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Random mass splitting and a quenched invariance principle

We will investigate a random mass splitting model and the closely related random walk in a random environment (RWRE). The heat kernel for the RWRE at time t is the mass splitting distribution at t. We prove a quenched invariance principle for the RWRE which gives us a quenched central limit theorem for the mass splitting model. Our RWRE has an environment which is changing with time. We follow the outline for proving a quenched invariant process for a random walk in a space-time random environment laid out by Rassoul-Agha and Sepp\"al\"ainen which in turn was based on the work of Kipnis and Varadhan and others.


Introduction
Imagine a one dimensional city (motivation from Abbott's Flatland) on the Y -axis with houses at the points (0, k). Suppose the city is stricken with an epidemic and things are getting worse by the day. As the death toll rises, each house (0, k) has only a Poisson(1) number of survivors v(0, k) when the long awaited medical breakthrough suddenly happens. The surviving residents (assume at least one survives) of house (0, 0) are scientists who were working on a cure for a while and they finally have an antidote! But the quantity that is produced is limited (say has mass 1 unit). So, they get out of their house carrying an equal proportion of the medicine (1/v(0, 0)), and start performing simple random walks, with time along the positive X-axis, to share it with the survivors. Meanwhile, the remaining survivors decide that staying inside the house in unhygienic conditions is dangerous, and they get out of their respective houses and start performing simple random walks at the same time as the scientists. Whenever a group of people meet on the way, say at (t, k), and at least one person in the group has some medicine, it gets equally divided among all the people in the group. This process continues. The question we ask in this article is: How does the distribution of this mass behave for large times t?
We call the above model the Random Mass Splitting model. In addition to the distribution of the medicine at time t we will study another property of the model. We use the movement of the villagers over time to define a random environment. Then we study the movement of a random walker in this random environment (RWRE). This defines a random walk in a random environment (which will be precisely defined in the next section). We are interested in this random walk in a random environment because it turns out that the heat kernel of the walk conditioned on the environment ω at time t is precisely the distribution of the medicine at time t when the villagers are moving according to ω. Random walk in a random environment models have been studied by many authors. This can be a very difficult field as even the simplest properties such as transience and recurrence are difficult to establish [12]. But a theory of random walk in random environments has been developed. We will harness that theory to find the asymptotic mass distribution.
In many examples, such as simple random walk on supercritical percolation clusters, we get a quenched invariance principle (i.e. an almost sure convergence of the RWRE to Brownian motion) [2] [21]. However in other examples we find behaviors that are very different from the usual diffusive behavior of simple random walk on Z d [22] [1]. Many of the proofs of invariance principles for random walks in random environments have their origins in the seminal work of Kipnis and Varadhan [13]. This paper laid down the foundation for quenched invariance principles for reversible Markov chains. Maxwell and Woodroofe [14] and Derriennic and Lin [6] subsequently extended their approach to the non-reversible set-up. Rassoul-Agha and Seppäläinen [18] developed further on these techniques to give a set of conditions under which a QIP holds for random walks in space-time random environments. They verified these conditions to prove a QIP for i.i.d. environments (see [18]). In this article, we follow the approach of [18]. Verifying their conditions for a QIP presents significantly more complications in our situation than it does with their i.i.d. environments. That these complications can be overcome demonstrates the robustness of the approach in [18].

