3.3. Let Z ¡ ¢ ¡ ¢ (3.8) sup L x, [0, r], dy L y, [0, r] × Ω < 1 x∈Ω

Ω

for sufficiently small r > 0. Then, for each signed transition function φ there exists a unique signed transition function ψ solving (3.4). The

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T. Lant, H.R. Thieme

solution has the series expansion ψ(t, x, D) =

∞ X

ψj (t, x, D)

j=0

with ψ0 = φ and

Z

ψj (t, x, D) =

L(x, d(s, y))ψj−1 (t − s, y, D),

j ∈ N.

[0,t)×Ω

In a first step, we get a transition function ψ on [0, σ] × Ω × B, but since this holds for all σ > 0 and uniqueness holds, ψ can be defined on [0, ∞) × Ω × B. We now assume that K is a Markov transition kernel. We define K∞ as the unique solution (which exists by Theorem 3.3) of Z L(x, d(s, y))K∞ (t − s, y, D). (3.9) K∞ (t, x, D) = K(t, x, D) + [0,t)×Ω

Theorem 3.4. Let K be a Markov transition kernel and L (or −L) be an output kernel for K satisfying (3.8). Then the unique solution K∞ of (3.9) is a (signed) Markov transition kernel. Proof. We first notice that K∞ (0, x, D) = K(0, x, D) = χD (x). It will be convenient to introduce the operator family notation Z ˜ (3.10) [S∞ (t)f ](x) = K∞ (t, x, dy)f (y), x ∈ Ω, f ∈ BM(Ω). Ω

˜ The Chapman-Kolmogorov and an analogous definition of a family S. equations are equivalent to the semigroup property for these operator families on the Banach space BM(Ω). We notice that, for t > 0, Z ˜ ˜ (3.11) [S∞ (t)f ](x) = [S(t)f ](x) + L(x, d(s, y))[S˜∞ (t − s)f ](y). R+ ×Ω

Let t, r > 0. By definition of K∞ , K∞ (t + r, x, D)

Z

=K(t + r, x, D) +

L(x, d(s, y))K∞ (t + r − s, y, D) [0,t+r)×Ω

Z

˜ + r)χD ](x) + =[S(t L(x, d(s, y))K∞ (t + r − s, y, D) [0,t)×Ω Z + L(x, d(s, y))g(s − t, y), [t,∞)×Ω

Perturbation of transition functions

9

with g(s, y) = K∞ (r − s, y, D)χ[0,r) (r − s). By Lemma 3.1, (3.10), and (3.11), Z L(x, d(s, y))g(s − t, y) [t,∞)×Ω Z Z = K(t, x, dz) L(z, d(s, y))g(s, y) Ω [0,∞)×Ω Z Z = K(t, x, dz) L(z, d(s, y))K∞ (r − s, y, D) Ω [0,r)×Ω Z ¡ ¢ ˜ = K(t, x, dz) [S˜∞ (r)χD ](z) − [S(r)χ D ](z) Ω

˜ S˜∞ (r)χD ](x) − [S(t) ˜ S(r)χ ˜ =[S(t) D ](x). We substitute this equation into the previous one and use that S˜ is a semigroup, K∞ (t + r, x, D) (3.12)

Z

˜ S˜∞ (r)χD ](x) + =[S(t)

L(x, d(s, y))K∞ (t + r − s, y, D). [0,t]×Ω

We fix r > 0 and define ˜ x, D) =K∞ (t + r, x, D), ψ(t, Z ¯ ψ(t, x, D) = K∞ (t, x, dy)K∞ (r, y, D) = [S˜∞ (t)fD ](x), (3.13) Ω

with fD = S˜∞ (r)χD . By (3.12),

Z

˜ x, D) =[S(t)f ˜ ψ(t, D ](x) +

˜ − s, y, D). L(x, d(s, y))ψ(t [0,t]×Ω

By (3.13) and (3.11),

Z

¯ x, D) =[S(t)f ˜ ψ(t, D ](x) +

L(x, d(s, y))[S˜∞ (t − s)fD ](y) Z

R+ ×Ω

¯ − s, y, D). L(x, d(s, y))ψ(t

˜ =[S(t)f D ](x) + R+ ×Ω

¯ Then Set ξ = ψ˜ − ψ.

