Show That the Image of a Measurable Set Under a Continuous Transformation Need Not Be Measurable

Measures on Algebraic-Topological Structures

Piotr Zakrzewski , in Handbook of Measure Theory, 2002

THEOREM 1.6

Suppose a group G acts by measurable transformations on a σ-finite measure space (X, ∑, m) and assume that the measure m is quasi-invariant. If P is a finite, finitely additive, invariant measure defined on ∑ with P ∼ m, then the function µ defined by

$\begin{array}{ll}\mu \left(A\right)=\inf \left\{{\displaystyle \sum _{n\in \mathbb{N}}P\left(A_n\right):A_n\in \sum ,A\subseteq {\displaystyle \underset{n\in \mathbb{N}}{\cup }{A}_n}}\right\}\hfill & forA\in \sum \hfill \end{array},$

is a finite, σ-additive, invariant measure on ∑ with µ ∼ m.

To formulate another necessary and sufficient condition for the existence of an invariant probability measure we need the notion of a weakly wandering set.

Suppose a group G acts by measurable transformations on a measurable space (X, ∑). A set A ∈ ∑ is called weakly wandering (under G) if there exist elements $g_n\in G,n\in \mathbb{N}$ , such that $g_{n}A\cap g_{m}A=0$ for $n\ne m$ . Clearly, every weakly wandering set is G-negligible.

The following strengthening of the Hopf-Kawada theorem was first proved by Hajian and Kakutani (1964) for ℤ-actions and then generalized by Hajian and Itô (1969) to arbitrary actions (partial generalizations were obtained by several authors – see the references in Hajian and Itô (1969)).

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Handbook of Dynamical Systems

Luis Barreira , ... Omri Sarig , in Handbook of Dynamical Systems, 2006

Lemma 5.13 [139]

Let f : X → X be a measurable transformation. If K :X → ℝ is a positive measurable function tempered on some subset Z ⊂ X, then for any ε > 0 there exists a positive measurable function K_ε :Z → ℝ such that $K\left(x\right)⩽K_{\epsilon }\left(x\right)$ and if x ∈ Z then

$e^{-\epsilon }⩽\frac{K_{\epsilon }\left(f\left(x\right)\right)}{K_{\epsilon }\left(x\right)}⩽e^{\epsilon }.$

Note that if f preserves a Lebesgue measure ν on the space X, then any positive function K :X → ℝ with log K ∈ L ¹(X, ν) satisfies (5.17). The following is now an immediate consequence of Theorem 5.12.

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Topics from Ergodic Theory

George Roussas , in An Introduction to Measure-Theoretic Probability (Second Edition), 2014

15.4 Measure-Preserving Ergodic Transformations, Invariant Random Variables Relative to a Transformation, and Related Results

The concept of invariance is also defined for r.v.s.

Definition 9

Let $T$ be a (measurable) transformation on $(\mathrm{\Omega }\text{,}\mathcal{A}\text{,}P)$ into itself, and let $X$ be a r.v. on the same space. We say that $X$ is invariant (relative to $T$ ), if $X(\omega )=X(T\omega )\text{,}\omega \in \mathrm{\Omega }.\blacksquare$

Remark 4

For an invariant r.v. $X$ , it is immediate that $X(\omega )=X(T^n\omega )\text{,}n\ge 0$ (and for $n=-1\text{,}-2\text{,}\dots$ , if $T$ is one-to-one onto). Indeed, let ${\omega }^{\prime }=T^{-1}\omega$ . Then

$X(T^{-1}\omega )=X({\omega }^{\prime })=X(T{\omega }^{\prime })=X(T(T^{-1}\omega ))=X(\omega )\text{,}$

and likewise, if $T^{-n}\omega ={\omega }^{\prime }$ , then

$\begin{array}{cc}\hfill & X(T^{-n}\omega )=X({\omega }^{\prime })=X(T{\omega }^{\prime })=X(T(T^{-n}\omega ))\hfill \\ \hfill & =X(T^{-(n-1)}\omega )=X(\omega )\text{,}\hfill \end{array}$

by the induction hypothesis.

The fact that $X(\omega )=X(T^n\omega )\text{,}\omega \in \mathrm{\Omega }$ , $n=0\text{,}\pm 1\text{,}\dots$ , means that $X(\omega )$ remains constant on the orbit $T^n\omega \text{,}n=0\text{,}\pm 1\text{,}\dots$

Now the question arises as to what r.v.s are invariant. The answer to this is given by the following result.

Proposition 16

Let $T$ be a (measurable) transformation on $(\mathrm{\Omega }\text{,}\mathcal{A}\text{,}P)$ into itself, and let $X$ be a r.v. on the same space. Then $X$ is invariant (relative to $T$ ), if and only if $X$ is $\mathcal{J}$ -measurable, where $\mathcal{J}$ is the $\sigma$ -field of ( $T$ -) invariant sets in $\mathcal{A}$ .

Proof

Let $X$ be invariant, and set $A(x)=\left\{\omega \in \mathrm{\Omega }\text{;}X(\omega )\le x\right\}\text{,}x\in \mathfrak{R}$ .

Then

$\begin{array}{cc}\hfill T^{-1}A(x)=& \left\{\omega \in \mathrm{\Omega }\text{;}T\omega \in A(x)\right\}=\left\{\omega \in \mathrm{\Omega }\text{;}X(T\omega )\le x\right\}\hfill \\ \hfill =& \left\{\omega \in \mathrm{\Omega }\text{;}X(\omega )\le x\right\}=A(x)\text{,}\hfill \end{array}$

since $X(T\omega )=X(\omega )$ . Thus, $T^{-1}A(x)=A(x)\text{,}x\in \mathfrak{R}$ , and this establishes $\mathcal{J}$ -measurability for $X$ .

Next, let $X$ be $\mathcal{J}$ -measurable. We shall show that $X$ is invariant. Since every $X$ is the pointwise limit of simple r.v.s, it suffices to show the result for the case that $X$ is only an indicator function. So let $X=I_A$ with $A\in \mathcal{J}$ . Then

$\begin{array}{cc}\hfill & X(T\omega )=I_A(T\omega )=I_{T^{-1}A}(\omega )=I_A(\omega )=X(\omega )\text{,}\hfill \\ \hfill & \mathrm{since}{T}^{-1}A=A.(\mathrm{See}\mathrm{also}\mathrm{Exercise}4.)\blacksquare \hfill \end{array}$

Ergodicity and invariant r.v.s are related as follows.

Proposition 17

Let $T$ be a (measurable) transformation on $(\mathrm{\Omega }\text{,}\mathcal{A}\text{,}P)$ into itself. Then $T$ is ergodic, if and only if every real-valued r.v. invariant relative to $T$ , defined on the same probability space, is a.s. equal to a finite constant.

Proof

Suppose that every invariant r.v. is a.s. equal to a constant, and let $A$ be an arbitrary set in $\mathcal{J}$ . If we set $X=I_A$ , then $X$ is $\mathcal{J}$ -measurable, and therefore invariant, by Proposition 16. Since $X$ is equal to 1 or 0 with probability $P(A)$ or $P(A^c)$ , respectively, we have that

$P(A)=1\mathrm{or}P(A^c)=1.\mathrm{So}P(A)=0\mathrm{or}P(A)=1\text{.}$

Next, suppose that $T$ is ergodic, so that $P(A)=0$ or $1$ for every $A\in \mathcal{J}$ . We shall prove that every invariant r.v. $X$ is a.s. equal to a constant. Since $X$ is invariant, it is $\mathcal{J}$ -measurable and hence $P(X<x)=0$ or $1$ for every $x\in \mathfrak{R}$ . On the other hand, $P(X<x)\to 1$ as $x\to \infty$ . Hence $P(X<x)=1$ for all $x\ge x_1$ , some sufficiently large $x_1$ . Set

$x_0=\inf \left\{x\in \mathfrak{R}\text{;}P(X<x)=1\right\}\text{;}$

$x_0$ is finite; i.e., $x_0>-\infty$ , because otherwise $P(X<x)=1$ for all $x$ would imply $1=P(X<x)\downarrow P(X=-\infty )$ as $x\downarrow -\infty$ , a contradiction.

Then

$\begin{array}{cc}\hfill & P\left(x_0-\epsilon <X<x_0+\epsilon \right)=P\left(X<x_0+\epsilon \right)\hfill \\ \hfill & -P\left(X\le x_0-\epsilon \right)=1-0=1\hfill \end{array}$

for every $\epsilon >0$ . Letting $\epsilon \downarrow 0$ and observing that

$\left(x_0-\epsilon <X<x_0+\epsilon \right)\downarrow \left(X=x_0\right)\text{,}$

we get $P(X=x_0)=1.\blacksquare$

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Lyapunov Exponents and Strange Attractors

M. Viana , in Encyclopedia of Mathematical Physics, 2006

Linear Cocycles

Let μ be a probability measure on some space M and $f:M\to M$ be a measurable transformation that preserves μ. Let $\pi :E\to M$ be a finite-dimensional vector bundle, endowed with a Riemannian metric ${\Vert .\Vert }_x$ on each fiber $E_x={\pi }^{-1}\left(x\right)$ . Let $\mathcal{A}:E\to E$ be a linear cocycle over f. What we mean by this is that

$\pi \text{o}A=f\text{o}\pi$

and the action $A\left(x\right):E_x\to E_{f\left(x\right)}$ of $\mathcal{A}$ on each fiber is a linear isomorphism. Notice that the action of the nth iterate ${\mathcal{A}}^n$ is given by

$A^n\left(x\right)=A\left(f^{n-1}\left(x\right)\right)\cdots A\left(f\left(x\right)\right).A\left(x\right)$

for every $n\ge 1$ .

