A note on extremal invariant measures for a Markov process^{1}

Nobuo Yoshida^{2}

1 Introduction

It is sometimes the case that a Markov process has more than one invariant measures and yet the process has certain ergodic property with respect to some invariant measures. In the field of interacting particle, the above phenomenon is very common ([Lig85]) and this note comes out of an attempt to understand such situation.

We begin by introducing the standard setup of the theory of time homogeneous Markov processes.

The state space: The state space X of our Markov process will be a separable metric space with the Borel σ-field B(X). We will introduce the folowing notations;

B_{b}(X) = the set of all bounded B(X)-measurable functions on X,  (1.1)
C_{b}(X) = the set of all bounded continuous functions on X,  (1.2)
P(X) = the set of all probability measures on (X, B(X)).  (1.3)

The set P(X) is topolized with Lévy-Prohorov metric;

ρ(μ, ν) = inf{ε > 0 | μ(F) ≤ ν(F^{ε}) + ε}  (1.4)

where the infimum is taken over all closed subset F of X and F^{ε} denotes the ε-neighbourhood of the set F. Then, P(X) is again a separable metric space ([EK86, page 101]).

The transition function: The Markovian transition function which we will work on is a family {p_{t}(x, ·) ∈ P(X)|x ∈ X, t > 0} which satisfies the following;

(P1) p_{0}(x, ·) = δ_{x} for all x ∈ X,

(P2) p_{t}(x, A) is jointly measurable in (t, x) ∈ [0, ∞) × X for any fixed A ∈ B(X),

(P3) p_{t+s}(x, ·) = ∫ p_{t}(x, dy)p_{s}(y, ·) for all t, s ∈ [0, ∞).

For t > 0, f ∈ B_{b}(X) and μ ∈ P(X), S_{t}f ∈ B_{b}(X) and μS_{t} ∈ P(X) are defined respectively by

S_{t}f = ∫ p_{t}(·, dy)f(y), μS_{t} = ∫ μ(dx)p_{t}(x, ·).  (1.5)

We will assume that C_{b}(X) is closed under (S_{t});

(P4) S_{t}f ∈ C_{b}(X) for all t > 0 and f ∈ C_{b}(X).

The set of invariant measure I is defined by

I = ⋂_{t>0}{μ ∈ P(X) | μS_{t} = μ}.  (1.6)

1Notes for a seminar talk on May 13, 1998 at Kyoto University

2Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan, email:nobuo@kusm.kyoto-u.ac.jp, URL:http://www.kusm.kyoto-u.ac.jp/ nobuo

The Markov process: W will stand for the set of all function from [0, ∞) to X. The value of w ∈ W at t ∈ [0, ∞) is denoted either by w(t) or w_{t}. For t > 0, B_{t}(W) denotes a σ-field on W generated by sets of the form ⋂_{i=1}^{n}{w ∈ W | w(t_{i}) ∈ B_{i}} with n = 1, 2, …, t_{1} < … < t_{n} ≤ t, and B_{1}, …, B_{n} ∈ B(X). B(W) is the σ-field generated by ⋃_{t>0} B_{t}(W). Time-shifts (θ_{t})_{t≥0} is a one parameter semi-group of transformations on W defiend by

θ_{t}w = w(t + ·).  (1.7)

A family {P_{x} ∈ P(W, B(W)) | x ∈ X} is said to be the Markov process with the transition function {p_{t}(x, ·) | x ∈ X, t > 0} if

P_{x} (w_{t+s} ∈ A|B_{t}(W)) = p_{s}(w_{t}, A) P_{x}-a.e.  (1.8)

for any x ∈ X, s, t ∈ [0, ∞) and A ∈ B(X). We will assume that the following;

(P5) The transition function {p_{t}(x, ·) | x ∈ X, t > 0} and a Markov process {P_{x} | x ∈ X} is related as (1.8).

This is indeed the case when X is complete ( See e.g., [EK86, page 157]) .