Z

ξ(t, x, D) =

L(x, d(s, y))ξ(t − s, y, D). R+ ×Ω

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T. Lant, H.R. Thieme

Obviously 0 is the unique solution for φ = 0 of (3.4) which exists ac¯ The Chapman-Kolmogorov cording to Theorem 3.3. So 0 = ξ = ψ˜ − ψ. equation for K∞ follows from (3.13). ¤ 4. The operator interpretation We now assume that K is a non-degenerate Markov transition kernel. Definition 4.1. [1, 3.5.1] A linear operator A in a Banach space is a Hille-Yosida operator, if there exist constants M ≥ 1, ω ∈ R such that (ω, ∞) is contained in the resolvent set of A and k(λ − A)−n k ≤ M (λ − ω)−n ,

n = 1, 2, . . .

See [1, 3.17] for background and history of Hille-Yosida operators. As shown in [13], there exists a Hille-Yosida operator A on M(Ω) such that A is uniquely determined by the Laplace transforms of K. Theorem 4.2. Let K be a non-degenerate Markov transition kernel. Then there exists a Hille-Yosida operator A in M(Ω) such that Z Z ∞ −1 [(λ − A) µ](D) = µ(dx) dte−λt K(t, x, D) Ω

0

for all λ > ω, D ∈ B, µ ∈ M(Ω). Through this formula, K and A determine each other in a unique way. Notice that the uniqueness of K does not just follow from uniqueness properties of the Laplace transform because K does not need to have any continuity properties. K is also uniquely determined by A as the solution of a forward equation [13]. Theorem 4.3. Let K be a non-degenerate transition kernel and A the Hille-Yosida operator associated with K in Theorem 4.2. Then Z t dsK(s, x, ·) K(t, x, ·) = δx + A 0

for all t ≥ 0, x ∈ Ω, where δx is the Dirac measure concentrated at x. K is uniquely determined by this equation. Even the following holds: if K is a transition kernel which satisfies this equation and sup

K(t, x, Ω) < ∞

∀σ > 0,

0≤t≤σ,x∈Ω

then K is a Markov transition kernel and the Laplace transform of K is related to the resolvent of A as in Theorem 4.2.

Perturbation of transition functions

11

We call A the Hille-Yosida operator associated with the Markov transition kernel K. How is the perturbation of a Markov transition function by an output kernel reflected on the operator level? Theorem 4.4. Let L be an output kernel for K satisfying (3.8). Then the perturbed Markov transition kernel K∞ is non-degenerate and has the associated Hille-Yosida operator A + B where B is a linear operator from D(A) to M(Ω) which satisfies Z Z −1 [B(λ − A) µ](D) = µ(dx) L(x, d(t, y))e−λt , (4.1) Ω R+ ×D µ ∈ M(Ω), D ∈ B. Proof. We apply Lemma 3.1 with g(s, ·) = e−λs χD , Z Z L(z, d(s, y))e−λs K(t, x, dz) R+ ×D Ω Z = L(x, d(s, y))e−λ(s−t) ∀t ≥ 0, x ∈ Ω. [t,∞)×D

We multiply this equation by e−νt and integrate over t from 0 to ∞, Z ∞ Z Z −νt dt e K(t, x, dz) L(z, d(s, y))e−λs 0 Ω R+ ×D Z ∞ Z = dt e−νt L(x, d(s, y))e−λ(s−t) Z

0

[t,∞)×D

=

L(x, d(s, y)) R+ ×D

e−λs − e−νs , ν−λ

∀t ≥ 0, x ∈ Ω.

We introduce the following operators F (λ) on M(Ω), Z Z (4.2) [F (λ)µ](D) = µ(dx) L(x, d(s, y))e−λs . Ω

R+ ×D

The previous equation can be rewritten in operator form as (4.3)

F (λ)(ν − A)−1 =

1 (F (λ) − F (ν)). ν−λ

This equation is called resolvent output identity and the operator family F a resolvent output [16]. By (4.3), the following definition does not depend on the choice of λ, (4.4)

Bx = F (λ)(λ − A)x,

x ∈ D(A).

