Stochastic homogenization of diffusions in turbulence driven by non-local symmetric Lévy operators

) where b(•; ω) is a divergence-free random field.

When L 0 corresponds to the Laplacian operator, this model is closely related to a fundamental paradigm in statistical fluid mechanics, which describes the dynamics of diffusive particles convected by a random incompressible velocity field. In particular, the long-time behavior of the Markov process generated by the random operator L ω is characterized via the stochastic homogenization framework (or the invariance principle). A core component of this theory lies in the derivation of the effective diffusivity, a key parameter that determines the coefficients of the limiting Brownian motion induced by the drift field b(x; ω) under appropriate regularity conditions.

Independently, Kozlov [30] and Papanicolaou and Varadhan [39] pioneered the so-called -seen from the particle -method to construct the associated corrector in ergodic random environments. This technique has since emerged as a cornerstone in the research of stochastic homogenization. Under the assumption that the drift field b(x; ω) admits a stream function (see Assumption 1.2 below), extensive studies have been conducted on properties of stochastic homogenization for L ω . For instance, Osada [37] established the quenched invariance principle under the condition of a bounded stream function. Landim, Olla and Yau [32] proved the corresponding result for time-dependent environments, assuming the existence of a (time-dependent) bounded stream. Oelschläger [36] relaxed the boundedness constraint and analyzed stochastic homogenization under finite p-moment conditions on the stream function for any p < ∞. With finite second moment assumptions on the stream function, Fannjiang and Papanicolaou [20] further demonstrated the L 2 -convergence for the density of the associated Markov process. Leveraging Moser’s iteration argument, Fannjiang and Komorowski [18,19] proved the quenched invariance principle for both static and time-dependent ergodic environments, respectively, under strengthened finite moment conditions on the stream function. Komorowski and Olla [29] proposed a criterion for stochastic homogenization in time-dependent models based on the spectral resolution of the drift field. Readers are referred to the monograph [28] and the references therein for details on related developments. Fehrman [22] explored stochastic homogenization in space-time ergodic settings, where the impact of temporal variables was completely analyzed, and the limiting process was shown to deviate from a standard Brownian motion in certain cases. Additionally, Fehrman [21] provided a simple proof of quenched stochastic homogenization, and also examined the large-scale Hölder regularity and the first-order Liouville principle of the operator L ω .

Recently, significant progress has been made for the critically correlated case, where the drift field b(x; ω) does not possess a globally defined stream function. Cannizzaro, Haunschmid-Sibitz and Toninelli [6] derived longtime second-moment estimates for a Brownian particle in R 2 , subject to a random, time-independent drift given by the curl of the two-dimensional Gaussian Free Field. By combining stochastic homogenization techniques with refined cutoff arguments, Chatzigeorgiou, Morfe, Otto and Wang [7] improved these estimates to achieve the optimal order in the large-time regime. Furthermore, Armstrong, Bou-Rabee and Kuusi [1] employed a renormalization group approach to analyze the coarse-grained diffusivity across different scales, proving both qualitative and quantitative versions of the quenched invariance principle with super-diffusive scaling. Notably, the limiting Brownian motion in their work is fully determined by the drift field. For additional properties of models with critically correlated divergence-free drift fields, we refer the reader to [2,33,34,35,38] and the references therein.

In contrast, results for the case where L 0 is a non-local operator remain relatively scarce. For symmetric stable-like operators L 0 , Chen and Yin [12] studied the stochastic homogenization of L ω , showing that the limiting process is an α-stable Lévy process with no effective diffusivity term. A feature of this work is that the proof does not require the construction of a corrector. To the best of our knowledge, it remains an unknown question whether the limiting process in the stochastic homogenization of non-local operators L 0 can be a Brownian motion with a non-trivial effective diffusivity. In this paper, we focus on a class of nonlocal Lévy operators L 0 whose Lévy measures are L 2 -integrable (see Assumption 1.1 for details). In fact, for Markov processes generated by symmetric non-local operators with L 2 -integrable jumping kernels, stochastic homogenization has been established by Biskup, Chen, Kumagai and Wang [4], Flegel, Heida and Slowik [23], and Piatni