We have the following;

Theorem 1.1 Suppose that the transition function satisfies (P1)–(P5). Then for ν ∈ I and for any fixed 1 ≤ p < ∞, following are equivalent;

(a1) ν ∈ I_{e} and there is an X_{0} ∈ B(X) such that ν(X_{0}) = 1 and (\frac{1}{t} ∫_{0}^{t} dτδ_{x}S_{τ} )_{t>0} is relatively compact in P(X) for all x ∈ X_{0}, where I_{e} is a subset of I defined by

I\I_{e} = {μ ∈ I | μ = αμ_{1} + (1 − α)μ_{2} for some α ∈ (0, 1) and μ_{i} ≠ μ}.  (1.9)

(b1) lim_{t→∞} ∥\frac{1}{t} ∫_{0}^{t} dτS_{τ}f − νf∥_{L^{p}(ν)} = 0 for all f ∈ L^{p}(ν).

(c1) lim_{t→∞} ν \{x ∈ X | |\frac{1}{t} ∫_{0}^{t} dτS_{τ}f(x) − νf| ≥ ε\} = 0 for all f ∈ C_{b}(X) and ε > 0.

(d1) lim_{t→∞} ν \{x ∈ X | ρ (\frac{1}{t} ∫_{0}^{t} dτδ_{x}S_{τ}, ν) ≥ ε\} = 0 for all ε > 0 (cf. (1.4)).

(e1) lim_{t→∞} ∫ ν(dx)F (\frac{1}{t} ∫_{0}^{t} dτδ_{x}S_{τ} ) = F (ν) for all F ∈ C_{b} (P(X)).

(f1) lim_{t→∞} \frac{1}{t} ∫_{0}^{t} dτν(S_{τ}f · g) = ν(f)ν(g) for all f, g ∈ L^{2}(ν).

(g1) If f ∈ L^{p}(ν) is such that ν{x ∈ X | S_{t}f(x) = f(x)} = 1 for all t > 0, then ν{x ∈ X | f(x) = νf} = 1.

(h1) P_{ν} is ergodic with respect to the time shift (θ_{t})_{t≥0}, i.e., if Φ ∈ L^{∞}(P_{ν}) is such that

P_{ν}{w ∈ W | Φ ∘ θ_{s}(w) = Φ(w)} = 1 for all s > 0,  (1.10)

then

P_{ν}{w ∈ W | Φ(w) = P_{ν}Φ} = 1.  (1.11)

Remark 1.1 If the state space X is compact (and hence so is P(X)), (a1) says nothing but ν ∈ I_{e}.

It is sometimes more imprtant to disscuss the convergence of (S_{t}f)_{t>0} itself, for which we have the following;

Theorem 1.2 Suppose that the transition function satisfies (P1)–(P5) and consider the following conditions for ν ∈ I and 1 ≤ p < ∞.

(b2) lim_{t→∞} ∥S_{t}f − νf∥_{L^{p}(ν)} = 0 for all f ∈ L^{p}(ν).

(c2) lim_{t→∞} ν {x ∈ X | |S_{t}f(x) − νf| ≥ ε} = 0 for all f ∈ C_{b}(X) and ε > 0.

(d2) lim_{t→∞} ν {x ∈ X | ρ (δ_{x}S_{t}, ν) ≥ ε} = 0 for all ε > 0 (cf. (1.4)).

(e2) lim_{t→∞} ∫ ν(dx)F (δ_{x}S_{t}) = F (ν) for all F ∈ C_{b} (P(X)).

(f2) lim_{t→∞} ν(S_{t}f · g) = ν(f)ν(g) for all f, g ∈ L^{2}(ν).

(h2) P_{ν} is mixing with respect to the time shift (θ_{t})_{t≥0}, i.e.,

lim_{t→∞} P_{ν}(F ∘ θ_{t} · G) = P_{ν}(F)P_{ν}(G) for any F, G ∈ L^{2}(P_{ν}).  (1.12)

Then, (b2) ⇔ (c2) ⇔ (d2) ⇔ (e2) ⇒ (f2) ⇔ (h2) ⇒ (g1).

Corollary 1.3 Suppose that ν ∈ I is such that (S_{t})_{t≥0} can be extended to a strongly continuous, self-adjoint semi-group on L^{2}(ν). Then, (a1)–(h1) in Theorem 1.1 and (b2)– (h2) in Theorem 1.2 are all equivalent.