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T. Lant, H.R. Thieme

In order to determine the generator of K∞ , we take the Laplace transform of equation (3.9) and apply Fubini’s theorem, Z ∞ e−λt K∞ (t, x, D)dt Z0 ∞ = e−λt K(t, x, D)dt 0 Z Z ∞ −λs + L(x, d(s, y))e e−λt K∞ (t, y, D)dt. R+ ×Ω

0

We define the operators Rλ on M(D) by Z Z ∞ (4.5) (Rλ µ)(D) = µ(dx) e−λt K∞ (t, x, D)dt. Ω

0

The previous equation translates into Rλ = (λ − A)−1 + Rλ F (λ). By (3.8) and Lemma 3.2, the spectral radius of F (λ) is strictly smaller than 1 for all sufficiently large λ > 0 and so I − F (λ) is invertible, ¡ ¢−1 Rλ = (λ − A)−1 (I − F (λ))−1 = (I − F (λ))(λ − A) = (λ − A − B)−1 . This shows that the Markov transition kernel K∞ is non-degenerate and, by Theorem 4.2 and (4.5), that A + B is the associated HilleYosida operator. ¤ Example 4.5. We return to the example (3.3) with supx∈Ω β(x, Ω) < ∞. Then L(x, [0, r] × Ω) ≤ r sup β(y, Ω) sup K(s, x, Ω). y∈Ω

0≤s≤r

By Definition 2.2 (d), L(x, [0, r]×Ω) → 0 as r → 0 and (3.8) is satisfied. Taking the Laplace transform of (3.3) we see from (4.1) and Theorem ˜ − A)−1 where B ˜ is the bounded linear 4.2 that B(λ − A)−1 = B(λ operator on M(Ω) given by Z ˜ [Bµ](D) = µ(dy)β(y, D). Ω

By Theorem 4.4, the Hille-Yosida operator associated with K∞ is given ˜ to D(A). by A∞ = A + B where B is the restriction of B

Perturbation of transition functions

13

5. Incorporating mortality Let K be a non-degenerate Markov transition kernel and A the associated Hille-Yosida operator (Theorems 4.2 and 4.3). Let ϑ(y) denote the mortality rate at state y (since µ is used for measure we choose ϑ because of the Greek word for death: ϑ´ ανατ oς, thanatos). Let us ¯ first assume that ϑ(y) = ϑ¯ is constant. Set ψ(t, x, D) = e−ϑt K(t, x, D). Then ψ is a non-degenerate Markov transition kernel. By Theorem ¯ Now consider 4.2, ψ has the associated Hille-Yosida operator A − ϑ. a general bounded mortality rate ϑ(y) ≥ 0. We choose some constant ϑ¯ ≥ ϑ. Starting from ψ above, we find a Markov transition function Kϑ which solves the following modification of (3.9), ¯

Kϑ (t, x, D) =e−ϑt K(t, x, D) Z t Z ¯ + ds e−ϑs K(s, x, dy)(ϑ¯ − ϑ(y))Kϑ (t − s, y, D). 0

Ω

By Example 4.5, with β(x, D) = ϑ¯ − ϑ(x), Kϑ is associated with the ¯ ϑ−ϑ ¯ Hille-Yosida operator A−ϑ+ = A−ϑ where (ϑµ)(dx) = µ(dx)ϑ(x). Since a non-degenerate transition function is uniquely determined by its associated Hille-Yosida operator, this construction does not depend ¯ on the choice of ϑ. Theorem 5.1. Let K be a Markov transition function with generator A and ϑ ∈ BM (Ω) be non-negative. Then there exists a Markov transition function Kϑ with associated Hille-Yosida operator A − ϑ. We have the following monotonicity result. Theorem 5.2. Let K be a Markov transition function with associated ˜ Hille-Yosida operator A and ϑ, ϑ˜ ∈ BM (Ω) be non-negative and ϑ ≤ ϑ. Then the Markov transition functions Kϑ and Kϑ˜ associated with A−ϑ and A − ϑ˜ respectively satisfy Kϑ ≥ Kϑ˜. ˜ + (ϑ˜ − ϑ), Kϑ and K ˜ are related by the Proof. Since A − ϑ = (A − ϑ) ϑ integral equation Z t Z ˜ − ϑ(y))Kϑ (t − s, y, D). Kϑ (t, x, D) = Kϑ˜ + ds Kϑ˜(s, x, dy)(ϑ(y) 0