Description of the model
We now present a more rigorous mathematical description of our model. Consider i.i.d. Poisson(1) random variables {v(0, k) : k ∈ Z} representing particles at points (0, k) of the Y -axis at time t = 0. Each particle starts performing a simple random walk {S (k,i) (·) : 1 ≤ i ≤ v(0, k), k ∈ Z}, with S (k,i) (0) = k and time represented along the positive X-axis. This gives a collection of random variables {e + (t, y), e − (t, y), v(t, y)} where e + (t, y) = #{(k, i) : S (k,i) (t) = y, S (k,i) (t + 1) = y + 1}, For technical convenience, we take the S (k,i) to be two-sided random walks and thus the above random variables are defined for all (t, y) ∈ Z 2 .
For t ≥ 0, p(t, y) denote the (random amount of) mass at (t, y). Then, for each t, t y=−t p(t, y) = 1.
To write down the explicit distribution of mass at time t, we define the environment to be the collection of 'random edge crossings' We call the space of environments Ω = ((Z + ∪ {0}) 2 ) Z 2 = {(a(t, y), b(t, y)) ∈ (Z + ∪ {0}) 2 : (t, y) ∈ Z 2 }. With these definitions we can inductively define the mass distribution with the equation The space Ω is equipped with the canonical product sigma field G and the natural shift transformation The random edge crossings induce a measure P on (Ω, G) and it is easy to see that P is invariant under the shift T (t ′ ,y ′ ) . It can be further proved that Ω, G, T (t ′ ,y ′ ) (t ′ ,y ′ )∈Z 2 , P is ergodic. We leave this verification to the reader. We shall denote the corresponding expectation by E. Now fix an ω. This generates a random mass distribution at (t, y) which we write as p(t, y; ω). Let Γ t (0, y) denote the set of all nearest neighbor paths γ : {0, 1, . . . , t} → Z with γ(0) = 0 and γ(t) = y. Then the mass at (t, y) is given by where a product in the above sum is interpreted as zero if one of the terms in the product vanishes.
Given any ω and any (t, y) we can define a random walk in a random environment which is started at y at time t and has transition probabilities generated by ω. We write X n (t, y) for the random variable that represents the location of the random walk after n steps (at time t + n). Its distribution is determined by P ω (t,y) which is defined as follows: P ω (t,y) (X 0 = (t, y)) = 1.
The X n are a time inhomogeneous Markov chain with transition probabilities Now we present the key observation that connects the random walk in a random environment with the random mass splitting model. Lemma 1. For t ∈ N and y ∈ Z p(t, y; ω) = P ω (0,0) (X t = (t, y)).
Proof. This follows by induction on t. Fix an environment ω. Then we have p(0, 0; ω) = P ω (0,0) (X 0 = (0, 0)) = 1 and p(0, y; ω) = P ω (0,0) (X 0 = (0, y)) = 0 for all y = 0. Now suppose the lemma is true for some t and all y ∈ Z. The transition probabilities for X t+1 given in (2.2) and (2.3) and the mass splitting rule in (2.1) show that for an arbitrary y Thus the induction hypothesis holds for t + 1 and all y ∈ Z. This completes the proof. ✷ This gives us a Markov chainX = (X n ) n≥0 for a fixed environment ω. P ω is called the quenched law. The joint annealed law is given bỹ The lawP (t,y) (dX, Ω) is called the marginal annealed law or just annealed law. Let Since the major breakthrough in [18], there has been a lot of work in RWRE. One of the major technical hurdles one has to overcome to prove QIP is to control the difference of two random walks driven by the same environment ω. In [18], the annealed law of this process is simply a random walk perturbed at the origin. In [19] this process becomes significantly more complicated due to the backtracking nature of such walks although the environment is i.i.d. Later, [10] applied the techniques of [19] in an environment with spatial correlations present. Besides these models, a large variety of RWRE in dynamic correlated environments have been studied in recent years, see for example [4] [5] [7] [8] [10] [20]. The methods used vary from Fourier series in [4] [5], to martingale approximation in [7] [8], to regeneration times in [20], although, to the best of our knowledge, a clear general idea about how to solve such problems seems to be missing as of now.
The major features that make our model different from the models considered in literature so far are the following: (i) Our model is not symmetric, even for fixed ω, as the first co-ordinate (time) always takes a deterministic step of 1 with each transition of the Markov chainX. This makes it different from the random conductance model, see [3]. This environment thus falls in the category of random walk in random space-time environments due to the deterministic increase in the time at each step.
(ii) The environment under consideration has high correlations between different points in spacetime. The correlated environment models considered so far are rather specialised and, as far as we know, the assumptions made in those works do not apply to our model.
(iii) Our model is not elliptic in the sense of condition (A5) of [7] as there are points in space-time where the walker's next move is entirely determined by the environment. These points (t, y) occur where all of the walkers in ω moved in the same direction (e + (t, y) > e − (t, y) = 0 or vice versa). A sub-case of this that creates more trouble is the possibility of having a long space-time path in ω. This is because, if the two copies of the random walk driven by ω get stuck in τ , they are forced to move together for a long time resulting in some kind of 'stickiness'. In Lemma 8, we show that such paths cannot be too long with high probability.
It is worth mentioning that [18] requires a weaker form of ellipticity, but their assumption of an i.i.d environment pushes the arguments through.
These differences from the models in the literature present some technical challenges, although we adapt the same rough outline as in [18] to prove a quenched invariance principle. Now we present the main theorem of this article: For almost all ω with respect toP, the process S (n) converges weakly to a standard Brownian motion B in the usual Skorohod topology under P ω (0,0) .
From this we deduce the answer to the question that was posed at the beginning of this paper.
Corollary 1.1. For a.e. ω with respect toP, the mass distribution µ t on R given by converges weakly to the standard normal distribution as t → ∞.
Proof. Fix an environment ω. As we have shown in Lemma 1, p(t, y; ω) = P ω (X t = (t, y)) for all t ∈ N and all y ∈ Z so the scaled mass distribution µ t at time t is given by the distribution of S (n) (t). The corollary follows directly from Theorem 1. ✷ Before we set off proving the theorem, we recall some of the machinery involved in proving QIP in general space-time random environments, mostly taken from [18].