Remark 1.2 A typical example which Corollary 1.3 is applied to is the stochastic Ising models ([HS76], [Lig85, Chapter4]) with ν being a Gibbs measure.

Proof of Corollary 1.3, given Theorem 1.1 and Theorem 1.2: It is enough to prove (g1) ⇒ (a2). Let E : L^{2}(ν) → L^{2}(ν) be the projection to the closed subspace: ⋂_{t>0}{f ∈ L^{2}(ν) | S_{t}f = f}. Then it follows from the spectral calculus that

lim_{t→∞} ∥S_{t}f − Ef∥_{L^{2}(ν)} = 0 for all f ∈ L^{2}(ν).  (1.13)

Since Ef is, by definition, (S_{t})_{t>0}-invariant and hence is a constant ν-a.e.. Therefore (1.13) proves (a2) for p = 2 (and thus for all 1 ≤ p < ∞). □

2 Proof of Theorem 1.1 and Theorem 1.2

Proof of Theorem 1.1: Before going into the proof, let us collect some facts which follow merely from the P_{ν}-invariance of the time-shift (θ_{t})_{t≥0}. First of all, we see by Birkoff’s ergodic theorem that for any f ∈ L^{p}(ν), there is a Φ_{f} ∈ L^{p}(P_{ν}) such that lim_{t→∞} \frac{1}{t} ∫_{0}^{t} dτf(w_{τ}) = Φ_{f}, P_{ν}-a.e. and in L^{p}(P_{ν}). Therefore, as t ↗ ∞,

∫ ν(dx) |\frac{1}{t} ∫_{0}^{t} dτS_{τ}f(x) − P_{x}Φ_{f} |^{p} ≤ ∫ P_{ν}(dw) |\frac{1}{t}∫ _{0}^{t}dτf(w_{t}) − Φ_{f}(w)|^{p} → 0  (2.1)

We have for any s > 0 that P_{ν}{w ∈ W | Φ_{f} ∘ θ_{s}(w) = Φ_{f}(w)} = 1, since the limit lim_{t→∞} \frac{1}{t} ∫_{0}^{t} dτf(w_{s+τ}) is independent of s. This implies that

ν{x ∈ X | P_{x}Φ_{f} ∘ θ_{s} = P_{x}Φ_{f}} = 1 for s > 0,  (2.2)

since

∫ ν(dx) |P_{x}Φ_{f} ∘ θ_{s} − P_{x}Φ_{f}| ≤ P_{ν} |Φ_{f} ∘ θ_{s} − Φ_{f}| .

The following implications are standard;

(g1) ⇐ (f1) ⇐ (b1) ⇔ (c1) ⇔ (d1) ⇔ (e1).  (2.3)

(a1) ⇒ (b1): To prove this, we may and will assume that X is totally bounded, since X is homeomprphic to a totally bounded metric space. Also, it is sufficient to prove the desired L^{p}- convergence for f ∈ C_{u}(X), the set of uniformly continuous function, since C_{u}(X) is dense in L^{p}(ν) (To see this, just note that the indicator function of any open subset in X is a increasing limit of a sequence in C_{u}(X → [0, 1]).). By (2.1), it is enough to prove that

ν{x ∈ X | P_{x}Φ_{f} = νf} = 1 for all f ∈ C_{u}(X).  (2.4)

This will be done by proving (2.5) and (2.8) below.