Ω

Since all ingredients are non-negative, the asserted inequality follows. ¤ 6. Incorporating mortality in the non-autonomous case Let Ω be a measurable space with σ-algebra B, I a finite or infinite interval and ∆ = {(t, r) ∈ I 2 ; r ≤ t}. BR+ denotes the Borel σ-algebra

14

T. Lant, H.R. Thieme

on R+ and B∆ the Borel σ-algebra on ∆ with the respective standard topologies. A function K : ∆ × Ω × B → R is called a (non-autonomous) transition kernel if the following hold: (a) K(t, r, x, ·) is a non-negative measure on B for all (t, r) ∈ ∆, x∈Ω (b) K(t, r, x, D) is a measurable function of (t, r, x) for all D ∈ B from B∆ × B to BR+ . (c) K(r, r, x, D) = 1 if x ∈ D and K(r, r, x, D) = 0 if x ∈ Ω \ D. (d) There exist δ, c > 0 such that K(t, r, x, Ω) ≤ c for all whenever (t, r) ∈ ∆, t ≤ r + δ, x ∈ Ω. A transition kernel K is called a Markov transition kernel if it satisfies the Chapman-Kolmogorov equations, Z K(t, r, x, D) = K(t, s, y, D)K(s, r, x, dy) (6.1) Ω x ∈ Ω, r ≤ s ≤ t, r, s, t ∈ I. We consider the following integral equation for non-autonomous transition functions K and K∞ with K being given and K∞ being wanted, K∞ (t, r, x, D) =K(t, r, x, D) Z t Z (6.2) − ds K(s, r, x, dy)ϑ(s, y)K∞ (t, s, y, D), r

Ω

x ∈ Ω, (r, t) ∈ ∆, D ∈ B. The non-autonomous situation can be reduced to the autonomous one by incorporating time into the state space [2, (2.2.9)], (6.3)

˜ (r, x), B × D) K(t, =χB (t + r)K(t + r, r, x, D);

r ∈ I, B ∈ BI , D ∈ B.

Here BI denotes the Borel σ-algebra on I. K(t + r, x, D) is not defined if t + r ∈ / I, but in this case χB (t + r) = 0 which makes clear that ˜ ∞ is the whole expression needs to be interpreted as 0 in this case. K defined from K∞ in complete analogy. ˜ is an autonomous Markov transition funcOne easily checks that K tion if and only if K is a non-autonomous Markov transition function. The subsequent result follows from standard measure-theoretic approximation arguments.

Perturbation of transition functions

15

Lemma 6.1. For all f ∈ BM(I × Ω), Z ˜ (r, x), d(s, y))f (s, y) K(t, ZI×Ω = K(t + r, r, x, dy)f (t + r, y)χI (t + r). Ω

By this lemma, the non-autonomous integral equation (6.2) can rewritten as an autonomous integral equation of the form (3.4), ˜ ∞ (t, (r, x), B × D) = χB (t + r)K∞ (t + r, r, x, D) K =χB (t + r)K(t + r, r, x, D) Z t+r Z − χB (t + r) ds K(s, r, x, dy)ϑ(s, y)K∞ (t + r, s, y, D) r

Ω

˜ ∞ (t, (r, x), B × D) =K Z t Z ds K(s + r, r, x, dy)ϑ(s + r, y) − χB (t + r) 0

Ω

K∞ (t − s + r + s, s + r, y, D) ˜ ∞ (t, (r, x), B × D) =K Zt Z ˜ (r, x), d(u, y))ϑ(u, y)K∞ (t − s + u, u, y, D) − χB (t + r) ds K(s, 0