Quenched invariance principle in general space-time random environments
Let For a bounded measurable function h on Ω define the operator The operator h → Πh − h defines the generator of the Markov process {T Xn (ω)} n≥0 of the environment as 'seen from the particle', i.e. keeping the particle fixed at the origin and shifting the environment around it. The transition probability of this Markov process is given by This Markov process, first conceived by [16] plays a key role in the arguments. Now suppose that this Markov process has a stationary ergodic measure P ∞ . Then we can extend the operator Π to a contraction on L p (Ω, P ∞ ) for any p ∈ [1, ∞]. Denote the joint law of (ω, T X 1 (ω)) by For a fixed ω, define the filtration (F ω n ) n≥0 where F ω n = σ(X 0 , X 1 , · · · , X n ). The main idea involved in proving QIP is to create a martingale M with respect to (P ω (0,0) , F ω n ) that is 'close' to the process X in some sense, and then to apply martingale central limit theorems to M and translate the results to X. The main challenge involved is in the last step: to control the error involved in estimating X by M . To prove convergence of X to a Brownian motion in the Skorohod topology, we need to show that the error is uniformly small along a 'typical' path of X. The way to do this, which we elaborate now, has its roots in the seminal work of Kipnis and Varadhan, and subsequently extended to the non-reversible set-up by Maxwell and Woodroofe [14] and Derriennic and Lin [6].
Towards this end, note that the process 0) can be expressed as a (P ω (0,0) , F ω n ) martingale plus some error that can be controlled, a quenched CLT can be established for (X n ) n≥0 . Now, a standard way to have this decomposition is to investigate whether there exists a solution h to Poisson's Equation If such a solution h did exist, we could write down Note that the first term in the above expression is a (P ω (0,0) , F ω n ) martingale and the second term is L 2 bounded with respect to P ∞ ⊗ P ω (0,0) . This would immediately give us the result. But unfortunately, a solution to the Poisson equation might not exist. To see this, note that if it did, then but the above does not hold for the i.i.d. space-time product environment where E ∞ E ω (0,0) (X n − (n, 0)) 2 grows like √ n. Rassoul-Agha and Seppäläinen [18] follow an extension of this idea, first introduced by [13], and later extended by Toth [24] and subsequently Maxwell and Woodroofe [14], to deal with the case where a solution to Poisson equation does not exist.
Call g = D − (1, 0). Then, the solution of the above can be written as Then, as our random walk has bounded range, Theorem 2 of [18] says the following: Theorem 2. Let P ∞ be any stationary ergodic probability measure for the Markov process on the environment Ω generated by Π − I. Also assume that there is α < 1 such that

2)
Then H = lim ǫ→0 H ǫ exists in L 2 (µ ∞ 2 ) and for P ∞ a.e. ω, S (n) converges weakly to a Brownian motion with variance So, to prove Theorem 1 with the aid of Theorem 2, we need to prove the existence of a stationary distribution P ∞ for the Markov process of the environment as seen by the particle (generated by Π − I), the moment bound (3.2) and the ergodicity of the Markov chain {T Xn (ω)} with respect to P ∞ . We prove these in the following section.

Proof of Theorem 1
Proof of Theorem 1. We break this proof up into three parts corresponding to the three conditions in Theorem 2. In Section 4.1, we show the existence of a stationary measure P ∞ for the Markov process of the environment with Lemma 2. In Section 4.2, we prove Lemma 3 which gives a moment condition that verifies (3.2). In Section 4.3, we prove the ergodicity of the Markov process {T Xn (ω)} n≥1 with respect to the stationary measure P ∞ . This is in Lemma 13. These three lemmas verify the hypothesis of Theorem 2 thus proving the desired almost sure invariance principle. The fact that σ 2 in Theorem 2 is one for our case follows from the easily verifiable fact that the annealed law of X n is that of (n, S n ), where S n is a simple random walk with steps +1, −1 with equal probability. ✷