Assume the function ${log}^{+}{\Vert A\left(x\right)\Vert }_x$ is μ-integrable:

[5] ${log}^{+}{\Vert \text{A}\left(x\right)\Vert }_x\in L^1\left(\mu \right)$

(we write ${log}^{+}\phi =\text{log max}\left\{\phi ,1\right\}$ , for any $\phi >0$ ). It is clear that the sequence of functions $a_n\left(x\right)=log{‖A^n\left(x\right)‖}_x$ satisfies

$a_{m+n}+n\left(x\right)\le a_m\left(x\right)+a_n\left(f^m\left(x\right)\right)$

for every m, n, and x. It follows from J Kingman's subadditive ergodic theorem that the limit

$\underset{n\to \infty }{lim}\frac{1}{n}{a}_n\left(x\right)$

exists for μ-almost all x. In view of [2], this means that the largest Lyapunov exponent ${\lambda }_k\left(x\right)$ of the sequence $A^n\left(x\right),n\ge 1$ is a limit, and not just a lim sup, at almost every point.

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On Products of Topological Measure Spaces

Grekas Stratos , in Handbook of Measure Theory, 2002

THEOREM 2.1

If μ is a Baire measure on a compact group G, then every automorphism of the measure algebra of μ can be induced by an invertible, completion Baire measurable transformation of G.

The proof uses an interesting method of combining the classical Weil-Pontrjagin theorem with ideas of Choksi (1972).

Choksi and Simha (1978) remarked that the proof of Theorem 2.1 "seems to depend on topological properties of homogeneity of the space". Confirming this vague assertion, these authors proved, among others, the following theorem.

THEOREM 2.2

Let G be a locally compact σ-compact group, L a closed subgroup of G and μ a σ-finite Baire measure on (the homogeneous space) G/L. Then, every automorphism of the measure algebra of μ is induced by an invertible, completion Baire measurable point transformation of G/L.

It is worthwhile to note, here, that the purpose of this section is not to survey the whole work on measures on compact groups, but, mainly, to focus the reader's interest on the similarity that exists between Haar measure and product measure.

In (1979), Choksi and Fremlin wrote an article in which a large class of results in topological measure theory, usually proved for Polish spaces, was shown to hold in any uncountable product ${\displaystyle {\prod }_{i\in I}{X}_i}$ of compact metric spaces. Two Radon probability measures on ${\displaystyle {\prod }_{i\in I}{X}_i}$ , whose measure algebras are homogeneous of the same Maharam type, where shown to be (completion) Baire isomorphic. Also, the authors, combining their results with Maharam's famous theorem, arrived at some surprising theorems, some of whose proofs do not assume any set-theoretic hypotheses. The main of the latter are included in the statement (for a precise statement, see Choksi and Fremlin (1979, Theorem 5)):

PROPERTY 2.3

For a large, in fact cofinal, class of cardinals I, any two completion regular measures on ${\displaystyle {\prod }_{i\in I}{X}_i}$ , are completion Baire isomorphic.

Choksi (1984) asks "whether analogues of the results of Choksi and Fremlin – without additional set-theoretic assumptions – are valid for compact groups (and possibly even for homogeneous spaces)". This important conjecture was based on the one hand, on Kakutani and Kodaire theorem, and on the other on Theorems 2.1 and 2.2. Furthermore, this conjecture implies the necessity to find an intimate connection between Haar measure and product measure.

Choban (1994) had already proved that every compact group is Baire isomorphic to a product of compact metric spaces. In a subsequent paper, the author rediscovered the same result and proved that such a Baire isomorphism can be constructed so that it takes Haar measure to a product measure (Grekas, 1994). Thus, the Haar measure on a compact group plays a similar role to that of a product measure in the structure theory of measure algebras on a product of compact metric spaces (similar results can be proved for compact homogeneous spaces (Grekas, 1995)).

Shortly before the previously mentioned, the author attempted, bearing in mind Choksi's question, to obtain "compact group analogues" of the results of Choksi and Fremlin (1979). Grekas (1992a) deals with the isomorphisms of measure algebras on compact groups and their inducing point maps. Modifying the proof of Theorem 2.1, the author first proves that two (Radon probability) measures on the group, whose measure algebras are homogeneous of the same type, are completion Baire isomorphic. The rest of the paper, as Choksi and Fremlin did, studies how many non-isomorphic completion regular measures can exist on the group. Using approximation by Lie groups, it is shown that the compact group analogue of Property 2.3 is valid (see Theorem 5.A and Remark 5.13(3) there); note that to carry out the extension to compact groups, it is used, in a crucial way, the fact – incidentally established there – that the Haar measure on any uncountable compact group G is homogeneous of Maharam type w(G). The author states nextly the following theorem.

THEOREM 2.4

If G is a compact group with $w\left(G\right)⩾w$ then the Maharam type of G is equal to the Maharam type of ${\left\{0,1\right\}}^{rl\left(G\right)}$ .

Although the statement was correct, there were some gaps in the proof; see also corrigendum (Grekas, 1994) (a detailed account of these problems is given in Godin (2000)).

REMARK 2.5

Theorem 2.4 leads to an affirmative answer to Choksi's question, not only regarding the results free on additional set-theoretic assumptions but also regarding the remaining results of Choksi and Fremlin.

A correct proof of Theorem 2.4 was done in Grekas and Mercourakis (1998), where the measure-theoretic structure of topological groups is further more investigated. More precisely. Theorem 2.4 is an immediate consequence of the following structure theorem (Grekas and Mercourakis, 1998, Theorems 1.1 and 1.4):

THEOREM 2.6

Let G be a compact group with $\omega \left(G\right)=\alpha ⩾\omega$ . There exist two families $G_{\xi },H_{\xi },\xi <\alpha$ of compact metric groupseach having at least two points if $\alpha ⩾{\omega }^{+}$ and two continuous open surjections

such that

where ${\lambda }_{\xi },\lambda ,{\mu }_{\xi }$ denotes the (normalized) Haar measure on $G_{\xi },G,H_{\xi }$ respectively.

Roughly speaking, the measure λ is compressed between ${\otimes }_{\xi }{\lambda }_{\xi }$ and ${\otimes }_{\xi }{\mu }_{\xi }$ so that the maps f, g are very close to being continuous epimorphisms (see Remarks 1.2 and Corollaries 1.3, 1.6 in the same paper). Theorem 2.6, whose proof is based on classical results of Pontrjagin-van Kampen and Mostert (see Price (1997); also references given in Grekas and Mercourakis (1998). extends and refines the well-known structure theorem of Kuzminov (1959): every compact group is a dyadic space (Comfort et al., 1992) contains many interesting insights into the structure of topological groups). The remaining part of Grekas and Mercourakis (1998) deals with the structure of the Jordan algebra of the Haar measure. As it is already mentioned, the Haar measure on a compact group G is Baire isomorphic to the Haar measure on some product $X={\displaystyle {\prod }_{\xi }<w\left(G\right)}{X}_{\xi }$ compact Lie groups. The authors, therefore, obtain a stronger result, in fact that the isomorphism can be chosen to be a "Jordan isomorphism" (see Definition 2.6 and Theorem 2.13); this, particularly, implies that the space of Riemann integrable functions on a compact group is isometric to that of Riemann integrable functions on some product of compact Lie groups (Corollary 2.15 there). Additionally, several applications of the above results are given. Among these, the authors obtain a direct proof of the existence of a strong – not necessarily invariant – lifting for the Haar measure on a compact group and a rather illuminating proof of the existence of uniformly distributed sequences on a compact separable group (Veech, 1971).

REMARKS AND QUESTIONS 2.7

(1)

Grekas (1995) showed, using Furstenberg's (1963) theorem that the phase space of any minimal distal flow is Baire isomorphic to a product of compact metric spaces; the isomorphism takes some invariant, completion regular probability measure to a product measure. According to the previously referred, the author believes that this isomorphism can be chosen to be also a Jordan isomorphism.

(2)

It is easy to verify that if two Radon probability spaces have isomorphic Jordan algebras, then they have isomorphic measure algebras are hence the same Maharam type (Grekas and Mercourakis, 1998). The converse is not true, as it follows from Examples (1) and (2) in the same paper. In view of these examples, it is natural to ask whether there exists a strictly positive measure μ on $X={\left\{0,1\right\}}^{\alpha },\alpha ⩾{\omega }^{+}$ , homogeneous of Maharam type or. such that its Jordan algebra is not isomorphic to the Jordan algebra of the Haar measure on X (of special interest is the case ${\omega }^{+}⩽\alpha ⩽c$ ; notice that a similar question has been posed in Mercourakis (1996)).

