Abstract
We provide sufficient conditions for the uniqueness of an invariant measure of a Markov process as well as for the weak convergence of transition probabilities to the invariant measure. Our conditions are formulated in terms of generalized couplings. We apply our results to several SPDEs for which unique ergodicity has been proven in a recent paper by Glatt-Holtz, Mattingly, and Richards and show that under essentially the same assumptions the weak convergence of transition probabilities actually holds true.
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Acknowledgements
The authors are deeply grateful to anonymous referees for the attention paid to the paper and for valuable comments and remarks. The authors thank Benjamin Gess for a fruitful discussion and instructive comments.
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Appendices
Appendix 1: proofs of Propositions 2 and 4
Proof of Proposition 2
I. Take an arbitrary \(\xi \in {\widehat{C}}^R_p(\mathbb {P},\mathbb {Q})\), and consider the sets
\(\gamma \in (0,1).\) Define the sub-probability measure \(\eta _\gamma \) on \((E^\infty \times E^\infty , \mathcal {E}^{\otimes \infty }\otimes \mathcal {E}^{\otimes \infty })\) by
Then the “marginal distributions” \(\pi _i(\eta _\gamma ), i=1,2\) (which now are sub-probability measures, as well) satisfy
Denote
then each of the measures \(\mathbb {P}-\pi _1(\eta _\gamma ), \mathbb {Q}-\pi _2(\eta _\gamma )\) has total mass \(1-\beta _\gamma .\) We put
which by construction belongs to \(C(\mathbb {P},\mathbb {Q})\). Let us show that \(\gamma \) can be chosen small enough, so that \(\zeta =\zeta _\gamma \) possesses the required property.
Let \(\alpha >0\). For \(A \in \mathcal {E} \otimes \mathcal {E}\) satisfying \(\xi (A)\ge \alpha \), we have
Next,
and by the definition of the sets \(B_\gamma ^i, i=1,2\)
Hence, if \(\gamma \) is taken small enough for \(4\gamma ^{p-1}R^p\le \alpha \), for every A with \(\xi (A)\ge \alpha \) we have for \(\zeta =\zeta _\gamma \)
which completes the proof of statement I.
II. We fix \(\xi \in {\widehat{C}}(\mathbb {P},\mathbb {Q})\) and modify slightly the construction from the previous part of the proof. Let \(B_\gamma ^i, i=1,2, C_\gamma \) be as above, then we define
We fix \(\gamma \in (0,1)\) small enough, so that
where \(\alpha \) is as in the statement of the lemma.
We have \({\widetilde{\eta }}_\gamma =\gamma ^{-1}\eta _\gamma \), and thus the total mass of the measure \({\widetilde{\eta }}_\gamma \) equals \(\gamma ^{-1}\beta _\gamma =\xi (C_\gamma )\le 1\). In addition,
and the total mass for each of the measures \(\gamma ^{-1}\mathbb {P}-\pi _1({\widetilde{\eta }}_\gamma ), \gamma ^{-1}\mathbb {Q}-\pi _2({\tilde{\eta }}_\gamma )\) equals \(\gamma ^{-1}(1-\beta _\gamma )\ge \gamma ^{-1}-1\). Then
is a probability measure with
that is, \({\widetilde{\zeta }}_\gamma (A)\ge \alpha /2\) as soon as \(\xi (A)\ge \alpha \). In addition, the marginal distributions of \({\widetilde{\zeta }}_\gamma \) equal
and their Radon-Nikodym densities w.r.t. \(\mathbb {P},\mathbb {Q}\) respectively are bounded by
hence \({\widetilde{\zeta }}_\gamma \in {\widehat{C}}_p^R(\mathbb {P},\mathbb {Q})\) for every \(p\ge 1\).
Proof of Proposition 4
There exist two increasing sequences \(K_1^n,\, K_2^n, n\ge 1\) of compact subsets of E such that \(K_1^n\cap K_2^n=\emptyset \), \(\nu _1(K_1^n)\ge 1-1/n\), and \(\nu _2(K_2^n)\ge 1-1/n\) for \(n\ge 1\). Let
Clearly \(\delta _n, n\ge 1\) is non-increasing and \(\delta _n>0\) for all \(n\ge 1\) since \(d:E \times E \rightarrow [0,\infty )\) is continuous with respect to \(\rho \otimes \rho \). On the other hand, for any \(\xi \in C(\nu _1,\nu _2)\) we have
proving the proposition.
Appendix 2: Jankov’s lemma and the proof of Proposition 1
Recall that a measurable space \((\mathbb {X}, \mathcal {X})\) is called (standard) Borel if it is measurably isomorphic to a Polish space equipped with its Borel \(\sigma \)-algebra. For any Borel space \((\mathbb {X}, \mathcal {X})\) and any set \(A\in \mathcal {X}\), this set endowed with its trace \(\sigma \)-algebra
is a Borel measurable space, see [14, Corollary 13.4].
Our proof of Proposition 1 is based on the following lemma.
