Skip to main content

Generalized couplings and convergence of transition probabilities

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.

This is a preview of subscription content, access via your institution.

References

  1. Bakhtin, Y., Mattingly, J.: Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math. 7, 553–582 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bessaih, H., Kapica, R., Szarek, D.: Criterion on stability for Markov processes applied to a model with jumps. Semigroup Forum 88, 76–92 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  4. Brzezniak, Z., Goldys, B., Imkeller, P., Peszat, S., Priola, E., Zabczyk, J.: Time irregularity of generalized Ornstein–Uhlenbeck processes. C. R. Acad. Sci. Paris Ser. Math. 348, 273–276 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  6. Dudley, R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  7. Dynkin, E.B., Yushkevich, A.A.: Controlled Markov Processes. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  8. Glatt-Holtz, N.E., Mattingly, J.C., Richards, G.: On unique ergodicity in nonlinear stochastic partial differential equations. J. Stat. Phys. 166(3–4), 618–649 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Hairer, M.: Ergodic Properties of Markov Processes. http://www.hairer.org/notes/Markov.pdf (2006)

  10. Hairer, M.: Ergodic Theory for Stochastic PDEs. http://www.hairer.org/notes/Imperial.pdf (2008)

  11. Hairer, M., Mattingly, J.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. Math. 164, 993–1032 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hairer, M., Mattingly, J.C.: Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. 36, 2050–2091 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hairer, M., Mattingly, J., Scheutzow, M.: Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Rel. Fields 149, 223–259 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kechris, A.S.: Classical Descriptive Set Theory. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  15. Komorowski, T., Peszat, S., Szarek, D.: On ergodicity of some Markov processes. Ann. Probab. 38, 1401–1443 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kulik, A., Scheutzow, M.: A coupling approach to Doob’s theorem. Rendiconti Lincei Mat. Appl. 26, 83–92 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes: I. General, 2nd edn. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  18. Mattingly, J.C.: Ergodicity of 2D Navier–Stokes equations with random forcing and large viscosity. Commun. Math. Phys. 206(2), 273–288 (1999)

    Article  MATH  Google Scholar 

  19. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, London (1993)

    Book  MATH  Google Scholar 

  20. Parthasarathy, K.R.: Introduction to Probability and Measure. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  21. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)

    MATH  Google Scholar 

  22. Wagner, D.H.: Survey of measurable selection theorems: an update. Lect. Notes Math. 794, 176–219 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexei Kulik.

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

$$\begin{aligned} B_\gamma ^1=\left\{ x: {\mathrm {d}\pi _1(\xi )\over \mathrm {d}\mathbb {P}}(x)\le \gamma ^{-1}\right\} ,\quad B_\gamma ^2=\left\{ x: {\mathrm {d}\pi _2(\xi )\over \mathrm {d}\mathbb {Q}}(x)\le \gamma ^{-1}\right\} ,\quad C_\gamma =B_\gamma ^1\times B_\gamma ^2, \end{aligned}$$

\(\gamma \in (0,1).\) Define the sub-probability measure \(\eta _\gamma \) on \((E^\infty \times E^\infty , \mathcal {E}^{\otimes \infty }\otimes \mathcal {E}^{\otimes \infty })\) by

$$\begin{aligned} \eta _\gamma (A)=\gamma \xi (A\cap C_\gamma ). \end{aligned}$$

Then the “marginal distributions” \(\pi _i(\eta _\gamma ), i=1,2\) (which now are sub-probability measures, as well) satisfy

$$\begin{aligned} \pi _1(\eta _\gamma )\le \mathbb {P}, \quad \pi _2(\eta _\gamma )\le \mathbb {Q}. \end{aligned}$$

Denote

$$\begin{aligned} \beta _\gamma =\eta _\gamma (E^\infty \times E^\infty )=\gamma \xi (C_\gamma )\le \gamma <1, \end{aligned}$$

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

$$\begin{aligned} \zeta _\gamma =\eta _\gamma +(1-\beta _\gamma )^{-1}\big (\mathbb {P}-\pi _1(\eta _\gamma )\big )\otimes \big (\mathbb {Q}-\pi _2(\eta _\gamma )\big ), \end{aligned}$$
(6.27)

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

$$\begin{aligned} \zeta _\gamma (A)\ge \eta _\gamma (A)\ge \gamma \Big (\alpha - \xi \big ((E^\infty \times E^\infty ){\setminus } C_\gamma \big )\Big ). \end{aligned}$$