I×Ω

˜ ∞ (t, (r, x), B × D) =K Z t Z ˜ (r, x), d(u, y))ϑ(u, y)K ˜ ∞ (t − s, (u, y), D). − ds K(s, 0

I×Ω

The subsequent result follows from Theorem 5.1 and Theorem 5.2. Theorem 6.2. Let K be a non-autonomous transition kernel and ϑ : I × Ω → R+ be bounded and measurable. Then there exists a unique non-autonomous (not necessarily non-negative) transition kernel P∞ K∞ which solves (6.2). K∞ has the series representation K∞ = n=0 Kn with K0 =K, Z t Z Kn+1 (t, r, x, D) = − ds Kn (t, s, y, D)ϑ(s, y)K(s, r, x, dy). r

Ω

If K satisfies the Chapman-Kolmogorov equations, so does K∞ . If K is non-degenerate, K∞ is non-negative and K∞ depends on ϑ in a monotone decreasing way.

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T. Lant, H.R. Thieme

˜ is non-degenerate. We have the Here we call K non-degenerate if K same topological examples as in the autonomous case. Proposition 6.3. A non-autonomous transition kernel K is a nondegenerate transition kernel if one of the following two assumptions hold: (a) Ω is a σ-compact space (i.e. a locally compact space and a Kσ set) and K is weakly stochastically continuous, i.e., K(t, r, x, U ) → 1 as t → r+ whenever U is an open Kσ -set in Ω with compact closure and x ∈ U . (b) Ω is a normal space and K is stochastically continuous, i.e., K(t, r, x, U ) → 1 as t → r+ whenever U is an open Baire set in Ω and x ∈ U . ˜ inherits the (weak) stochastic continuity from It is easy to see that K K and Proposition 6.3 follows from Proposition 2.5. 7. A Feynman-Kac type formula Let Xt , t ∈ I, be a non-autonomous Markov process on a finite or infinite interval I that is associated with the Markov transition function ˜ S, P ) with a set K. More precisely there are a probability space (Ω, ˜ Ω, a σ-algebra S and a probability measure P and random variables ˜ → Ω, t ∈ I, such that Xt : Ω K(t, r, Xr , D) =E Xr χD (Xt ) = P Xr (Xt ∈ D),

(7.1)

D ∈ B, (t, r) ∈ ∆.

Here E Xr denotes the conditional expectation given Xr [3, 10.12] and P Xr the conditional probability given Xr . Recall that ∆ = {(t, r) ∈ I 2 ; t ≥ r}. We also assume that (t, y) 7→ Xt (y) is measurable from BR+ × B to B. This is the case, e.g., if Ω is normal and Xt (y) is a ˜ [13, Prop.B.1]. right-continuous function of t ∈ I for each y ∈ Ω Theorem 7.1. Let K∞ be the Markov transition kernel solving (6.2) with a non-negative bounded measurable function ϑ : I × Ω → R+ . Then the following Feynman-Kac type formula holds P -almost surely for all f ∈ BM(Ω), (t, r) ∈ ∆, µ Z ³ Z t ´¶ Xr K∞ (t, r, Xr , dy)f (y) = E f (Xt ) exp − ϑ(s, Xs )ds . Ω

r

Remark 7.1. We obtain the additional information that, if K is degenerate and we do not know whether K∞ is non-negative, we at least know that K∞ (t, r, Xr , D) ≥ 0, P -almost surely.

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17

f and set g∞ (t, r, x) = R To prove the theorem, we fix the function P K∞ (t, r, x, dy)f (y). By Theorem 6.2, g∞ = ∞ n=0 gn with Ω Z g0 (t, r, x) = K(t, r, x, dy)f (y), Ω (7.2) Z tZ gn+1 (t, r, x) = − gn (t, s, y)ϑ(s, y)K(s, r, x, dy). r

Ω

We show by induction, (−1)n Xr gn (t, r, Xr ) = E n!