(3)

Associate with any topological space X is the Boolean algebra $\Omega \left(X\right)$ of Baire sets modulo sets of first category. Given an automorphism $\tau$ of $\Omega \left(X\right)$ it may not be possible to find a point transformation of X that induces $\tau$ (Choksi et al., 1987). However, when X is a product of complete separable metric spaces, it is shown in Maharam (1979) that any automorphism of $\Omega \left(X\right)$ can be realized by an invertible, Baire measurable transformation of X. By combining Maharam's result with ideas of Grekas and Mercourakis (1998), it is possible to prove the same result in the case when X is any compact group (it is a natural question whether category results are valid for compact groups).

(4)

Methods which were developed in the proof of Theorem 2.6 can lead to some results concerning the "homeomorphic measure problem" for compact connected topological groups (Grekas and Mercourakis, 2002). For more information on this problem, see references in Prasad (1981), Morris and Peck (1983, 1984), Gryllakis (1989).

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Mathematical Statistical Physics

Alain-Sol Sznitman , in Les Houches, 2006

5 Diffusions in random environment

We now discuss the continuous set-up and consider a diffusion on ℝ ^d with local characteristics, i.e. covariance matrix and drift, that are bounded stationary functions a(x, ω), b(x, ω), x ∈ ℝ ^d , ω ∈ Ω, where Ω is endowed with a group ${\left(t_x\right)}_{x\in {ℝ}^d}$ of jointly measurable transformations preserving the probability ℙ on Ω. The local characteristics are supposed to satisfy a Lipschitz condition, i.e. for some K > 0,

(5.1) $\left|a\left(x,\omega \right)-a\left(y,\omega \right)\right|+\left|b\left(x,\omega \right)-b\left(y,\omega \right)\right|\le K\left|x-y\right|,\text{for}x,y\in {ℝ}^d,\omega \in \Omega ,$

and a(·, ·) is uniformly elliptic. Further they satisfy a finite-range dependence condition, i.e. for some R > 0,

(5.2) $\begin{array}{l}\sigma \left(a\left(x,\cdot \right),b\left(x,\cdot \right),x\in A\right)\text{and}\sigma \left(a\left(y,\omega \right),b\left(y,\omega \right),y\in B\right)\text{are}\mathbb{P}\text{-independent,}\\ \text{when}A,B\subseteq {ℝ}^d\text{lie at mutual distance at least}R.\end{array}$

The diffusion in the random environment ω is then the solution of the martingale problem attached to

(5.3) $L=\frac{1}{2}\sum _{i,j=1}^d{a}_{ij}\left(y,\omega \right){\partial }_{ij}^2+\sum _{i=1}^d{b}_i\left(y,\omega \right){\partial }_i.$

The main issue is now to understand its asymptotic behavior. When b ≡ 0, or L = ½ ▽(a(·, ω) ▽), or L = ½Δ − ▽ V(·, ω) · ▽, with V a bounded stationary function, the method of the environment viewed from the particle applies and enables to prove a functional central limit theorem for the particle, see for instance Kozlov [51], Olla [58], [59], Osada [60], Papanicolaou-Varadhan [61], [62], Yurinsky [91], and also for divergence-free drifts Landim-Olla-Yau [53], Oelschläger [57]. However, the application of the method is rigid, and for instance breaks down if one replaces ▽ V(·, ω) with ▽ V(·, ω) + u, with u ≠ 0 a constant vector, in the case of L = ½Δ − ▽ V(·, ω)· ▽. This case was only very recently treated in Shen [73], using methods stemming from the progresses made in the study of ballistic RWRE.

The investigation of diffusions in random environment attached to (5.3) in the general framework described above, constitutes a continuous analogue of the study of RWRE. In particular when d > 1, the model is massively non-self adjoint and few explicit calculations are available. Building up on the recent advances concerning ballistic random walk in random environment, progress has also been achieved in the analysis of diffusions in random environment with ballistic behavior (i.e. with non-degenerate limiting velocity), cf. Komorowski-Krupa [49], [50], Schmitz [70], [71], Shen [72], and also Goergen [34].

However diffusive behavior has remained quite poorly understood, we will now describe one recent advance. When d > 1, there is no explicit formula for what the limiting velocity ought to be, and in view of the discussion below (4.19)-(4.21), it is not a straightforward matter to predict whether the limiting velocity vanishes or not.

One convenient way to "center" the process when trying to investigate diffusive behavior, is to impose the following (restricted) isotropy condition:

(5.4) $\begin{array}{l}\text{for any rotation matrix}r\text{preserving the union of coordinate axes of}{ℝ}^d\text{,}\\ \left(a\left(r\cdot ,\omega \right),b\left(r\cdot ,\omega \right)\right)\text{has same law as}\left(ra\left(\cdot ,\omega \right)r^T,rb\left(\cdot ,\omega \right)\right).\end{array}$

One then has the following result concerning small isotropic perturbation of Brownian motion:

Theorem 5.1. (d ≥ 3, [85], [86])

There is η ₀(d, K, R) > 0, such that when

$\left|a\left(x,\omega \right)-I\right|\le {\eta }_0,\left|b\left(x,\omega \right)\right|\le {\eta }_0,forallx\in {ℝ}^d,\omega \in \Omega ,$

then for ℙ-a.e. ω,

(5.5) $\begin{array}{l}\frac{1}{\sqrt{t}}{X}_{t}convergesin{P}_{0,\omega }\text{−}distributionast\to \infty ,toaBrownianmotion\\ withvariance{\sigma }^2>0,\left(functionalcentrallimittheorem\right),\end{array}$

(5.6) $forallx\in {ℝ}^d,P_{x,\omega }-a.s.,\underset{t\to \infty }{lim}\left|X_t\right|=\infty ,\left(transience\right),$

(5.7) $\text{for any bounded}f\text{,}g\text{on}{ℝ}^d\text{, that are}$

with u_∈, ∈ ≥ 0, the solution of:

$\{\begin{array}{c}{\partial }_t{u}_{\varepsilon }=L_{\varepsilon }{u}_{\varepsilon }+gin\left(0,\infty \right)\times {ℝ}^d\\ u_{\varepsilon |t=0}=f\end{array}$

and

$\begin{array}{l}{L}_{\varepsilon }=\frac{1}{2}\sum _{i,j=1}^d{a}_{ij}\left(\frac{x}{\varepsilon },\omega \right){\partial }_{ij}^2+\frac{1}{\varepsilon }\sum _{i=1}^d{b}_i\left(\frac{x}{\varepsilon },\omega \right){\partial }_i,for\varepsilon >0,\\ L_0=\frac{1}{2}{\sigma }^2\Delta .\end{array}$

The situation under study is a continuous counterpart of the problem considered by Bricmont-Kupiainen [19] for RWRE (see also [14] for a control of exit distributions). The proof is however different, and provides very quantitative couplings of the diffusion in random environment with Brownian motion, cf. Proposition 6.2 of [86]. It uses a renormalization scheme together with a sequence of Hölder norms, coupling, and wavelets. The cases of not necessarily small isotropic perturbations, or of dimension 2, are presently open.

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Introduction

In Inference for Heavy-Tailed Data Analysis, 2017

1.1 Basic Probability Theory

Let Ω be a space, which is an arbitrary, nonempty set. Write $\omega \in \Omega$ if ω is an element of Ω, and write $A\subseteq \Omega$ if A is a subset of Ω.

Definition 1.1

A nonempty class $\mathcal{A}$ of subsets of Ω is called an algebra if

i)

the complementary set $A^c\in \mathcal{A}$ whenever $A\in \mathcal{A}$ ; and

ii)

the union $A_1\cup A_2\in \mathcal{A}$ whenever $A_1\in \mathcal{A}$ and $A_2\in \mathcal{A}$ .

Moreover, $\mathcal{A}$ is called a σ -algebra or a σ -field if, in addition to i) and ii),

iii)

${\cup }_{i=1}^{\infty }{A}_i\in \mathcal{A}$ whenever $A_i\in \mathcal{A}$ for $i\ge 1$ .

Definition 1.2

If $\mathcal{A}$ is a σ-algebra with respect to the space Ω, then the pair $(\Omega ,\mathcal{A})$ is called a measurable space. The sets of $\mathcal{A}$ are called measurable sets.

Definition 1.3

The elements of the σ-algebra $\mathcal{B}$ generated by the class of infinite intervals of the form $[-\infty ,x)$ for $-\infty <x<\infty$ are called Borel sets. The measurable space $(\overline{\mathbb{R}}=[-\infty ,\infty ],\mathcal{B})$ is called Borel space.

Definition 1.4

If $({\Omega }_1,{\mathcal{A}}_1)$ and $({\Omega }_2,{\mathcal{A}}_2)$ are two measurable spaces and f is a mapping from ${\Omega }_1$ to ${\Omega }_2$ , then f is said to be a measurable transformation/mapping if $f^{-1}(A)\in {\mathcal{A}}_1$ for any $A\in {\mathcal{A}}_2$ , where $f^{-1}(A)=\{\omega :\omega \in {\Omega }_1,f(\omega )\in A\}$ .