Next,

$$\begin{aligned} \begin{aligned} \xi \big ((E^\infty \times E^\infty ){\setminus } C_\gamma \big )&\le \xi \big ((E^\infty {\setminus } B_\gamma ^1)\times E^\infty \big )+ \xi \big (E^\infty \times (E^\infty {\setminus } B_\gamma ^2)\big )\\ {}&=\pi _1(\xi )\big (E^\infty {\setminus } B_\gamma ^1\big )+\pi _2(\xi )\big (E^\infty {\setminus } B_\gamma ^2\big ) \end{aligned} \end{aligned}$$

and by the definition of the sets \(B_\gamma ^i, i=1,2\)

$$\begin{aligned}&\pi _1(\xi )\big (E^\infty {\setminus } B_\gamma ^1\big )=\int _{E^\infty {\setminus } B_\gamma ^1} {\mathrm {d}\pi _1(\xi )\over \mathrm {d}\mathbb {P}}\,\mathrm {d}\mathbb {P}\le \gamma ^{p-1} \int _{E^\infty }\left( {\mathrm {d}\pi _1(\xi )\over \mathrm {d}\mathbb {P}}\right) ^p\,\mathrm {d}\mathbb {P}\le \gamma ^{p-1}R^p,\\&\pi _2(\xi )\big (E^\infty {\setminus } B_\gamma ^2\big )=\int _{E^\infty {\setminus } B_\gamma ^2} {\mathrm {d}\pi _2(\xi )\over \mathrm {d}\mathbb {Q}}\,\mathrm {d}\mathbb {Q}\le \gamma ^{p-1} \int _{E^\infty }\left( {\mathrm {d}\pi _2(\xi )\over \mathrm {d}\mathbb {Q}}\right) ^p\,\mathrm {d}\mathbb {Q}\le \gamma ^{p-1}R^p. \end{aligned}$$

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 \(\zeta =\zeta _\gamma \)

$$\begin{aligned} \zeta (A)\ge {\gamma \alpha \over 2}=:\alpha ', \end{aligned}$$

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

$$\begin{aligned} {\widetilde{\eta }}_\gamma (A)=\xi (A\cap C_\gamma ). \end{aligned}$$

We fix \(\gamma \in (0,1)\) small enough, so that

$$\begin{aligned} \xi \big ((E^\infty \times E^\infty ){\setminus } C_\gamma \big )\le \alpha /2, \end{aligned}$$

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,

$$\begin{aligned} \pi _1({\widetilde{\eta }}_\gamma )\le \gamma ^{-1}\mathbb {P}, \quad \pi _2({\widetilde{\eta }}_\gamma )\le \gamma ^{-1}\mathbb {Q}, \end{aligned}$$

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

$$\begin{aligned} {\widetilde{\zeta }}_\gamma :={\widetilde{\eta }}_\gamma +(1-\gamma ^{-1}\beta _\gamma )\Big (\gamma ^{-1}(1-\beta _\gamma )\Big )^{-2}\Big (\gamma ^{-1}\mathbb {P}-\pi _1 ({\widetilde{\eta }}_\gamma )\Big )\otimes \Big (\gamma ^{-1}\mathbb {Q}-\pi _2({\widetilde{\eta }}_\gamma )\Big ) \end{aligned}$$

is a probability measure with

$$\begin{aligned} {\widetilde{\zeta }}_\gamma (A)\ge \xi (A)-\xi \big ((E^\infty \times E^\infty ){\setminus } C_\gamma \big )\ge \xi (A)-{\alpha \over 2}; \end{aligned}$$

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

$$\begin{aligned}&(1-\beta _\gamma )^{-1}\Big ((1-\gamma ^{-1}\beta _\gamma )\mathbb {P}+(1-\gamma )\pi _1({\widetilde{\eta }}_\gamma )\Big ),\quad (1-\beta _\gamma )^{-1}\Big ((1-\gamma ^{-1}\beta _\gamma )\mathbb {Q}\\&\quad + (1-\gamma )\pi _2({\widetilde{\eta }}_\gamma )\Big ), \end{aligned}$$

and their Radon-Nikodym densities w.r.t. \(\mathbb {P},\mathbb {Q}\) respectively are bounded by

$$\begin{aligned} R:=(1-\beta _\gamma )^{-1}\Big (\Big (1-\gamma ^{-1}\beta _\gamma \Big ) +(1-\gamma )\gamma ^{-1}\Big )=\gamma ^{-1}, \end{aligned}$$

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

$$\begin{aligned} \delta _n=d\left( K_1^n,K_2^n\right) , \quad n\ge 1. \end{aligned}$$

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

$$\begin{aligned} \xi \big ( d(X,Y)< \delta _n\big )\le {2\over n}, \end{aligned}$$

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

$$\begin{aligned} \mathcal {X}_A:=\{A\cap B, B\in \mathcal {X}\} \end{aligned}$$

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

$$\begin{aligned} \varXi :=\{\xi \in \mathcal {P}(E^\infty \times E^\infty ):\theta (\xi )\text { is a}\,\, \delta \hbox {-measure}\} \end{aligned}$$