We first prove that there is X_{0} ∈ B(X) with ν(X_{0}) = 1, a function x ↦ ν_{x} from X_{0} to I such that

ν{x ∈ X | P_{x}Φ_{f} = ν_{x}f} = 1 for all f ∈ C_{u}(X).  (2.5)

Note that the space C_{u}(X) has a dense countable subset. By this and (2.1), we can pick an X_{1} ∈ B(X) and a sequence t_{n} ↗ ∞ such that ν(X_{1}) = 1 and

lim_{n→∞} \frac{1}{t_{n}} ∫ _{0}^{t_{n}}dτS_{τ}f(x) = P_{x}Φ_{f} for all x ∈ X_{1} and f ∈ C_{u}(X).  (2.6)

Furthermore, since (\frac{1}{t} ∫_{0}^{t} dτδ_{x}S_{τ} )_{t>0} is relatively compact ν(dx)-a.e., there is an X_{2} ∈ B(X) with ν(X_{2}) = 1 and a map x ↦ ν_{x} from X_{1} to P(X) such that

lim_{n→∞} \frac{1}{t_{n}(x)}∫ _{0}^{t_{n}(x)}dτS_{τ}f(x) = ν_{x}f for all x ∈ X_{2} and f ∈ C_{u}(X),  (2.7)

where t_{n}(x) ↗ ∞ is a subsequence of t_{n} ↗ ∞. Note that the choice of t_{n}(x) ↗ ∞ is independent of f, hence so is the choice of ν_{x}. Note also that ν_{x} ∈ I for all x ∈ X_{2}, as can easily be seen by using (P4) (cf. [Lig85, page 10, Proposition 1.8]). It follows from (2.6) and (2.7) that P_{x}Φ_{f} = ν_{x}f for all x ∈ X_{1} ∩ X_{2} =: X_{0} and f ∈ C_{u}(X), which proves (2.5).

We will next prove that

ν{x ∈ X | ν_{x} = ν} = 1.  (2.8)

We see from (2.4) and (2.6) that for any f ∈ C_{u}(X),

∫ ν(dx)ν_{x}f = lim_{n→∞} \frac{1}{t_{n}} ∫ _{0}^{t_{n}}dτ ∫ ν(dx)S_{τ}f(x)
= νf.  (2.9)

Suppose now that (2.8) is not true. Then it follows that there is h ∈ C_{u}(X) and A ∈ B(X) such that 0 < ν(A) < 1 and that

∫_{A}ν(dx)ν_{x}h ≠ ν(A)νh.  (2.10)

If we define ν_{A}, ν_{A^{c}} ∈ I by ν_{A} = ν(A)^{−1}∫_{A}ν(dx)ν_{x}(·) and ν_{A^{c}} = ν(A^{c})^{−1}∫_{A^{c}}ν(dx)ν_{x}(·), we then have that

ν = ν(A)ν_{A} + ν(A^{c})ν_{A^{c}},  (2.11)
ν ∉ {ν_{A}, ν_{A^{c}}}.  (2.12)

In fact (2.11) follows from (2.9) and (2.12) from (2.10). Since (2.11) and (2.12) contradict with ν ∈ I_{e}, we have proved (2.8).

(b1) ⇒ (a1): The relative compactness alluded in (a1) is obvious by (e1) (cf. (2.3)). To prove that ν ∈ I_{e}, suppose the contrary: there is α ∈ (0, 1) and ν_{1}, ν_{2} ∈ I\{ν} such that ν = αν_{1} + (1 − α)ν_{2}. Then it follows for all f ∈ B_{b}(X) that

lim_{t→∞} ∥\frac{1}{t} ∫_{0}^{t} dτS_{τ}f − νf∥_{L^{p}(ν_{1})} = 0

and hence, ν_{1}f = lim_{t→∞} \frac{1}{t} ∫_{0}^{t} dτν_{1}(S_{τ}f) = νf, which contradicts with what we have assumed.

(g1) ⇒ (h1): The proof we present here is taken from [HS76, (1.11)]. Suppose that Φ ∈ L^{∞}(P_{ν}) satisfies (1.10). We will prove (1.11). We first see that Φ satisfies ν{x ∈ X | P_{x}Φ∘θ_{s} = P_{x}Φ} = 1 in the same way as Φ_{f} did in (2.2). Then it follows from (g1) that ν{x ∈ X | P_{x}Φ = P_{ν}Φ} = 1. Here, we have used the Markov property of the process to check that x ↦ P_{x}Φ is (S_{t})_{t>0}-invariant. Now pick s > 0 and A ∈ B_{s}(W) arbitrarily and define a measure μ_{s} on (W, B(W)) by μ_{s} = P_{ν}(A∩{w_{s} ∈ ·}). Since μ_{s} ≤ ν, we have that μ_{s}{x ∈ X | P_{x}Φ ≠ P_{ν}Φ} = 0. Noting this, we conclude that

