A Sobolev Inequality and the Individual Invariance Principle for Diffusions in a Periodic Potential

Abstract

Publication issue de la thèse de Moustapha BaInternational audienceWe consider a diffusion process in $\mathbb{R}^d$ with a generator of the form $ L:=\frac 12 e^{V(x)}div(e^{-V(x)}\nabla ) $ where $V$ is measurable and periodic. We only assume that $e^V$ and $e^{-V}$ are locally integrable. We then show that, after proper rescaling, the law of the diffusion converges to a Brownian motion for Lebesgue almost all starting points. This pointwise invariance principle was previously known under uniform ellipticity conditions (when $V$ is bounded), and was recently proved under more restrictive $L^p$ conditions on $e^V$ and $e^{-V}$. Our approach uses Dirichlet form theory to define the process, martingales and time changes and the construction of a corrector. Our main technical tool to show the sub-linear growth of the corrector is a new weighted Sobolev type inequality for integrable potentials. We heavily rely on harmonic analysis technics

Quenched invariance principle for random walks among random degenerate conductances

“…Proof of Theorem 2. The choice γ = 1 2p ′ in Corollary 1 yield (7) for y = 0 ∈ Z d and by translation we obtain the general claim.…”

“…To the best of our knowledge Theorem 1 is the first quenched invariance principle in the general stationary & ergodic setting under less restrictive moment condition compared to (4) valid in d ≥ 3. Optimality of condition (2) in Theorem 1 is not clear to us, since in particular in [7] a quenched invariance principle for diffusion in R d with a locally integrable periodic potential is proven. However, we emphasize that condition (2) is essentially optimal for the everywhere sublinearity of the corrector, see Proposition 2 and Remark 4.…”

Invariance principle for symmetric diffusions in a degenerate and unbounded stationary and ergodic random medium

“…In periodic environment this has been proven recently in [3] using ideas coming from harmonic analysis and Muckenhaupt's weights. The authors consider a generator in divergence form given by Lu = e V ∇ · (e −V ∇u), where V : R d → R is periodic and measurable such that e V + e −V is locally integrable.…”

“…Recently, a lot of efforts has been put into extending this result beyond the uniform elliptic case. For example [14] consider a non-symmetric situation with uniformly elliptic symmetric part and unbounded antisymmetric part and the recent paper [3] proves an invariance principle for divergence form operators Lu = e V ∇ · (e −V ∇u) where V is periodic and measurable. They only assume that e V + e −V is locally integrable.…”

Invariance principle for the random conductance model in a degenerate ergodic environment

“…Recently, in [4], Ba and Mathieu have established a QFCLT for diffusions in R d with a locally integrable periodic potential. Their approach is also based on a Sobolev-type inequality, where the sublinearity of the corrector is only obtained along the path of the process.…”