Definition 1.5

For a measurable space $(\Omega ,\mathcal{A})$ , a set function P defined on $\mathcal{A}$ is called a probability if

i)

$P(\varnothing )=0$ , where ∅ denotes the empty set;

ii)

$P(A\cup B)=P(A)+P(B)$ for disjoint events $A,B\in \mathcal{A}$ (i.e., $A\cap B=\varnothing$ );

iii)

$P({\cup }_{i=1}^{\infty }{A}_i)={\sum }_{i=1}^{\infty }P(A_i)$ for disjoint events $A_i\in \mathcal{A}$ , $i=1,2,\cdots$ .

In this case, $(\Omega ,\mathcal{A},P)$ is called a probability space.

Definition 1.6

A real-valued measurable function X on a probability space $(\Omega ,\mathcal{A},P)$ is called a random variable. The function

$F(x)=P(X\le x):=P(\{\omega \in \Omega :X(\omega )\le x\})\text{for}x\in \mathbb{R}=(-\infty ,\infty )$

is called the cumulative distribution function or distribution function of X.

Definition 1.7

A sequence of random variables ${\{X_n\}}_{n=1}^{\infty }$ defined on a probability space $(\Omega ,\mathcal{A},P)$ is called independent if for any $m\ge 1$ , $1\le i_1<\cdots <i_m<\infty$ and $-\infty <x_1,\cdots ,x_m<\infty$

$P(X_{i_1}\le x_1,\cdots ,X_{i_m}\le x_m)=\prod _{j=1}^{m}P(X_{i_j}\le x_j).$

Definition 1.8

If ${\{X_n\}}_{n=0}^{\infty }$ is a sequence of random variables on a probability space $(\Omega ,\mathcal{A},P)$ , then ${\{X_n\}}_{n=1}^{\infty }$ is said to converge in probability to $X_0$ (notation: $X_n\stackrel{p}{\to }{X}_0$ ) if for any $ϵ>0$

$\underset{n\to \infty }{\lim }P(|X_n-X_0|>ϵ)=0.$

Definition 1.9

If ${\{X_n\}}_{n=0}^{\infty }$ is a sequence of random variables on a probability space $(\Omega ,\mathcal{A},P)$ with corresponding cumulative distribution functions ${\{F_n(x)\}}_{n=0}^{\infty }$ , then ${\{X_n\}}_{n=1}^{\infty }$ is said to converge in distribution to $X_0$ (notation: $X_n\stackrel{d}{\to }{X}_0$ or $X_n\stackrel{d}{\to }{F}_0$ ) if for any continuity point x of $F_0$

$\underset{n\to \infty }{\lim }{F}_n(x)=F_0(x).$

Definition 1.10

A sequence of random variables $\{X_n\}$ on a probability space $(\Omega ,\mathcal{A},P)$ is said to be bounded in probability if for any $ϵ>0$ , there exist constants $C>0$ and integer N such that

$P(|X_n|>C)\le ϵ\text{for all}n\ge N.$

Let $\{X_n\}$ be a sequence of random variables on a probability space $(\Omega ,\mathcal{A},P)$ and $\{b_n\}$ be a sequence of positive constants. We write $X_n=o_p(b_n)$ if $X_n/b_n\stackrel{p}{\to }0$ , and write $X_n=O_p(b_n)$ if $X_n/b_n$ is bounded in probability.

Definition 1.11

A stochastic process is a collection $\{X_t:t\in T\}$ , where T is a subset of $\mathbb{R}$ and $X_t$ is a random variable on a probability space $(\Omega ,\mathcal{A},P)$ .

Definition 1.12

A Wiener process $\{W(t):t\ge 0\}$ is a continuous-time stochastic process satisfying

i)

$W(0)=0$ ;

ii)

$W(t+u)-W(t)$ is independent of the σ-algebra generated by $\{W(s):0<s\le t\}$ for any $u>0$ ;

iii)

$W(t+u)-W(t)$ has a normal distribution with mean zero and variance u for any $u>0$ .

Definition 1.13

If $W(t)$ for $t\ge 0$ is a Wiener process, then $B(t)=W(t)-\frac{t}{T}W(T)$ is called a Brownian Bridge on $[0,T]$ . In this case,

$B(0)=B(T)=0\text{and}E(B(s)B(t))=s(T-t)$

for $0\le s<t\le T$ , but the increments are no longer independent.

Definition 1.14

The space $D[0,1]$ denotes the space of functions on $[0,1]$ that are right continuous and have left-hand limits.

For the space $(E,\epsilon )$ , let $C_K(E)$ be the set of all continuous, real valued functions on E with compact support, and $C_K^{+}(E)$ be the subset of $C_K(E)$ consisting of continuous, nonnegative functions with compact support. Let $M_{+}(E)$ be the set of all nonnegative Radon measures on $(E,\epsilon )$ and define ${\mu }_{+}(E)$ to be the smallest σ-field of subsets of $M_{+}(E)$ making the maps $m\to m(f)={\int }_{E}fdm$ from $M_{+}(E)\to \mathbb{R}$ measurable for all $f\in C_K^{+}(E)$ . Here Radon means the measure of compact sets is always finite.

Definition 1.15

ξ is a random measure if it is a measurable map from a probability space $(\Omega ,\mathcal{A},P)$ into $(M_{+}(E),{\mu }_{+}(E))$ .

Definition 1.16

For ${\mu }_n,\mu \in M_{+}(E)$ , we say ${\mu }_n$ converges vaguely to μ (written ${\mu }_n\stackrel{v}{\to }\mu$ ) if ${\mu }_n(f)\to \mu (f)$ for all $f\in C_K^{+}(E)$ .

Definition 1.17

$C\subset {\mathbb{R}}^d$ is a cone if $t\mathit{x}\in C$ for every $t>0$ and $\mathit{x}\in C$ .

Let Λ denote the class of strictly increasing, continuous mappings of $[0,1]$ onto itself with $\lambda (0)=0$ and $\lambda (1)=1$ for each $\lambda \in \Lambda$ . Given x and y in the space $D[0,1]$ , define $d(x,y)$ to be the infimum of those positive ϵ for which there exists a $\lambda \in \Lambda$ such that

$\underset{t}{\sup }|\lambda (t)-t|\le ϵ\text{and}\underset{t}{\sup }|x(t)-y(\lambda (t))|\le ϵ.$

In this way, $d(x,y)$ defines the Skorohod topology.

Definition 1.18

Let F be the cumulative distribution function of a random variable X. Then the generalized inverse of F is defined as

(1.1) $F^{-}(u)=\inf \{t:F(t)\ge u\}\text{for}0<u<1.$

Lemma 1.1

Let F be a cumulative distribution function.

i)

For any $x\in \mathbb{R}$ and $u\in (0,1)$ , $F^{-}(u)\le x$ if and only if $u\le F(x)$ .

ii)

If U is a random variable with uniform distribution over $(0,1)$ , then the distribution function of $F^{-}(U)$ is $F(x)$ .

iii)

If F is continuous, then $F(F^{-}(u))=u$ for $0<u<1$ .

Proof

i) If $u\le F(x)$ , then $x\in \{t:F(t)\ge u\}$ , which implies that

$x\ge \inf \{t:F(t)\ge u\}=F^{-}(u).$

Next assume that $F^{-}(u)\le x$ . Since F is right continuous, we have

$F(x)\ge F(F^{-}(u))=F(\inf \{t:F(t)\ge u\})=\inf \{F(t):F(t)\ge u\}\ge u.$

Hence part i) follows.

ii) It follows from part i) that $P(F^{-}(U)\le x)=P(U\le F(x))=F(x)$ for any x. That is, $F^{-}(U)$ has the distribution function $F(x)$ .

iii) It follows obviously. □

Like Csörgő et al. [25], we use the following conventions concerning integrals.

When $a<b$ and g is a left-continuous and f is a right-continuous function, then

(1.2) ${\int }_a^{b}fdg={\int }_{[a,b)}fdg\text{and}{\int }_a^{b}gdf={\int }_{(a,b]}gdf,$

whenever these integrals make sense as Lebesgue–Stieltjes integrals. In this case the usual integration by parts formula

(1.3) ${\int }_a^{b}fdg+{\int }_a^{b}gdf=g(b)f(b)-g(a)f(a)$

is valid.

For any Brownian bridge $\{B(s):0\le s\le 1\}$ , and with $0\le a<b\le 1$ and the functions f and g as above we define the following stochastic integral

(1.4) ${\int }_a^{b}f(s)dB(s)=f(b)B(b)-f(a)B(a)-{\int }_a^{b}B(s)df(s)$

and the same formula for g replacing f.

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TRANSIENT RANDOM WALKS ON DYNAMICALLY ORIENTED LATTICES

NADINE GUILLOTIN-PLANTARD , RENÉ SCHOTT , in Dynamic Random Walks, 2006

4 EXAMPLES

The main motivation of this work is the generalization of the transience of the i.i.d. case of [23] to dependent or inhomogeneous orientations. Depending on the original dynamical systems, we obtain various extensions corresponding to well known examples of dynamical systems such that Bernoulli and Markov shifts, Gibbs measures, SRB measures (Manneville-Pomeau maps), rotations on the torus, etc., our framework is very general from this point of view. Nevertheless, to get the transience of the walk, we need to generate the orientations by choosing a suitable function f satisfying (10.4). In some sense, this condition requires the model not to be too close to the deterministic case because to satisfy the condition, f should not be "μ-too often" 0 or 1. We describe now the examples providing extensions of the i.i.d. case to various disordered orientations.