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

$$\begin{aligned} \mathbb {Y}_1= & {} \Big \{\xi \in \varXi : \varrho _1(\xi )=x, \varrho _2(\xi )\in M, \pi _1(\xi )\sim \mathbb {P}_x, \pi _2(\xi )\ll \mathbb {P}_{\varrho _2(\xi )}\Big \},\\ \mathbb {Y}_2= & {} \bigcap _{m=1}^\infty \Big \{\lim _{n}\xi \big ( d(X_n,Y_n) \le 1/m\big )=1\Big \}. \end{aligned}$$

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

$$\begin{aligned} \{\xi \in \varXi : \varrho _1(\xi )=x\}, \quad \{\xi \in \varXi : \varrho _2(\xi )\in M\} \end{aligned}$$

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

$$\begin{aligned} \mathbb {P}(A)\le \varepsilon \quad \hbox {for any}\,\, A\in \mathcal {E}^{\otimes \infty }\,\, \hbox {such that}\quad \mathbb {Q}(A)\le \delta . \end{aligned}$$

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

$$\begin{aligned} \mathbb {P}(A\triangle A_\gamma )<\gamma , \quad \mathbb {Q}(A\triangle A_\gamma )<\gamma . \end{aligned}$$

Then in the above characterization of the absolute continuity the class \(\mathcal {E}^{\otimes \infty }\) can be replaced by \(\mathcal {A}\). Hence

$$\begin{aligned} \{\xi \in \varXi : \pi _2(\xi )\ll \mathbb {P}_{\varrho _2(\xi )}\}=\bigcap _{m=1}^\infty \bigcup _{k=1}^\infty \bigcap _{A\in \mathcal {A}}B_{m,k}(A), \end{aligned}$$

where

$$\begin{aligned} B_{m,k}(A)=\left\{ \xi : \pi _2(\xi )(A)\le m^{-1}, \mathbb {P}_{\varrho _2(\xi )}(A)\le k^{-1}\right\} \bigcup \left\{ \xi : \mathbb {P}_{\varrho _2(\xi )}(A)> k^{-1}\right\} . \end{aligned}$$

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

$$\begin{aligned} \mathcal {P}(E^\infty )\ni \mathbb {P}\mapsto \mathbb {P}(A)\in \mathbb {R}, \quad A\in \mathcal {A} \end{aligned}$$

are measurable, each of the sets \(B_{m,k}(A)\) is measurable. Therefore the set

$$\begin{aligned} \{\xi \in \varXi : \pi _2(\xi )\ll \mathbb {P}_{\varrho _2(\xi )}\} \end{aligned}$$

is measurable, as well. Finally, a similar and simpler argument shows that the set

$$\begin{aligned} \{\xi \in \varXi : \pi _1(\xi )\sim \mathbb {P}_{x}\} \end{aligned}$$

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,

$$\begin{aligned} \eta \in C_{\mathrm {opt}}(\mu , \nu )\quad \Leftrightarrow \quad \eta \in C(\mu , \nu ), \quad \int _{S_2\times S_2}h(u,v)\eta (\mathrm {d}u, \mathrm {d}v)=h(\mu , \nu ). \end{aligned}$$

We prove the following simple facts.

Lemma 2

For any \(\mu ,\nu \in \mathcal {P}(S_2)\):

  1. 1.

    the set \(C_{\mathrm {opt}}(\mu , \nu )\) is non-empty;

  2. 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

$$\begin{aligned} \mathcal {P}(S_2\times S_2)\ni \eta \mapsto I_h(\eta ):=\int _{S_2\times S_2}h(u,v)\eta (\mathrm {d}u, \mathrm {d}v) \in [0,1] \end{aligned}$$

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

$$\begin{aligned} \mathbb {E}h(X)\le \mathbb {E}\liminf _nh(X_n)\le \liminf _n \mathbb {E}h(X_n), \end{aligned}$$

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

$$\begin{aligned} \varPhi \big ((x,y)\big )=C_{\mathrm {opt}}(Q(x), Q(y)), \quad (x,y)\in E. \end{aligned}$$

We represent \(\varPhi \) as a composition of \(\varPsi \) and \(\varUpsilon \), where

$$\begin{aligned} \varPsi \big ((x,y)\big )=C(Q(x), Q(y)), \quad (x,y)\in E \end{aligned}$$

and

$$\begin{aligned} \varUpsilon (K)=\left\{ \eta \in \mathbb {X}: I_h(\eta )=\min _{\zeta \in K}I_h(\zeta )\right\} \in {\mathrm {comp}}\,(\mathbb {X}), \quad K\in {\mathrm {comp}}\,(\mathbb {X}). \end{aligned}$$

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. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • 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