Lemma 1
(Jankov’s lemma, [7, Appendix 3 §1]). Let \((\mathbb {X}, \mathcal {X})\), \((\mathbb {Y}, \mathcal {Y})\) be Borel measurable spaces and let \(f:\mathbb {Y}\rightarrow \mathbb {X}\) be a measurable mapping with \(f(\mathbb {Y})=\mathbb {X}\).
Then for any probability measure \(\nu \) on \((\mathbb {X}, \mathcal {X})\) there exists a measurable function \(\varphi : \mathbb {X}\rightarrow \mathbb {Y}\) such that \(f(\varphi (x))=x\) for \(\nu \)-a.a. \(x\in \mathbb {X}\).
In the framework of Proposition 1, we put \(\mathbb {X}=M, \mathcal {X}=\mathcal {E}_M\) (the trace \(\sigma \)-algebra), then \((\mathbb {X}, \mathcal {X})\) is a Borel space. We define \(\nu \) as the measure \(\mu \) conditioned by M.
Before proceeding with the construction, we mention several simple facts we will use. First, let \(\mathbb {S}\) be a Polish space and \(\mathcal {P}(\mathbb {S})\) be endowed by the corresponding Kantorovich–Rubinshtein metric. Then the subset \(\varDelta \subset \mathcal {P}(\mathbb {S})\) consisting of all \(\delta \)-measures (that is, measures concentrated in one point) is closed, and \(\mathbb {S}\) and \(\varDelta \) are isomorphic. Second, the mapping \(\theta \) from \(\mathcal {P}(E^\infty \times E^\infty )\) to \(\mathcal {P}(E\times E)\) which maps the law of \(\{(X_n, Y_n), n\ge 0\}\) to the law of \((X_0, Y_0)\) is (Lipschitz) continuous. Hence, the subset
is closed. In addition, the mapping \(\varrho :\varXi \rightarrow E\times E\) which transforms \(\xi \in \varXi \) to the (unique) point \((x,y)\in E\) such that \(\theta (\xi )=\delta _{(x,y)}\), is continuous. Then \(\varXi \) endowed with the trace \(\sigma \)-algebra is a Borel space and \(\varrho \) is a measurable mapping on this space with \(\varrho (\varXi )=E\times E\). Denote by \(\varrho _{1,2}\) the (measurable) mappings \(\varXi \rightarrow E\) such that \(\varrho (\xi )=(\varrho _1(\xi ), \varrho _2(\xi )), \xi \in \varXi .\)
Now we can proceed with the construction which deduces Proposition 1 from Jankov’s lemma. We fix \(x\in E\), put \( \mathbb {Y}= \mathbb {Y}_1\cap \mathbb {Y}_2\), and \(f(\xi )=\varrho _2(\xi ), \xi \in \mathbb {Y}\), where
Clearly, \(f(\mathbb {Y})=M\) and f is a restriction on \(\mathbb {Y}\) of a measurable mapping \(\varXi \rightarrow E\) (the projection of \(\varrho \) on the second coordinate). Hence in order to be able to apply Jankov’s lemma we need only to show that \(\mathbb {Y}\) is a measurable subset of \(\varXi \). Because \(\varrho _{1,2}\) are measurable and \(\{x\}, M\in \mathcal {E}\), the sets
are measurable.
Next, recall that for any two probability measures \(\mathbb {P}, \mathbb {Q}\) on \((E^\infty , \mathcal {E}^{\otimes \infty })\) one has \(\mathbb {P}\ll \mathbb {Q}\) if and only if, for every \(\varepsilon >0\) there exists \(\delta >0\) such that
Because \(E^\infty \) is a Polish space, there exists a countable algebra \(\mathcal {A}\) which generates \(\mathcal {E}^{\otimes \infty }\), and then for any \(\gamma >0, A\in \mathcal {E}^{\otimes \infty }\) there exists \(A_\gamma \in \mathcal {A}\) such that
Then in the above characterization of the absolute continuity the class \(\mathcal {E}^{\otimes \infty }\) can be replaced by \(\mathcal {A}\). Hence
where
Since the mappings \(\varrho _2:\varXi \rightarrow E, \pi _2:\varXi \rightarrow \mathcal {P}(E^\infty )\), \(E\ni v\mapsto \mathbb {P}_v\in \mathcal {P}(E^\infty )\), and
are measurable, each of the sets \(B_{m,k}(A)\) is measurable. Therefore the set
is measurable, as well. Finally, a similar and simpler argument shows that the set
is measurable (we omit the explicit expression for this set here). This proves measurability of \(\mathbb {Y}_1\). The proof of measurability for \(\mathbb {Y}_2\) is simpler and is omitted.
Summarizing, we have that \(\mathbb {Y}\) is a measurable subset of \(\varXi \) and therefore, being endowed with the trace \(\sigma \)-algebra, is a Borel space. We finish the proof of Proposition 1 by applying Jankov’s lemma to the Borel spaces \(\mathbb {X}\), \(\mathbb {Y}\), the mapping f, and the measure \(\nu \) specified above.