µ

³Z t ´n ¶ f (Xt ) ϑ(u, Xu )du , r

and the formula will follow. By (7.1), for n = 0, g0 (t, r, Xr ) = E Xr f (Xt ) and, by (7.2), Z (7.3)

t

gn+1 (t, r, Xr ) = −

dsE Xr gn (t, s, Xs )ϑ(s, Xs ).

r

By induction hypothesis, for r ≤ s ≤ t, P -almost surely,

(7.4)

− E Xr [gn (t, s, Xs )ϑ(s, Xs )] · · ³Z t ´n ¸¸ (−1)n Xs Xr =E −ϑ(s, Xs ) E f (Xt ) ϑ(u, Xu )du n! s ¶n ¸¸ · · µZ t (−1)n+1 Xr Xs ϑ(u, Xu )du E E ϑ(s, Xs )f (Xt ) = n! s ¶n ¸ · µZ t (−1)n+1 Xr = E ϑ(s, Xs )f (Xt ) ϑ(u, Xu )du . n! s

Here we have used various properties of conditional expectations [3, Sec.10.1] and the assumption that Xt is a Markov process [3, 12.4.1]. We combine (7.3) and (7.4), · Z t ³Z t ´n ¸ (−1)n+1 Xr gn+1 (t, r, x) = E f (Xt )ϑ(s, Xs ) ϑ(u, Xu )du ds n! r s · µZ t ¶n ¸ Z t (−1)n+1 Xr E = f (Xt ) ϑ(s, Xs ) ϑ(u, Xu )du ds n! r s " µZ t ¶n+1 # (−1)n+1 Xr . E f (Xt ) ϑ(s, Xs )ds = (n + 1)! r

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T. Lant, H.R. Thieme

7.1. Unbounded mortality rates. We now assume that ϑ : (a, b) × Ω → R+ is unbounded and measurable. We choose a sequence of bounded measurable ϑn such that ϑn (t, x) % ϑ(t, x), e.g., ϑn (t, x) = min{n, ϑ(t, x)}. We also assume that the non-autonomous Markov transition function is non-degenerate. By Theorem 6.2, we obtain a decreasing sequence of non-negative transition functions Kn∞ which satisfy the Feynman-Kac relations µ Z ³ Z t ´¶ ∞ Xr Kn (t, r, Xr , dy)f (y) = E f (Xt ) exp − ϑn (s, Xs )ds Ω

r

for all f ∈ BM(Ω), P -almost surely. The limits K∞ (t, r, x, D) = lim Kn∞ (t, r, x, D) n→∞

exist, form a Markov transition function, and satisfy µ Z ´¶ ³ Z t Xr ϑ(s, Xs )ds . K∞ (t, r, Xr , dy)f (y) = E f (Xt ) exp − r

Ω

Here we have used that conditional expectations also satisfy the usual convergence theorems [3, (10.1.13)]. It is not clear whether K∞ is independent of the choice of the approximating sequence ϑn , but K∞ (t, r, Xr , D) is independent P -almost surely. Appendix A. Non-autonomous perturbation by output kernels We add for the interested reader how non-autonomous Markov transition kernels can be perturbed by general output kernels. This has already been done in [5], but we feel that the reduction to the autonomous case and the perturbation on the level of transition kernels rather than output kernels presents a more accessible route. As a trade off, we lose the information about the perturbed output kernels obtained in [5]. Let I be a finite or infinite interval. For r ∈ I, let Ir = I ∩ [r, ∞). Recall that ∆ = {(t, r) ∈ I 2 ; t ≥ r}. Assumption A.1. Let K : ∆ × Ω × B → R+ be a non-autonomous transition kernel. A measure kernel L : (I × Ω) × (BI × B) → R+ is called an output kernel for K, if the following hold: (a) For all r ∈ I and x ∈ Ω, L(r, x, I × Ω) = L(r, x, Ir × Ω). (b) There exists some s > 0 such that ¡ ¢ sup L r, x, ([r, r + s] ∩ I) × Ω < ∞. r∈I,x∈Ω

