This paper enhances the result of the work [G. Kozma, B. Tóth, Ann. Probab. vol. 45 (2017) 4307-4347] . We prove the central limit theorem (in probability w.r.t. the environment) for the displacement of a random walker in divergence-free (or, doubly stochastic) random environment, with substantially relaxed ellipticity assumptions. Integrability of the reciprocal of the symmetric part of the jump rates is only assumed (rather than their boundedness, as in previous works on this type of RWRE). Relaxing ellipticity involves substantial changes in the proof, making it conceptually elementary in the sense that it does not rely on Nash's inequality in any disguise.
In the work [14] the weak CLT (that is, in probability with respect to the environment) was established for random walks in doubly stochastic (or, divergence-free) random environments, under the conditions of (ı) strict ellipticity assumed for the symmetric part of the drift field, and, (ıı) H -1 (|∆|) assumed for the antisymmetric part of the drift field. The proof relied on a martingale approximation (a la Kipnis-Varadhan) adapted to the non-self-adjoint and non-sectorial nature of the problem, the two substantial technical components being:
• A functional analytic statement about the unbounded operator formally written as
, where L is the infinitesimal generator of the environment process, as seen from the position of the moving random walker.
• A diagonal heat kernel upper bound which follows from Nash’s inequality, valid only under the assumed strict ellipticity.
The assumption of strict ellipticity, however, is conceptually restrictive and excludes relevant applications. In this paper we relax the strict ellipticity assumption, replacing it by an integrability condition on the reciprocals of the conductances. This can be done only by “de-Nashifying” the proof. On the other hand the functional analytic elements are refined. These changes are also of conceptual importance: the present proof (of a stronger result) does not invoke a conceptually higher level (than the CLT) element like a local heat kernel estimate. Altogether, it is conceptually simpler than that in [14]. (This was already demonstrated in [30] where the result of [14] was re-proved along a baby version of the arguments in the present paper.) To the best of our knowledge this is the first time a CLT is proved for a random walk in a non-reversible random environment without a strong ellipticity assumption. The integrability conditions imposed on the random jump rates seem to be close to optimal.
Let (Ω, F, π, (τ z : z ∈ Z d )) be a probability space with an ergodic Z d -action. Denote by N := {k ∈ Z d : |k| = 1} the set of unit elements generating Z d as an additive group. These will serve as the set of elementary steps of a continuous time nearest neighbour random walk on Z d . Let p : Ω → [0, ∞) N satisfy (π-a.s.) the following bi-stochasticity condition
and p : Z d × Ω → [0, ∞) N be its lifting to a random field over Z d , p k (x, ω) := p k (τ x ω).
(Throughout the paper, measurable functions f : Ω → R and their lifting to a random field f : Z d × Ω → R, defined as f (x, ω) := f (τ x ω), will be denoted by the same symbol.) Given the random field p : Z d ×Ω → [0, ∞) N , define the continuous-time random walk in random environment (RWRE), t → X(t) ∈ Z d as the Markovian nearest neighbour random walk with jump rates
and initial position X(0) = 0.
We use the notation P ω (•), E ω (•) and Var ω (•) for quenched probability, expectation and variance. That is: probability, expectation, and variance with respect to the distribution of the random walk X(t), conditioned on fixed environment ω ∈ Ω. The notation P • := Ω P ω (•) dπ(ω), E • := Ω E ω (•) dπ(ω) and Var • := Ω Var ω (•) dπ(ω) + Ω E ω (•) 2 dπ(ω) -E • 2 will be reserved for annealed probability, expectation and variance. That is: probability, expectation and variance with respect to the random walk trajectory X(t) and the environment ω, sampled according to the distribution π.
The environment process (as seen from the position of the random walker) is, t → η t ∈ Ω, defined as
This is a pure jump Markov process on the state space Ω. The infinitesimal generator of its Markovian semigroup is
The linear operator L is well defined for all measurable functions f : Ω → R, not just a formal expression.
It is well known (and easy to check, see e.g. [13]) that bi-stochasticity (1) of the jump rates p is equivalent to stationarity (in time) of the a priori distribution π of the environment process t → η t ∈ Ω. Moreover, under the conditions (1) and ( 8) (see below) spatial ergodicity of (Ω, F, π, (τ z : z ∈ Z d )) also implies time-wise ergodicity of the (time-wise) stationary environment process process t → η t ∈ (Ω, F, π). See [13] for details. Hence it follows that under these conditions the random walk t → X(t) will have stationary and ergodic annealed increments. Though, in the annealed setting the walk is obviously not Markovian.
It is convenient to separate the symmetric and antisymmetric parts of the jump rates:
and note that
and ( 1) is equivalent to
We assume for now weak ellipticity and finiteness of the conductances: π-a.s. for all k ∈ N ,
Later somewhat stronger integrability conditions will be imposed (see (18) further below). Accordingly, we decompose the infinitesimal generator into Hermitian, and anti-Hermitian parts (with respect to the stationary measure π) as
The right hand sides of ( 4) and ( 9) make perfect sense for arbitrary measurable functions f : Ω → R. So, disregarding for the time being topological issues in function spaces, the linear operators L, S
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