1 Bernoulli shift

The first considered dynamical system S is the Bernoulli shift on the product space E = [0, 1]^z endowed with the Borel σ-algebra, the bilatere shift transformation T defined by

$\begin{array}{c}T:E\to E\\ x={(x_y)}_{y\in Z}\mapsto {(Tx)}_y=x_{y+1},\forall y\in Z.\end{array}$

The product Lebesgue measure μ = λ^⊗z of the Lebesgue measure λ on [0, 1] is T-invariant and we choose as generating function f the projection on the zero coordinate:

$\begin{array}{c}f:E\to [0,1]\\ x\mapsto x_0.\end{array}$

For all y ∈ Z, we then have

$fоT^y(x)=x_y=\xi (y)\in [0,1].$

We consider this ξ′s as new random variables on E whose indepen dence is inherited from the product structure of μ. The condition

${\displaystyle {\int }_E\frac{d\mu }{\sqrt{f(1-f)}}<\infty }$

becomes

${\displaystyle {\int }_0^1\frac{d\lambda (x)}{\sqrt{x(1-x)}}<\infty }$

and the transience holds in this particular case. In fact, this product form of μ allows in this annealed case another description of the i.i.d. case of [23], for which we check $\xi (y)=\frac{1}{2}$ for all y ∈ ℤ and

(10.27) ${\text{Cov}}_{\mu }[X_0,X_y]=\mathbb{E}\mu [X_0{X}_y]=4\mathbb{E}[\xi (0)\xi (y)]-1=0.$

The result is also valid in the quenched case, for which the distribution of the orientation has an inhomogeneous product form.

2 Markov shift

If one considers a measure μ with correlations, then the same holds for P_μ. It is the case when one considers a Markovian measure instead of a product one on the space [0,1]^Z with stationary distribution π, whose correlations are given by (10.3).

The transience of the simple random walk on this particular dynamically oriented lattice holds for ℙ_μ-a.e. environment as soon as the following condition is satisfied:

${\displaystyle {\int }_0^1\frac{d\pi (x)}{\sqrt{x(1-x)}}<\infty .}$

It is the case when the usual Lebesgue measure or Lebesgue measure of index p is the invariant measure.

In the quenched case, there are no correlations by construction and the law of the orientations depends on the measurable transformation only. This case is nevertheless different from this of the Bernoulli shift because the typical set of points x for which the transience holds depends on the measure μ.

3 Translation-invariant Gibbs measures

We consider now a measurable space of the form E = ∑^z where ∑ is a finite alphabet and T is again the bilatere shift defined above. We focus on the Ising model for which ∑ = {−1, +1}, and the function f used to generate the transition probabilities and to come back in [0, 1] is a dyadic transformation. In ergodic theory, Gibbs measures can be defined as equilibrium states or directly in term of an energy function Ψ: E → R, regular enough and chosen here to be Hölder continuous (more details can be found in [93]). A Borel probability measure on E is a Gibbs measure for Ψ if for every homeomorphism τ that affects only finitely many coordinates,

$\tau \text{ψ=}\mu e^{{\text{ψ}}_{\tau }}$

where

${\text{ψ}}_{\tau }\text{=}\underset{n}{\text{lim}}{\text{ψ}}_nо{\tau }^{-1}-{\text{ψ}}_n$

and Ψ _n , is the restriction of Ψ on ∑^{−n,…,n}. There exist many equivalent definitions of Gibbs measures in ergodic theory, see [93]. We focus here on the example of the Ising model where the energy function is

$\text{ψ(}w\text{)=}-J{w}_0{w}_1$

with a coupling J ∈ ℝ. The Gibbs measures are very different depending on the sign of the coupling; if J > 0, the model is said to be ferromagnetic and has positive correlations (one orientation is likely to agree with its neighbors), while in the antiferromagnetic case (J < 0) the sign of the correlations can differ. The case J = 0 correspond to the i.i.d. case, already known to be a transient case. To go back in [0, 1], we introduce f = d о π with π: {−1, + 1}^ℤ → {0,1}^z; ω ↦ σ with ${\sigma }_y=\frac{1+{\omega }_y}{2}$ for all y ∈ ℤ, and

$\begin{array}{ll}d:{\{0,+1\}}^Z\hfill & \to [0,1]\hfill \\ \omega \hfill & \mapsto {\displaystyle \sum _{y\in Z}\frac{{\sigma }_y}{2^y}.}\hfill \end{array}$

Due to the absence of phase transition in this one dimensional model, the average of f under μ is $\frac{1}{2}$ . Condition (10.4) is believed to be true as soon as the energy function is finite range. This would extend in particular the transience of the i.i.d. case to more general models with exponential decays of (positive or negative) correlations.

4 SRB measures, Manneville-Pomeau maps

SRB measures provide another source of examples for dependent orientations. When E is the interval [0, 1], a measure μ of the dynamical system S is said to be an SRB measure if the empirical measure $\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{\delta }_{T^i(x)}}$ converge weakly to μ for Lebesgue a.e. x. There exist many other definitions of SRB measures, see e.g. [172, 89]. In particular, it has the Bowen boundedness property in the sense that it is close to a Gibbs measure on some increasing cylinder, i.e. there exists a constant C > 0 such that for all x ∈ [0,1] and every n ≥ 1

$\frac{1}{C}\le \frac{\mu (I_{i_1,\dots ,i_n}(x))}{\exp ({\displaystyle {\sum }_{k=0}^{n-1}\Phi (T^k(x)))}}\le C$

where Φ = − log|T′| and I_i1, …, i_n is the interval of monotonicity for Tⁿ which contains x.

In some cases, it is possible to control the correlations for SRB measures and we detail now an example where our transience result holds, the Manneville-Pomeau maps. These maps have been introduced in the 1980′s to study intermittency phenomenon in the study of turbulence in chaotic systems ([7] and references therein) and has been recently identified as weakly Gibbsian measures, see [126]. They are expanding interval maps and in this example we describe the original MP map. The measurable space E is the unit interval [0,1] and for α ∈]0, 1[ the map is given by

$\begin{array}{l}T:[0,1]\to [0,1]\hfill \\ x\mapsto T(x)=x+x^{1+\alpha }\mathrm{mod}1.\hfill \end{array}$

The existence of an absolutely continuous (w.r.t. the Lebesgue measure on [0, 1]) SRB invariant measure μ has been established by [150] and the following bounds of Radon-Nikodym derivative $h=\frac{d\mu }{d\lambda }$ has been proved (see [126, 185]):

(10.28) $\exists C_{★},C^{★}>0\text{s}\text{.t}\text{.}\frac{{\text{C}}_{★}}{{\text{x}}^{\alpha }}<\text{h(x)<}\frac{{\text{C}}_{★}}{{\text{x}}^{\alpha }}.$

This measure is known to be mixing, and a polynomial decay of correlation, with a power β > 0, has even been proved for g regular enough ([85, 118, 126, 189]):

(10.29) $|C_{\mu }^g(y)|=\mathcal{O}(|y{|}^{-\beta }).$

The map T is not invertible but we use Theorem 10.2. It remains to find suitable function f who generates orientations for which the simple random walk is transient. By (10.28), a sufficient condition for the condition (10.4) to hold is

${\displaystyle {\int }_0^1\frac{dx}{x^{\alpha }\sqrt{f(x)(1-f(x))}}<\infty }$

and this is for example true for the function $f(x)=\frac{1}{2}(1+x-T(x))$ and the choice of an $\alpha <\frac{1}{3}$ .

5 The rotation on the torus

We consider the dynamical system S = ([0, 1], B([0, 1]), λ, T _α) where T _α is the rotation on the torus [0, 1] with angle α ∈ ℝ defined by

and λ is the Lebesgue measure on [0, 1]. For every function f: [0, 1] → [0, 1] such that ${\displaystyle {\int }_0^{1}f(x)dx=\frac{1}{2}}$ and

${\displaystyle {\int }_0^1\frac{dx}{\sqrt{f(x)(1-f(x))}}<\infty }$

conclusions of Theorem 10.1 hold. Such functions are called admissible. Every function uniformly bounded from 0 and 1, with integral $\frac{1}{2}$ is admissible. We also allow functions f to take values 0 and 1: for instance, f ₁(x) = x is admissible although f ₂(x) = cos²(2πx) is not. We actually have no explanations about this phenomenon, moreover we do not know the behavior (recurrence or transience) of the simple random walk on the dynamically oriented lattice generated by f ₂. Nevertheless, we can construct particular angles and functions for which the random walk on the corresponding lattice is recurrent. Take α = $\frac{1}{2q}$ for q an integer larger or equal to 1 and f = 1 _[0,1/2[, then the lattice we obtain is Z² with undirected vertical lines and horizontal strips of height q, alternatively oriented to the left then to the right. The simple random walk on this deterministic and periodic lattice is known to be recurrent, see [23]. A deeper study is needed for this particular choice of dynamical system.