Appendix 3: Kuratovskii and Ryll-Nardzevski’s theorem and the proof of Proposition 3
Our proof of Proposition 3 is based on measurability and measurable selection results discussed in [21], Chapter 12.1. Let us survey the required results briefly.
Let \(\mathbb {X}\) be a Polish space with complete metric \(\rho \). Denote by \({\mathrm {comp}}\,(\mathbb {X})\) the space of all non-empty compact subsets of \(\mathbb {X}\), endowed with the Hausdorff metric.
Theorem 5
([21, Theorem 12.1.10] Let \((E, \mathcal {E})\) be a measurable space and \(\varPhi :E \rightarrow {\mathrm {comp}}\,(\mathbb {X})\) be a measurable map. Then there exists a measurable map \(\varphi : E\rightarrow \mathbb {X}\) such that \(\varphi (q)\in \varPhi (q), q\in E\).
The above theorem is a weaker version of the Kuratovskii and Ryll-Nardzevski’s theorem on measurable selection for a set-valued mapping which takes values in the space of closed subsets of \(\mathbb {X}\); e.g. [22].
In the set-up of Proposition 3, for \(\mu ,\nu \in \mathcal {P}(S_2)\), we denote by \(C_{\mathrm {opt}}(\mu , \nu )\) the subset of \(C(\mu , \nu )\) consisting of all couplings which minimize the integral for the function h; that is,
We prove the following simple facts.
Lemma 2
For any \(\mu ,\nu \in \mathcal {P}(S_2)\):
-
1.
the set \(C_{\mathrm {opt}}(\mu , \nu )\) is non-empty;
-
2.
the sets \(C(\mu , \nu )\), \(C_{\mathrm {opt}}(\mu , \nu )\) are compact.
Proof
Since \(\pi _1, \pi _2: \mathcal {P}(S_2\times S_2)\rightarrow \mathcal {P}(S_2)\) are continuous, any weak limit point of a sequence from \(C(\mu , \nu )\) belongs to \(C(\mu , \nu )\). Because the marginal distributions of all \(\eta \in C(\mu , \nu )\) are the same, the set \(C(\mu , \nu )\) is tight, which by the Prokhorov theorem completes the proof of compactness of \(C(\mu , \nu )\).
Next, the mapping
is lower semicontinuous. To see that, consider a sequence \(\eta _n\Rightarrow \eta \) and use the Skorokhod “common probability space principle”: there exist random elements \(X_n, n\ge 1, X\) with \(\mathrm {Law}\,(X_n)=\eta _n, \mathrm {Law}\,(X)=\eta \) such that \(X_n\rightarrow X\) a.s. (see [6, Theorem 11.7.2]). Since h is bounded and lower semicontinuous, we have
which proves the required semicontinuity of \(I_h\). By this semicontinuity the function \(I_h\) attains its minimum on the compact set \(C(\mu , \nu )\), i.e. \(C_{\mathrm {opt}}(\mu , \nu )\) is non-empty. The semicontinuity of \(I_h\) also yields that the set \(C_{\mathrm {opt}}(\mu , \nu )\) is closed, and since it is a subset of the compact set \(C(\mu , \nu )\), it is compact.
To prove Proposition 3, we apply Theorem 5 in the following setting: \(E=S_1\times S_1\), \(\mathbb {X}=\mathcal {P}(S_2\times S_2)\), and
We represent \(\varPhi \) as a composition of \(\varPsi \) and \(\varUpsilon \), where
and
Clearly, the minimization of \(I_h\) is equivalent to maximization of \(1-I_h\), and \(1-I_h\) is upper semicontinuouus. Hence the mapping \(\varUpsilon :{\mathrm {comp}}\,(\mathbb {X})\rightarrow {\mathrm {comp}}\,(\mathbb {X})\) is measurable by [21], Lemma 12.1.7. On the other hand, for any sequence \((x_n,y_n)\rightarrow (x,y)\) and \(\eta _n\in \varPsi ((x_n, y_n))\) we have that the marginal distributions of \(\eta _n\) weakly converge to Q(x), Q(y) respectively. Then by the Prokhorov theorem there exist a weakly convergent subsequence \(\eta _{n_k}\), and in addition the weak limit has the marginal distributions Q(x), Q(y), that is, belongs to \(\varPsi ((x,y))\). Then the mapping \(\varPsi :E\rightarrow {\mathrm {comp}}\,(\mathbb {X})\) is measurable by [21, Lemma 12.1.8]. Hence \(\varPhi \) is measurable, as well, and we obtain the statement of Proposition 3 as a straightforward corollary of Theorem 5. \(\square \)
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Kulik, A., Scheutzow, M. Generalized couplings and convergence of transition probabilities. Probab. Theory Relat. Fields 171, 333–376 (2018). https://doi.org/10.1007/s00440-017-0779-8
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DOI: https://doi.org/10.1007/s00440-017-0779-8
Keywords
- Markov process
- Invariant measure
- Generalized coupling
- Convergence of transition probabilities
Mathematics Subject Classification
- 60J05
- 60J25
- 37L40