Boundedness properties of operators associated with non-degenerate symmetric $α$-stable, $α\in (1,2)$, probability measures on $\mathbb{R}^d$ are investigated on appropriate, Euclidean or otherwise, $L^p$-spaces, $p \in (1,+\infty)$. Our approach is based on first obtaining Bismut-type formulae which lead to useful representations for various operators. In the Euclidean setting, the method of transference and one-dimensional multiplier theory combined with fine properties of stable distributions provide dimension-free estimates for the fractional Laplacian. In the non-Euclidean setting, we obtain boundedness results for the non-singular cases as well as dimension-free estimates when the reference measure is the rotationally invariant $α$-stable probability measure.
We apply improved elliptic regularity results to a concrete symmetric Dirichlet form and various non-symmetric Dirichlet forms with possibly degenerate symmetric diffusion matrix. Given the (non)-symmetric Dirichlet form, using elliptic regularity results and stochastic calculus we show weak existence of the corresponding singular stochastic differential equation for any starting point in some subset E of R^d. As an application of our approach we can show the existence of n-regularized Liouville Brownian motion only via Dirichlet form theory starting from all points in R^2.
We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients $a^ω$. The diffusion is formally associated with $L^ωu = \nabla\cdot(a^ω\nabla u)$, and we make sense of it through Dirichlet forms theory. We prove for $X$ a quenched invariance principle, under some moment conditions on the environment; the key tool is the sublinearity of the corrector obtained by Moser's iteration scheme.
We study a symmetric diffusion process on $\mathbb{R}^d$, $d\geq 2$, in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also shown for a natural choice of speed measure, under an additional mixing assumption on the environment. Using these estimates, a scaling limit for the Green's function is proven.
For a contraction $C_0$-semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincaré inequalities for the symmetric and anti-symmetric part of the generator. As applications, non-exponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.
We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic H{ö}lder spaces. Our approach mainly relies on a perturbative method based on forward parametrix expansions and, due to the low regularizing properties on the degenerate variables and to some integrability constraints linked to the stability index, it also exploits duality results between appropriate Besov Spaces. In particular, our…
This paper studies homogenization of symmetric non-local Dirichlet forms with $α$-stable-like jumping kernels in one-parameter stationary ergodic environment. Under suitable conditions, we establish homogenization results and identify the limiting effective Dirichlet forms explicitly. The coefficients of the jumping kernels of Dirichlet forms and symmetrizing measures are allowed to be degenerate and unbounded; and the coefficients in the effective Dirichlet forms can be degenerate.
We obtain explicit criteria for both exponential ergodicity and strong ergodicity for one-dimensional time-changed symmetric stable processes with $α\in(1,2)$. Explicit lower bounds for ergodic convergence rates are given.
We study the asymptotics, for small and large values, of the supremum of a product of symmetric stable processes. We show in particular that the persistence exponent remains the same as for only one process, up to some logarithmic terms.
We determine the limiting distribution and the explicit tail behavior for the maximal displacement of a branching symmetric stable process with spatially inhomogeneous branching structure. Here the branching rate is a Kato class measure with compact support and can be singular with respect to the Lebesgue measure.
We consider the ferromagnetic q-state Potts model on a finite grid graph with non-zero external field and periodic boundary conditions. The system evolves according to Glauber-type dynamics described by the Metropolis algorithm, and we focus on the low temperature asymptotic regime. We analyze the case of negative external magnetic field. In this scenario there are $q-1$ stable configurations and a unique metastable state. We describe the asymptotic behavior of the first hitting time from the metastable to the set of the stable states as $β\to\infty$ in probability, in expectation, and in dist…
In this paper, we prove two limit laws for functionals of self-intersection symmetric alpha-stable processes with alpha\in(1,2). The results are obtained based on the method of moments, the sample configuration and the chaining argument introduced in (Nualart and Xu 2013) are employed.
In this paper we investigate functions that are harmonic with respect to the non-symmetric strictly $α$-stable Lévy processes on an open set $D \in \mathbb{R}^d$. We obtain the explicit formula for their boundary decay rate at parts of the boudary of $D$ outside of which they vanish.
We estimate up to universal constants tails of symmetric and totally asymmetric 1-dimensional $α$-stable distributions in terms of functions of the parameters of these distributions. In particular, for values of $α$ close to $2$ we specify where exactly the tail changes from being Gaussian and starts to behave like in the Pareto distribution
The L2-approximation of occupation and local times of a symmetric $α$-stable L{é}vy process from high frequency discrete time observations is studied. The standard Riemann sum estimators are shown to be asymptotically efficient when 0 < $α$ $\le$ 1, but only rate optimal for 1 < $α$ $\le$ 2. For this, the exact convergence of the L2-approximation error is proven with explicit constants.
In this paper we consider weak Harnack inequality and Hölder regularity estimates for symmetric $α$-stable Lévy process in $\mathbb{R}^d$, $α\in (0,2)$, $d\geq 2$. We consider a symmetric $α$-stable Lévy process $X$ for which a spherical part $μ$ of the Lévy measure is a spectral measure. In addition, we assume that $μ$ is absolutely continuous with respect to the uniform measure $σ$ on the sphere and impose certain bounds on the corresponding density. Eventually, we show that the weak Harnack inequality holds, which we apply to prove Hölder regularity results.