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Mathematical Statistical Physics

Frank Redig , in Les Houches, 2006

4 Towards infinite-volume: the basic questions

In the previous subsection we showed how to compute probabilities of several subconfigurations in the thermodynamic limit. The computation of

(4.1) $\underset{V\uparrow {\mathbb{Z}}^2}{{\displaystyle lim}}{\mu }_V\left({\eta }_0=k\right)$

for k = 2, 3, 4 is still possible explicitly, but already requires a "tour de force" performed in a seminal paper of Priezzhev [34]. For d > 2 the height probabilities 2, …, 2d cannot be computed explicitly, and probabilities of more complicated local events cannot be computed explicitly either (not even for ℤ²).

However the interesting features of this model, like power-law decay of correlations, avalanche tail distribution, etc., are all statements about the large volume behavior. A natural question is therefore whether the thermodynamic limit

(4.2) $\underset{V\uparrow {\mathbb{Z}}^d}{{\displaystyle lim}}{\mu }_V$

exists as a weak limit (we will define this convergence more accurately later on). Physically speaking this means that enlarging the system more and more leads to converging probabilities for local events, i.e., events like η(x ₁) = k ₁, … η (x_n ) = k_n . A priori this is not clear at all. Usually, e.g. in the context of models of equilibrium statistical mechanics, existence of the thermodynamic limit is based on some form of locality. More precisely existence of thermodynamic limits in the context of Gibbs measures is related to "uniform summability" of the "local potential". This "uniform locality" is absent in our model (the burning algorithm is non-local, and application of the addition operator can change the configuration in a large set).

Let us now come to some more formal definitions. For V ⊆ ℤ ^d a finite set we denote by Ω _V = {1, … 2d} ^V the set of stable configurations in V. The infinite-volume stable configurations are collected in the set $\Omega ={\left\{1,\dots ,2d\right\}}^{{\mathbb{Z}}^d}$ . We will always consider Ω with the product topology. Natural σ-fields on Ω are F_V = σ{ψ _x : η → η _x , x ∈ V}, and the Borel-σ-field $\mathit{F}=\sigma \left({\cup }_{V\subseteq {\mathbb{Z}}^d}{\mathit{F}}_V\right)$ . If we say that μ is a measure on Ω we always mean a measure defined on the measurable space (Ω, F). For a sequence a_V indexed by finite subsets of ℤ ^d and with values in a metric space (X, d) we say that a_V → a if for every ∈ > 0 there exists V ₀ such that for all V ⊃ V ₀ finite, d(a_V, a) < ∈. Remind that Ω with the product topology is a compact metric space. An example of a metric generating the product topology is

(4.3) $d\left(\eta ,\xi \right)=\underset{x\in {\mathbb{Z}}^d}{\Sigma }{2}^{-\left|x\right|}\left|\eta \left(x\right)-\xi \left(x\right)\right|$

where for $x=\left(x_1,\dots ,x_d\right)\in {\mathbb{Z}}^d,\left|x\right|={\sum }_{i=1}^d\left|x_i\right|$ .

A function f : Ω → ℝ is said to be local if there exists a finite set V such that $f\left({\eta }_V{\xi }_{V^c}\right)=f\left({\eta }_V{\zeta }_{V^c}\right)$ for all η, ξ, ζ, i.e., the value of the function depends only of the heights at a finite number of sites. Of course a local function can be viewed as a function defined on Ω _V if V is large enough. Local functions are continuous and the set of local functions is uniformly dense in the set of continuous functions (by Stone-Weierstrass theorem). So in words, continuity of a function f : Ω → ℝ means that the value of the function depends only weakly on heights far away (uniformly in these far-away heights).

Definition 4.1

Let ${\left({\mu }_V\right)}_{V\subseteq {\mathbb{Z}}^d}$ be a collection of probability measures on Ω _V and μ a probability measure on Ω. We say that μ _V converges to μ if for all local functions f

(4.4) $\underset{V\uparrow {\mathbb{Z}}^d}{{\displaystyle lim}}{\mu }_V\left(f\right)=\mu \left(f\right)$

Remark

If for all local functions the limit in the lhs of (4.4) exists, then by Riesz representation theorem, this limit indeed defines a probability measure on Ω.

We denote byℛ _V the set of recurrent (allowed) configurations in finite volume V ⊆ ℤ ^d . The set ℛ ⊆ Ω is the set of allowed configurations in infinite-volume, i.e.,

(4.5) $ℛ=\left\{\eta \in \Omega :\forall V\subseteq {\mathbb{Z}}^d\text{finite},{\eta }_V{ℛ}_{\text{V}}\right\}$

For η ∈ Ω we define a_x,V :

(4.6) $a_x,V\eta =\left(a_x,V\eta V\right)\eta V^c$

For a measure μ concentrating on ℛ we say that a_x,V → a_x μ-a.s. if there exists a set Ω′ ⊆ Ω with μ(Ω′) = 1 such that for all η ∈ Ω′, a_x,V η converges to a configuration a_x (η) (in the metric (4.3)) as V ↑ ℤ ^d . Ifa_x,V → a_x μ a.s, then we say that the infinite-volume addition operator a_x is well-defined μ-almost surely. It is important to realize that we cannot expect a_x,V η to converge for all η ∈ ℛ.. For instance consider the maximal configuration (in d = 2 e.g.) η_max ≡ 4. Then it is easy to see that a_x,V η does not converge: taking the limit V ↑ ℤ ^d along a sequence of squares or along a sequence of triangles gives a different result. Since we will show later on that we have a natural measure μ on ℛ, we can hope that "bad" configurations like η_max are exceptional in the sense of the measure μ.

We now present a list of precise questions regarding infinite-volume limits.

1.

Do the stationary measures ${\mu }_V=\frac{1}{\left|{ℛ}_V\right|}{\sum }_{\eta \in {ℛ}_V}{\delta }_{\eta }$ converge to a measure μ as V ↑ ℤ ^d , concentrating on ℛ? Is the limiting measure μ translation invariant, ergodic, tail-trivial?

2.

Is a_x well-defined μ-a.s.? Is μ invariant under the action of a_x ? Does abelianness still hold (i.e., a_xa_y = a_ya_x on a set of μ-measure one)?

3.

What remains of the group structure of products of a_x in the infinite volume? E.g., are the a_x invertible (as measurable transformations)?

4.

Is there a natural stationary (continuous time) Markov process {η _t : t ≥ 0} with μ as an invariant measure?

5.

Has this Markov process good ergodic properties?

Regarding question 4, a natural candidate is a process generated by Poissonian additions. In words it is described as follows. At each site x ∈ ℤ ^d we have a Poisson process $N_t^{{\varphi }_x}$ with intensity ϕ _x , for different sites these processes are independent. Presuppose now that questions 1−2 have a positive answer. Then we can consider the formal product

(4.7) $\underset{x\in {\mathbb{Z}}^d}{\Pi }{a}_x^{}$

More precisely we want conditions on the addition rates ϕ _x such that

$\underset{V\uparrow {\mathbb{Z}}^d}{lim}\underset{x\in V}{\Pi }{a}_x{}^{}\left(\eta \right)$

exists for μ almost every η ∈ ℛ (or preferably even for a bigger class of initial configurations η ∈ Ω). It turns out that a constant addition rate will not be possible (for the non-dissipative model). A sufficient condition is (as we will prove later)

(4.8) $\underset{x\in {\mathbb{Z}}^d}{\Sigma }{\varphi }_{x}G\left(0,x\right)<\infty$

where G(0, x) is the Green function of the lattice Laplacian on ℤ ^d . This implies that for questions 4−5, we have to restrict to transient graphs (i.e., d ≥ 3). We remark here that probably the restriction d ≥ 3 is of a technical nature, and a stationary dynamics probably exists also in d = 2, but this is not proved. On the other hand, we believe that the condition (4.8) is necessary and sufficient for the convergence of the formal product (4.7). Results in that direction are presented in the last section of these notes.

4.1 General estimates

Suppose that we have solved problem 1 from our list in the previous section, i.e., we know that μ _V converges to μ. Then we can easily solve problem 2.

Proposition 4.2

Suppose that μ_V converges to μ in the sense of definition 4.1. Suppose that d ≥ 3. Then $a_x={\text{lim}}_{V\uparrow {\mathbb{Z}}^d}{a}_{x,V}$ is μ-almost surely well-defined. Moreover there exists a μ-measure one set Ω′ ⊆ ℛℛ such that for all η ∈ Ω′, for all V ⊆ ℤ ^d finite and n_x ≥ 0, x ∈ V the products ${\prod }_{x\in V}{a}_x^{n_x}\left(\eta \right)$ are well-defined.

$a_x^{-1}={lim}_{V\uparrow {\mathbb{Z}}^d}{a}_{x,V}^{-1}$ is μ-almost surely well-defined. Moreover there exists a μ-measure one set Ω′ ⊆ℛ such that for all η ∈ Ω′, for all V ⊆ ℤ ^d finite and n_x ≥ 0, x ∈ V the products ${\prod }_{x\in V}{a}_x^{-n_x}\left(\eta \right)$ are well-defined.