Stationarity
In Lemma 2, we show that the measureP is in fact a stationary measure for the Markov process of the environment.
Lemma 2. The measureP is stationary for the Markov process on Ω as seen from the particle, that is, the Markov process {T Xn (ω)} n≥1 .
Proof. LetẼ denote the corresponding expectation. Take ψ measurable with respect to G. It will suffice to show thatẼ(Πψ) =Ẽ(ψ). Note that This proves the lemma. ✷

Moment condition
In this section we verify the moment condition by proving the following lemma.
Sketch of Proof of Lemma 3. Our strategy is as follows. Fix an environment ω. Let X and X be two independent copies of the random walk starting from (0, 0) and running in the same environment ω. Also define Y i by (0, Y i ) = X i −X i and letP * be the law of the Markov process Finally letẼ * be its expectation.
Step 1 Our first task is to establish Step 2 Next we show that Y i behaves like (two times) a lazy random walk when Y i = 0. But when Y i = 0 the distribution of Y i+1 has a complicated dependence structure.
Step 3 We use the sequence {Y i } n i=0 to divide the interval [0, n] into excursions (maximal connected intervals where Y i = 0) and holding times (maximal connected intervals where Y i = 0). By the previous step we know the distribution of excursion lengths. Much of the work of this proof is in estimating the distribution of the lengths of the holding times.
Step 4 We bound the right hand side of (4.1) by breaking it up into two parts. One is when the longest holding time is at most k log 3 (n) and the other is when the longest holding time is longer than k log 3 (n). Using the fact that excursions have the same distribution as a lazy random walk we are able to show that for a typical sequence Y i there are about √ n holds (and also about √ n excursions). So the first part contributes approximately √ n log 3 (n) to the expectation. We show that the probability of a long holding time is going to zero sufficiently quickly so the second part is contributing very little to the expectation (provided that k is sufficiently large). Adding up these two bounds completes the proof.
If i > j, we can condition on to get using the fact that Also, the terms in the above sum corresponding to i = j and y = z vanish as (e + (i, y), e − (i, y)) and (e + (i, z), e − (i, z)) are conditionally independent given F i implying Thus the only terms in the above sum which do not vanish are those with i = j and y = z. For these terms, we see that

by (4.4). Thus we get
✷ Also, notice that although the two walks X andX are conditionally independent given ω, the annealed joint law of (X,X) is far from being the product of the annealed marginal laws of X and X, as the subsequent calculations will indicate. This is what creates a major technical obstacle. Now we start to understand the annealed law of the process Y i . We will show that it behaves like a lazy random walk provided that Y i = 0, but at 0 it behaves very differently. The following lemma describes the behavior of Y i away from 0.