P_{ν}(1_{A}Φ) = P_{ν}(1_{A}Φ ∘ θ_{s}) = ∫_{A}P_{ν}(dw)P_{w(s)}Φ
= ∫ μ_{s}(dx)P_{x}Φ = P_{ν}(A)P_{ν}Φ.  (2.13)

Since s > 0 and A ∈ B_{s}(W) were arbitrary, (2.13) proves (1.11).

(h1) ⇒ (b1): It follows from (h1) that Φ_{f} in (2.1) is a constant P_{ν}-a.e. and hence P_{x}Φ_{f} is a constant ν(dx)-a.e.. Thus, (b1) follows from (2.1). □

Proof of Theorem 1.2: The following implications are standard;

(h2) ⇒ (f2) ⇐ (b2) ⇔ (c2) ⇔ (d2) ⇔ (e2), (f2) ⇒ (g1).

It is thus, enough to prove that (f2) implies (1.12). To prove this, we will assume that F and G are given by

F (w) = ∏_{i=1}^{m} f_{i}(w(r_{i})), G(w) = ∏_{j=1}^{n} g_{j}(w(s_{j}))  (2.14)

where r_{1} < … < r_{m} and s_{1} < … < s_{n} and f_{i}, g_{j} ∈ B_{b}(X → [0, ∞)). Since the functions of the form given as above span a dense subspace in L^{p}(P_{ν}), we may assume (2.14) without loss. Now, for h_{1}, h_{2}, … ∈ B_{b}(X) and τ_{1}, τ_{2}, … ∈ [0, ∞) we define S_{τ_{1},...,τ_{n}}^{h_{1},...,h_{n}} ∈ B_{b}(X) by

S_{τ_{n},...,τ_{1}}^{h_{n},...,h_{1}} = \{ \begin{matrix}S_{τ_{1}}h_{1} & if n = 1 \\ S_{τ_{n}} (h_{n}S_{τ_{n−1},...τ_{1}}^{h_{n−1},...,h_{1}}) & if n ≥ 2. \end{matrix}  (2.15)

We then have by a repeated use of the Markov property that for F, G in (2.14)

P_{ν}(F ∘ θ_{t} · G) = P_{ν} (GS_{r_{1}+t−s_{n}}(f_{1}S_{τ_{2},...,τ_{m}}^{f_{2},...,f_{m}})(w(s_{n})))
= ∫ μ(dx)S_{r_{1}+t−s_{n}}(f_{1}S_{τ_{2},...,τ_{m}}^{f_{2},...,f_{m}})(x)  (2.16)

where τ_{j} = r_{j} − r_{j−1} (j = 2, …, m) and μ = P_{ν}(G1_{w(s_{n})∈·}). Since the measure μ is bounded by a constant multiple of ν, there is h ∈ L^{∞}(ν) such that dμ = hdν. Therefore, it follows from (2.16) and (f2) that

lim_{t→∞} P_{ν}(F ∘ θ_{t} · G) = lim_{t→∞} ν (hS_{r_{1}+t−s_{n}}(f_{1}S_{τ_{2},...,τ_{m}}^{f_{2},...,f_{m}}))
= lim_{t→∞} ν(h)ν (S_{r_{1}+t−s_{n}}(f_{1}S_{τ_{2},...,τ_{m}}^{f_{2},...,f_{m}}))
= P_{n}(G)P_{n}(F),  (2.17)

which proves (1.12) for F, G in (2.14). □

References

[EK86] Ethier, N. and Kurtz, T. : Markov Processes–Characterization and Convergence: John Wiley and sons (1986).

[HS76] Holley, R. and Stroock, D. W. : L_{2} theory for the stochastic Ising model: Z. Warsch. verw. Gebiete 35, 87–101, (1976).

[Lig85] Ligget, T. M. : Interacting Particle Systems”, Springer Verlag, Berlin-Heidelberg-Tokyo (1985).