Proof

Let us prove that a_x is μ-almost surely well-defined. The other statements of the proposition will then follow easily.

Call N_V (x, y, η) the number of topplings at y needed to stabilize η = δ _x in the finite volume V (so throwing away the grains falling off the boundary of V).

It is easy to see that for V′ ⊃ V and all y ∈ ℤ ^d , $N_V\left(x,y,\eta \right)\le N_{V^{\prime }}\left(x,y\eta \right)$ (this follows from abelianness). Therefore we can define $N\left(x,y,\eta \right)={lim}_{V\uparrow {\mathbb{Z}}^d}{N}_V\left(x,y,\eta \right)$ (which is at this stage possibly +∞). Clearly N_V (x, ·, η) is only a function of the heights η(z), z ∈ V, and thus it is a local function of η.

Moreover, we have, μ _V (N_V (x, y, η)) = G_V (x, y) (see (2.21), (3.20)). Therefore, using that μ _V → μ,

(4.9) $\begin{array}{l}\int d\mu \left(N\left(x,y,\eta \right)\right)=\int d\mu \left(\underset{V\uparrow {\mathbb{Z}}^d}{lim}{N}_V\left(x,y,\eta \right)\right)=\underset{V}{lim}\int d\mu \left(N_V\left(x,y,\mu \right)\right)\\ \underset{V}{lim}\underset{W}{lim}\int d{\mu }_W\left(N_V\left(x,y,\eta \right)\right)\\ \le \underset{W}{lim}\int d{\mu }_W\left(N_W\left(x,y,\eta \right)\right)=\underset{W}{lim}{G}_W\left(x,y\right)=G\left(x,y\right)\end{array}$

where we used d ≥ 3, so that G(x, y) < ∞ (this works in general for transient graphs).

This proves that N (x, y, η) is μ-a.s. well-defined and is an element of L ¹ (μ) (so in particular finite, μ-a.s.). Therefore we can define

(4.10) $a_x\left(\eta \right)=\eta +{\delta }_x-\Delta N\left(x,\cdot ,\eta \right)$

Then we have a_x = lim _V a _x,V μ-almost surely, so a_x is well-defined μ-a.s.

Let us now prove the a.s. existence of inverses $a_x^{-1}$ . In finite volume we have, for η ∈ ℛ _V

(4.11) $a_{x,V}^{-1}\eta =\eta -{\delta }_x+\Delta n_x^V\left(\cdot ,\eta \right)$

where now $n_x^V\left(y,\eta \right)$ denotes the number of "untopplings" at y in order to make η − δ _x recurrent. Upon an untoppling of a site y, the site y receives 2d grains and all neighbors z ∈ V of y lose one grain. Upon untoppling of a boundary site some grains are gained, i.e., the site receives 2d grains but only the neighbors in V lose one grain.

The inverse $a_{x,V}^{-1}$ can be obtained as follows. If η − δ _x is recurrent (∈ ℛ _V ), then it is equal to $a_{x,V}^{-1}\eta$ . Otherwise it contains a FSC with support V ₀. Untopple the sites of V ₀. If the resulting configuration is recurrent, then it is $a_{x,V}^{-1}\eta$ . Otherwise it contains a FSC with support V ₁, untopple V ₁, etc. It is clear that in this way one obtains a recurrent configuration $a_{x,V}^{-1}\eta$ such that (4.11) holds. Integrating (4.11) over μ _V gives

(4.12) ${\mu }_V\left(n_x^V\left(y,\eta \right)\right)=G_V\left(x,y\right)$

Proceeding now in the same way as with the construction of a_x , one obtains the construction of $a_x^{-1}$ in infinite volume. The remaining statements of the proposition are obvious.

The essential point in proving that a_x is well-defined in infinite volume is the fact that the toppling numbers N (x, y, η) are well-defined in ℤ ^d , by transience.

For x ∈ ℤ ^d and η ∈ ℛ, we define the avalanche initiated at x by

(4.13) $A\upsilon \left(x,\eta \right)=\left\{y\in {\mathbb{Z}}^d:N\left(x,y,\eta \right)>0\right\}$

This is possibly an infinite set, e.g. if η is the maximal configuration. In order to proceed with the construction of a stationary process, we need that μ is invariant under the action of the addition operator a_x . This is the content of the following proposition.

Proposition 4.3

Suppose that avalanches are almost surely finite. Then we have that μ is invariant under the action of a_x and $a_x^{-1}$ .

Proof.

Let f be a local function. We write, using invariance of μ _W under the action of a_x,W :

(4.14) $\begin{array}{l}\int fd\mu -\int a_{x}fd\mu \\ =\left(\int \left(a_{x}f-a_{x,V}f\right)d\mu \right)+\left(\int \left(a_{x,V}f\right)d\mu -\int \left(a_{x,V}f\right)d{\mu }_W\right)\\ +\left(\int \left(a_{x,V}f\right)d\mu -\int \left(a_{x,W}f\right)d{\mu }_W\right)+\left(\int fd{\mu }_W-\int fd\mu \right)\\ :=A_V+B_{V,W}+C_{V,W}+D_W\end{array}$

For ∈ > 0 given, using that a_x,V → a_x, A can be made smaller than ∈/4 by choosing V large enough. The second and the fourth term B and D can be made smaller than ∈/4 by choosing W big enough, using μ _W → μ, the fact that f is local and the fact that for fixed V, a_x,V f is also a local function. So the only problematic term left is the third one. If a_x,V ≠a_x,W , then the avalanche is not contained in V. Therefore

$C_{V,W}\le 2\Vert f{\Vert }_{\infty }{\mu }_W\left(A\upsilon \left(x,\eta \right)⊈V\right)$

Notice that for fixed V, the event Aυ(x, η) ⊈ V is a local event (depends only on heights in V together with its exterior boundary). Therefore, we can choose W big enough such that

${\mu }_W\left(A\upsilon \left(x,\eta \right)⊈V\right)-\mu \left(A\upsilon \left(x,\eta \right)⊈V\right)<\varepsilon /8$

and by the assumed finiteness of avalanches, we can assume that we have chosen V large enough such that

$\mu \left(A\upsilon \left(x,\eta \right)⊈V\right)<\varepsilon /8$

Since ∈ > 0 was arbitrary, and f an arbitrary local function, we conclude that μ is invariant under the action of a_x .

By the finiteness of avalanches, for μ almost every η ∈ ℛ, there exists V = V(η) such that $a_{x,V}\left(\eta \right)=a_{x,V\left(\eta \right)}\left(\eta \right)$ for all V ⊃ V(η). It is then easy to see that $a_x^{-1}\left(\eta \right)=a_{x,V}^{-1}\left(\eta \right)$ for all V ⊃ V(η). One then easily concludes the invariance of μ under $a_x^{-1}$ .

4.2 Construction of a stationary Markov process

In this subsection we suppose that $\mu ={lim}_{V\uparrow {\mathbb{Z}}^d}$ exists, and that d ≥ 3, so that we have existence of a_x and invariance of μ under a_x . In the next section we will show how to obtain this convergence μ _V → μ.

The essential technical tool in constructing a process with μ as a stationary measure is abelianness. Formally, we want to construct a process with generator

(4.15) $Lf\left(\eta \right)=\underset{x}{\Sigma }{\varphi }_x\left(a_{x}f-f\right)$

where a_x f (η) = f (a_x η). For the moment don't worry about domains, etc. For a finite volume V, the generator

(4.16) $L_{V}f\left(\eta \right)=\underset{x\in V}{\Sigma }{\varphi }_x\left(a_{x}f-f\right)$

defines a bounded operator on the space of bounded measurable functions. This is simply because a_x are well-defined measurable transformations. Moreover it is the generator of a pure jump process, which can explicitly be represented by

(4.17) $n_t^V=\underset{x\in V}{\Pi }{a}_x^{{}_t^{\varphi x}}\left(\eta \right)$

where $N_t^{{\varphi }_x}$ are independent (for different x ∈ ℤ ^d ) Poisson processes with rate ϕ _x .

The Markov semigroup of this process is

(4.18) $S_V\left(t\right)=e^{t{L}_V}f=\underset{x\in V}{\Pi }{e}^{{\varphi }_x(a_x-I)t}$

Notice that since μ is invariant under a_x , it is invariant under S_V (t) and S_V (t) is a semigroup of contractions on L^p (μ) for all 1 ≤ p ≤ ∞. We are interested in the behavior of this semigroup as a function of V.

Theorem 4.4

Suppose that (4.8) is satisfied. Then, for all p > 1, and every local function f, S_V (t) f is a L^p (μ) Cauchy net, and its limit $S\left(t\right)f:={lim}_{V\uparrow {\mathbb{Z}}^d}{S}_V\left(t\right)f$ extends to a Markov semigroup on L^p (μ).