✷
Note that the presence of the term 1 + 1 v(n, y) in the summand makes this process depend on its entire past while making transitions from zero, thus destroying its Markov property. This is the striking difference from its analogue in the i.i.d. case studied by [18], where the process Y is a homogeneous Markov process perturbed at zero.
To deal with this problem, we decompose the path {Y i } into excursions away from zero and holding times at zero. Denote the j-th excursion by e j . Let the time interval spanned by e j be [α j , β j ], i.e., |e j (0)| = |Y α j | = 2 and β j = inf{k > α j : Y k = 0}. Let the first holding time at 0 be γ 0 = α 1 − 1 and let γ j = α j+1 − β j − 1 be the holding time at zero between e j and e j+1 . Define a(n) = sup{j ≥ 1 : α j ≤ n} with a(n) = 0 if the above set is empty. With these defined, we can write down Let the time duration of the j-th excursion be denoted by T j = β j − α j .
Denote by R the homogeneous random walk starting from zero with probabilities of increments 2, 0, −2 being 1/4, 1/2, 1/4 respectively, and let Denote the law of R by P R and the corresponding expectation by E R . Let It is easy to see that T 1 + ... + T k has the same distribution as the hitting time of level 2k by R. Thus {A n : n ≥ 1} and {R * n /2 : n ≥ 1} have the same distribution. Note that a(n) ≤ A n + 1.
for some constant C < ∞. The last step above follows from reflection principle arguments, see [23].
The excursions away from zero have the same law as those of a lazy random walk R so their distribution is well understood. Our main challenge is to provide an upper bound on the supremum of the holding times γ j . For a lazy random walk R, the holding times are i.i.d. geometric random variables and we can derive an upper bound on the probability that a holding time is at least log n. The γ j are far from i.i.d. and they depend on the entire past till that time. In spite of this dependence we will bound the probability that a holding time γ j is bigger than (log n) k for some positive number k. Our bound will be uniform in j and independent of all previous holding times. Using these bounds we will be able to bound the probability that the supremum of the holding times is large.
To do this we note by Lemma 6 that if Y i = 0 and v(X i ) ≥ 2 then there is at least a probability of 1/4 that Y i+1 = 0, independent of anything in the process or the environment up to time i.
We now define a set of stopping times to indicate when these times occur. These stopping times will give us our bound on the holding times.
We now define some stopping times for the Markov process (X i ,X i , T X i , TX i ). For each j ≥ 1, define: Thus we get a bi-indexed family {I (j) i : 1 ≤ i < ∞} which is well defined with probability one. Note that for each j, the sequence indexed by i eventually becomes constant.
For j ≥ 1 with β j < n, these stopping times indexed by i represent the times t ∈ [β j , α j+1 − 1] ∩ [0, n] (the portion of the holding time at zero between e j and e j+1 lying in [0, n]) where there are at least two particles at X t (=X t ) and the sequence becomes constant if either time n or time α j+1 − 1 is reached.
Call the corresponding stopped sigma fields {F * i } with k j = 1 for a constant sequence. With this notation, γ j = I (j) Control over the holding times γ j is obtained through the following two lemmas. (4.8) Proof. Note that, by the strong Markov property applied at these stopping times, where the last step follows by induction. ✷ Lemma 8. For sufficiently large M > 0, Proof. For each u ≤ n, let Note that by the stationarity of {T Xn } underP * , UnderP * , v(0, k) ∼ Poi(1) for k = 0 and v(0, 0) has a size-biased Poi(1) distribution. Let where {S (k,i) } are the walkers defining the environment ω as discussed in Section 2. It is easy to where S is a simple random walk starting from zero. It follows from (4.10) that there is 0 < C < ∞ such that µ t ≥ C √ t for every t > 0.
Note that if both N + t and N − t are non-zero, then the process X starting from (0, 0) must intersect at least one random walk S (2k,·) , k = 0 at or before time t, and at the time of intersection, say l, v(X l ) ≥ 2. Thus,P Thus,P * (τ 0 ≥ M 2 log 2 n) ≤ 2e −CM log n = 2 n CM .

Consequently,
for all n.
Proof. For sufficiently large M and any 0 ≤ j, k ≤ n, we have: where the last step follows from Lemmas 7 and 8.
The assertion then follows by taking k = CM log(4/3) log n and the union bound. ✷ Now, to prove Lemma 3, notice that by (4.6), choosing M sufficiently large.

Ergodicity from the point of view of a tagged particle
In this section we prove the ergodicity of the shift from the point of view of a tagged particle. Note that the measureP on Ω along with the transition probabilities π(ω, ·) define a unique measure P on Ω Z which is invariant with respect to the shift θ : Ω Z → Ω Z given by where ω = (ω 1 , ω 2 , . . . ) ∈ Ω Z . We sayP is ergodic for the Markov process {T Xn (ω)} n≥0 if the measure P is ergodic for θ. See [9] for more details.
A measure preserving system (Y, S, ν) is said to be totally ergodic if (Y, S k , ν) is ergodic for all k. Any measure preserving system that is strong mixing is also totally ergodic (see [17]). Lemma 9. The vertical shift (Ω, T (0,1) , P) is strong mixing and thus totally ergodic. In particular (Ω, T (0,2) , P) is ergodic.
Proof. For any cylinder set R ⊂ Ω which is defined by the values in a finite rectangle we have that for any sufficiently large M the sets T (0,M ) (R) and R are independent. As these cylinder sets generate the σ-algebra, the shift T (0,1) is mixing and totally ergodic. Thus T (0,2) is ergodic. A (very slightly) more involved analysis shows that T (0,1) is weak Bernoulli and isomorphic to the shift on an infinite entropy i.i.d. measure. ✷ Now we want to define a new transformation built from (Ω, T (0,2) , P). This transformation is constructed from (Ω, T (0,2) , P) by inducing on the subset {v(0, 0) > 0} and then building a "Kakutani skyscraper" of height v(0, 0). The techniques of inducing and building skyscrapers are standard in ergodic theory and have been well studied in the theory of Kakutani equivalence [15] [11] [17]. We start by describing the state space and the measure.
Also note that as every point in π −1 (π(A)) is in theT -orbit of a point in A we have A ⊂ π −1 (π(A)) ⊂ ∞ m=−∞T m (A) = A, (4.12) where the last containment is by the invariance of A.
Lemma 12. If A ⊂ Ω is invariant for T X 1 , thenÂ is invariant forT .