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19

(c) L satisfies the output relation with respect to K: if r, s ∈ ∆ and Γ ∈ BI × B, then Z K(s, r, x, dy)L(s, y, Γ) = L(r, x, Γ). Ω

Theorem A.1. Let K be a non-autonomous Markov transition kernel and L an output kernel for K. Assume that, for some ² > 0, Z ¡ ¢ ¡ ¢ (A.1) sup L r, x, d(s, y) L s, y, ([s, s + ²] ∩ I) × Ω < 1. r∈I,x∈Ω ([r,r+²]∩I)×Ω

Then there exists a unique non-autonomous Markov transition kernel K∞ such that Z (A.2) K∞ (t, r, x, D) = K(t, r, x, D) + L(r, x, dz)K∞ (t, z, D) [r,t)×Ω

whenever (t, r) ∈ ∆, x ∈ Ω, D ∈ B. The rest of this appendix is devoted to the proof of this theorem which consists in a transformation to the autonomous situation. To ˜ = I ×Ω this end we incorporate time into the state space and set Ω ˜ with the product σ-algebra BI × B. We have already and endow Ω explained in (6.3) how to translate a non-autonomous transition kernel into an autonomous one. Now we do the translation for the output kernels. ˜ of BR+ × BI × B, we define For any r ∈ I, x ∈ Ω and every subset Γ Z ¡ ¢ ˜ (r, x), Γ ˜ = (A.3) L L(r, x, d(s, y))χΓ˜ (s − r, s, y). Ir ×Ω

If r ∈ I and B0 is a Borel set in R+ and B1 a Borel set in I, this formula specializes to ³ ´ ¡ ¢ ¡ ¢ ˜ (A.4) L (r, x), B0 × B1 × D = L r, x, (B0 + r) ∩ B1 × D , where B0 + r = {s + r; s ∈ B0 }. Further, for every measurable function f : R+ × I × Ω → R which is non-negative or bounded, Z ¡ ¢ ˜ (r, x), d(u, s, y) f (u, s, y) L R ×I×Ω Z + (A.5) ¡ ¢ = L r, x, d(s, y) f (s − r, s, y), Ir ×Ω

where Ir = I ∩ [r, ∞). We define a non-autonomous convolution between K and L as follows: Z (A.6) (K ∗ L)(t, r, x, D) = L(r, x, d(s, y))K(t, s, y, D), Ir ×Ω

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T. Lant, H.R. Thieme

where Ir = I ∩ [r, ∞). This convolution is compatible with the analogous autonomous convolution between the autonomized kernels. ˜ = I × Ω, Γ ∈ BI × B, Lemma A.2. For all x˜ = (r, x) ∈ Ω Z ˜ x, d(u, z))K(t ˜ − u, z, Γ). ^ K ∗ L(t, x˜, Γ) = L(˜ ˜ [0,t)×Ω

Proof. By the definition of product σ-algebras, it is sufficient to consider Γ = B × D with B ∈ BI and D ∈ B. Further let t ≥ 0, r ∈ I, x ∈ Ω. By (6.3), ^ K ∗ L(t, (r, x), B × D) =χB (t + r)(K ∗ L)(t + r, r, x, D) Z =χB (t + r) L(r, x, d(s, y))K(t + r, s, y, D) [r,t+r)×Ω Z ˜ − (s − r), (s, y), B × D). = L(r, x, d(s, y))K(t [r,t+r)×Ω

By (A.5), ^ K ∗ L(t, (r, x), B × D) Z ˜ ˜ − u, (s, y), B × D). = L((r, x), d(u, s, y))K(t [0,t)×I×Ω