Proof

We use abelianness and the fact that S_V (t) are L^p (μ) contractions to write, for V ⊆ W ⊆ ℤ ^d

(4.19) $\begin{array}{cc}{\Vert S_V\left(t\right)f-S_W\left(t\right)f\Vert }_p& ={\Vert S_V\left(t\right)\left(I-S_{W\backslash V}\left(t\right)\right)f\Vert }_p\hfill \\ & \hfill ={\Vert S_V\left(t\right){\int }_0^t{S}_{W\backslash V}(s)\left(L_{W\backslash V}f\right)ds\Vert }_p\\ & \le {\int }_0^t{\Vert S_{W\backslash V}\left(s\right)\left(L_{W\backslash V}f\right)\Vert }_{p}ds\hfill \\ & \le t{\Vert L_{W\backslash V}f\Vert }_p\hfill \end{array}$

Now, suppose that f is a local function with dependence set D_f .

(4.20) $\begin{array}{cc}\left|L_{W\backslash V}f\right|& \le \underset{x\in W\backslash V}{\Sigma }{\varphi }_x\left|\left(a_{x}f-f\right)\right|\hfill \\ & \hfill \le \underset{x\in W\backslash V}{\Sigma }2{\varphi }_x\Vert f{\Vert }_{\infty }I\left(\exists y\in -\overline{D_f}:N\left(x,y,\eta \right)>0\right)\end{array}$

Where $\overline{D_f}$ is the union of D_f with its external boundary. Hence

(4.21) $\begin{array}{cc}{\Vert L_{W\backslash V}f\Vert }_p& \le 2{\Vert f\Vert }_{\infty }{\Vert \underset{x\in W\backslash V}{\Sigma }{\varphi }_{x}I\left(\exists y\in \overline{D_f}:N\left(x,y,\eta \right)>0\right)\Vert }_p\hfill \\ & \le 2{\Vert f\Vert }_{\infty }\underset{x\in W\backslash V}{\Sigma }{\varphi }_x\mu \left(\left\{\eta :\exists y\in \overline{D_f}:N\left(x,y,\eta \right)>0\right\}\right)\hfill \\ & \le 2{\Vert f\Vert }_{\infty }\underset{x\in W\backslash V}{\Sigma }{\varphi }_x\underset{y\in \overline{D_f}}{\Sigma }G\left(x,y\right)\hfill \end{array}$

This converges to zero as V, W ↑ ℤ ^d by assumption (4.8) (remember that $\overline{D_f}$ is a finite set).

Therefore, the limit

(4.22) $S\left(t\right)f=\underset{V\uparrow {\mathbb{Z}}^d}{lim}{S}_V\left(t\right)f$

exists in L ^∞ (μ) for all f local and defines a contraction. Therefore it extends to the whole of L ^∞ (μ) by density of the local functions. To verify the semigroup property, note that

(4.23) $\begin{array}{c}\Vert S\left(t+s\right)f-S\left(t\right)\left(S\left(s\right)f\right)\Vert =\underset{W\uparrow {\mathbb{Z}}^d}{lim}\underset{V\uparrow {\mathbb{Z}}^d}{lim}\Vert S_V\left(t+s\right)f-S_W\left(t\right)S_V\left(s\right)f\Vert \\ =\underset{W\uparrow {\mathbb{Z}}^d}{lim}\underset{V\uparrow {\mathbb{Z}}^d}{lim}\Vert S_V\left(s\right)\left(S_V\left(t\right)-S_W\left(t\right)\right)f\Vert \\ \le \underset{W\uparrow {\mathbb{Z}}^d}{lim}\underset{V\uparrow {\mathbb{Z}}^d}{lim}\Vert \left(S_V\left(t\right)-S_W\left(t\right)\right)f\Vert =0\end{array}$

It is clear that S(t) 1 = 1, and S(t) f ≥ 0 for f ≥ 0, since for all V ⊆ ℤ ^d these hold for S_V (t), i.e., S(t) is a Markov semigroup.

So far, abelianness delivered us a simple proof of the fact that under (4.8) we have a natural candidate semigroup with stationary measure μ. Kolmogorov's theorem gives us the existence of a Markov process with semigroup S(t). The following explicit representation can be used in order to show that this process has a decent (cadlag) version with paths that are right-continuous and have left limits.

Theorem 4.5

Let the addition rate satisfy (4.8). Denote by ℙ the joint distribution of the independent Poisson processes $N_t^{{\varphi }_x}$ . Then μ × ℙ almost surely, the product

(4.24) $\underset{x\in V}{\Pi }{a}_x^{N_t^{\varphi x}}\eta$

converges as V ↑ ℤ ^d to a configuration η_t ∈ ℛ. The process {η _t : t ≥ 0} is a version of the process with semigroup S(t) defined in Theorem 4.4, i.e., for all t ≥ 0, η ∈ Ω′, with μ(Ω′) = 1,

(4.25) $S\left(t\right)f\left(\eta \right)={\text{E}}_{\eta }f\left({\eta }_t\right)$

Moreover this version concentrates on D([0, ∞), ℛ).

As a corollary of this theorem, one recovers the generator of the semigroup.

Proposition 4.6

Define

(4.26) ${ℬ}_{\varphi }=\left\{f\in L^1\left(\mu \right):\underset{x\in {\mathbb{Z}}^d}{\Sigma }{\varphi }_x\int \left|a_{x}f-f\right|d\mu <\infty \right\}$

For ϕ satisfying (4.8) all local functions are in ℬ_ϕ. For f ∈ ℬ_ϕ, the expression

(4.27) $Lf=\underset{X\in {\mathbb{Z}}^d}{\Sigma }\left(a_{x}f-f\right)$

is well-defined, i.e., the series converges in L^p (μ) (for all 1 ≤ p ≤ ∞) and moreover,

(4.28) $\underset{t\downarrow 0}{lim}\frac{S\left(t\right)f-f}{t}=Lf$

in L^p(μ).

The following theorem shows that we can start the process from a measure stochastically below μ. We remind this notion briefly here, for more details, see e.g. [24] chapter 2. For η, ξ ∈ Ω we define η ≤ ξ if for all x ∈ ℤ ^d , η _x ≤ ξ _x . Functions f: Ω → ℝ preserving this order are called monotone. Two probability measures ν₁, ν₂ on Ω are ordered (notation ν₁ ≤ ν₂) if for all f monotone, the expectations are ordered, i.e., ν₁ (f) ≤ ν₂ (f). This is equivalent with the existence of a coupling ν₁₂ of ν₁ and ν₂ such that

$v\left\{\left(\eta ,\xi \right):\eta \le \xi \right\}=1$

Theorem 4.7

Let ν ≤ μ, and ℙ be as in Theorem 4.5. Then, ν × ℙ almost surely the products ${\prod }_{x\in V}{a}_x^{N_t^{{\varphi }_x}}$ (η) converge, as V ↑ ℤ ^d , to a configuration η_t . {η _t, t ≥ 0} is a Markov process with initial distribution ν.

Proof

For V ⊆ ℤ ^d finite and η ∈ Ω′, we define

(4.29) ${\eta }_t^V=\underset{x\in V}{\Pi }{a}_x^{N_t^{\varphi x}}\left(\eta \right)$

Then we have the relation

(4.30) ${\eta }_t^V=\eta +N_V^t-\Delta n_t^V\left(\eta \right)$

where $N_t^V$ denotes ${\left(N_t^{{\varphi }_x}\right)}_{x\in V}$ , and $n_t^V$ collects for all sites in ℤ ^d the number of topplings caused by addition of $N_t^V$ to η. Since η ∈ Ω′, these numbers are well-defined. Moreover for η ∈ Ω′, the toppling numbers are non-decreasing in V and converge as V ↑ ℤ ^d to their infinite-volume counterparts n_t which satisfy

(4.31) ${\eta }_t=\eta +N^t-\Delta n_t\left(\eta \right)$

For all V and η ≤ ξ it is clear that $n_t^V\left(\eta \right)\le n_t^V\left(\xi \right)$ . Choose now ν₁₂ to be a coupling of ν and μ such that ν₁₂ (η₁ ≤ η₂) = 1. Clearly, η₁ belongs to Ω′ with ν₁₂ probability one. Therefore for ν₁₂ almost all η₁, the limit ${lim}_{V\uparrow {\mathbb{Z}}^d}{n}_t^V\left({\eta }_1\right)={\left({\eta }_1\right)}_t$ is well-defined, and the corresponding process η _t , starting from η = η₁ is then defined via (4.31).

A trivial example of a possible starting configuration is the minimal configuration η ≡ 1. Less trivial examples are measures concentrating on minimal recurrent configurations (i.e., minimal elements of ℛ.).

Remark

In general we do not know whether νS(t) converges to μ as t → ∞, unless, as we will see later, ν is absolutely continuous w.r.t. μ. With respect to that question, it would be nice to find a "successfull" coupling, i.e., a coupling ℙ_ζ,ξ of two versions of η _t , starting from ζ, ξ such that lim _t→∞ ℙ_ζ,ξ(ζ _V (t) = ξ _V (t)) = 1 for all finite V ⊆ ℤ ^d . Less ambitious but also interesting would be to obtain this result for ζ ∈ Ω′ and ξ = a_xζ. This would yield that all limiting measures lim _n→∞ νS(t_n ) along diverging subsequences t_n ↑ ∞ are a_x -invariant. Uniqueness of a_x -invariant measures can then possibly be obtained by showing that μ is a Haar measure on some decent group consisting of products of a_x .

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