¤ By Lemma A.2, equation (A.2) translates to equation (3.4) with ˜ ∞, φ = K ˜ and L ˜ replacing L. In order to have Theorem 3.3 ψ = K imply the existence of the transition function K∞ in Theorem A.1 it remains to check that assumption (A.1) implies assumption (3.8). This is the case, indeed, as can be seen by the following equality. By (A.3), Z ˜ ˜ L((r, x), [0, ²], d(s, y))L((s, y), [0, ²] × I × Ω) I×Ω Z ˜ = L(r, x, d(s, y))χ[0,²] (s − r)L((s, y), [0, ²] × I × Ω) Ir ×Ω Z = L(r, x, d(s, y))L(s, y, ([s, s + ²] ∩ I) × Ω). (I∩[r,r+²])×Ω

Finally K∞ is a Markov transition function by Theorem 3.4 and the following result. ˜ is a cumulative output for K. ˜ Lemma A.3. L

Perturbation of transition functions

21

Proof. Let r ∈ I, x ∈ Ω, u ≥ 0, B1 ∈ BI and D ∈ B. By Lemma 6.1, Z ˜ (r, x), d(s, y))L((s, ˜ K(t, y), [0, u] × B1 × D) I×Ω Z ¡ ¢ ˜ (t + r, y), [0, u] × B1 × D χI (t + r). = K(t + r, r, x, dy)L Ω

By (A.3), (A.4) and the output relation for L, this equals Z ¡ ¢ = K(t + r, r, x, dy)L t + r, y, ([t + r, t + r + u] ∩ B1 ) × D Ω ³ ´ £ ¢ ¡ ¢ ˜ (r, x), [t, t + u] × B1 × D . =L r, x, t + r, t + r + u] ∩ B1 ×D = L So, for all Γ ∈ BI × B, Z ¡ ¢ ˜ (r, x), dz)L(z, ˜ [0, u] × Γ) = L ˜ (r, x), [t, t + u] × Γ , K(t, I×Ω

˜ and K. ˜ (3.1) follows from Assumption A.1 and (3.2) is satisfied for L (b) and (A.4). ¤ Acknowledgement. The authors thank Doug Blount, Mats Gyllenberg, and an anonymous referee for useful comments. References [1] W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander, Vectorvalued Laplace Transforms and Cauchy Problems, Birkh¨auser, Basel 2001. [2] L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley, New York 1974 [3] H. Bauer, Probability Theory and Elements of Measure Theory, Academic Press, London 1981 [4] M. Demuth, J.A. van Casteren, Stochastic Spectral Theory for Selfadjoint Feller Operators, Birkh¨auser, Basel 2000 [5] O. Diekmann, M. Gyllenberg, J.A.J. Metz, H.R. Thieme, On the formulation and analysis of deterministic structured population models. I. Linear theory, J. Math. Biol. 43 (1998), 349-388 [6] J. Dugundji, Topology, Allyn and Bacon, Boston 1966, Prentice-Hall of India, New Delhi 1975 [7] S.N. Ethier, T.G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, New York 1986 [8] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Wiley 1965 [9] A. Friedman, Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York 1975 [10] J.A. Goldstein, Semigroups of Linear Operators and Application, Oxford University Press, New York 1985

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[11] M. Gyllenberg, T. Lant, H.R. Thieme, Perturbing dual evolutionary systems by cumulative outputs, Differential Integral Equations, 19 (2006), 401436 [12] T. Lant, Transition Kernels, Integral Semigroups on Spaces of Measures, and Perturbation by Cumulative Outputs, Dissertation, Arizona State University, Tempe, December 2004 [13] T. Lant, H.R. Thieme, Markov transition functions and semigroups of measures, Semigroup Forum, to appear [14] B. Øksendal, Stochastic Differential Equations, Springer, Berlin Heidelberg 1985 [15] K. Taira, Semigroups, Boundary Value Problems, and Markov Processes, Springer, Berlin Heidelberg 2004 [16] H.R. Thieme, Positive perturbations of dual and integrated semigroups, Adv. Math. Sci. Appl. 6, 445-507, 1996 [17] J.A. van Casteren, Generators of Strongly Continuous Semigroups, Pitman, Boston 1985 † present address: Decision Center for a Desert City, Arizona State University, PO Box 878209, Tempe, AZ 85287-8209, U.S.A. E-mail address: [email protected] †¦

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, U.S.A. E-mail address: [email protected]