In this paper, we study the stochastic homogenization for a class of symmetric random walks in random conductance model, whose one-step transition probability from x to y is proportional to |x-y|^-d-2. As the associated jumping kernel fails to be L^2-integrable yet admits a finite α-th moment for all α∈ (0,2), we refer to the corresponding process (X^_t)_t≥0 as a long-range random walk with critical jump index. In this critical regime, the scaled process (k^-1X_k^2(log k)^-1t)_t≥ 0, whose scaling order is different from the diffusive scaling and the α-stable scaling, converges to a Brownian motion. Besides characterizing the limiting Brownian motion, we will give a convergence rate for associated scaled resolvents, which obeys the order (log k)^-1/2+1/2(d-2)+ε with any ε>0 for all d>3.

Brox's diffusion is a typical one-dimensional singular diffusion, which was introduced by Brox (1986) as a continuous analogue of Sinai's random walk. In this paper, we will establish quenched heat kernel estimates for short time and annealed heat kernel estimates for large time of Brox's diffusion. The proofs are based on Brox's construction via the scale-transformation and the time-change arguments as well as the theory of resistance forms for symmetric strongly recurrent Markov processes. We emphasize that, since the reference measure of Brox's diffusion does not satisfy the so-called volume doubling conditions neither for the small scale nor the large scale, the existing methods for heat kernel estimates of diffusions in ergodic media do not work, and new techniques will be introduced to establish both quenched and annealed heat kernel estimates of Brox's diffusions, which take into account different oscillation properties for one-dimensional Brownian motion in random environments.

We investigate the stochastic homogenization of a class of turbulent diffusions generated by non-local symmetric Lévy operators with divergence-free drift fields in ergodic random environments, where neither the drift fields nor their associated stream functions are assumed to be bounded. A pivotal step in our proof is the establishment of W_loc^1,q estimates with q∈ (1,2) for the corresponding correctors, under mild prior regularity conditions imposed on the Lévy measure and the stream function.

In this paper we will study the homogenization of stable-like processes with divergence-free drifts in ergodic environments. In particular, neither the drifts nor the stream functions are required to be bounded.

We establish the quenched local limit theorem for reversible random walk on Zd (with d >= 2) among stationary ergodic random conductances that permit jumps of arbitrary length. The proof is based on the weak parabolic Harnack inequalities and on-diagonal heat-kernel estimates for long-range random walks on general ergodic environments. In particular, this partly solves (Probab. Theory Related Fields 180 (2021) 847-889), Open Problem 2.7, where the quenched invariance principle was obtained. As a byproduct, we prove the maximal inequality with an extra tail term for long-range reversible random walks, which in turn yields the everywhere sublinear property for the associated corrector.

In this paper we will study the equivalence between super-Poincaré inequality and some log-Sobolev type inequalities, including weak log-Sobolev inequality and super log-Sobolev inequality. The explicit relations between associated rate functions will also be established.

In this paper, we establish higher-order convergence rates of the periodic homogenizatio for symmetric Lévy-type operators, encompassing the subcritical α-stable regime, critical regime, and supercritical diffusive regime. To this end, we develop a systematic framework to decompose the contributions of the underlying jumping kernel across small, intermediate, and large spatial scales – a strategy tailored to all the aforementioned regimes. To the best of our knowledge, this work represents the first comprehensive study of higher-order convergence rates in the homogenization of non-local operators.

We establish global two-sided heat kernel estimates (for full time and space) of the Schrodinger operator -1/2 Delta + V on & Ropf;(d), where the potential V (x) is locally bounded and behaves like c|x |(-alpha) near infinity with alpha is an element of (0, 2) and c > 0, or with alpha > 0 and c < 0. Our results improve all known results in the literature, and it seems that the current paper is the first one where two-sided matching heat kernel bounds for the long range potentials are established. The results of the paper mostly rely on probabilistic approaches.

In this paper we will introduce a class of forward-backward stochastic differential equations on tensor fields of Riemannian manifolds, which are related to semilinear parabolic partial differential equations on tensor fields. Moreover, we will use these forward-backward stochastic differential equations to give a stochastic representation of incompressible Navier-Stokes equations on Riemannian manifolds, where some extra conditions used in (Potential Anal. 48 (2018) 181-206) are not required.

We show anchored versions of the Nash inequality for discrete non-local divergence-form operators with degenerate weights. They allow to control the L^2-norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide on-diagonal heat kernel upper bounds for a class of random conductance models with degenerate jump rates allowing long-range jumps. The results are established on a class of graphs including the integer lattice and possibly correlated supercritical percolation clusters.

We give two-sided, global (in all variables) estimates of the heat kernel and the Green function of the fractional Schrödinger operator with a non-negative and locally bounded potential V such that V(x) →∞ as |x| →∞. We assume that V is comparable to a radial profile with the doubling property. Our bounds are sharp with respect to spatial variables and qualitatively sharp with respect to time. The methods we use combine probabilistic and analytic arguments. They are based on the strong Markov property and the Feynman–Kac formula.

We establish quantitative homogenization results for time-dependent random conductance models with stable-like long range jumps on ^d, where the transition probability from x to y is given by w_t, x,y|x-y|^-d-α with α∈ (0,2). In particular, time-dependent random coefficients {w_t,x,y: t∈_+, (x,y)∈ E} are uniformly bounded from above (but may be degenerate), and satisfy the Kolmogorov continuous condition, where E={(x, y): x ≠ y ∈^d} is the set of all unordered pairs on ^d. The proofs are based on L^2-estimates and energy estimates for solutions to regionalparabolic equations and multi-scale Poincaré inequalities associated with time-dependent symmetric stable-like random walks with random coefficients.

In this paper, we study the quenched invariance principle for L 2-integrable long-range random walk in balanced random environments. As a key ingredient for the proof, we will also prove a kind of ABP type maximal inequality for a general non-local operator in Z d.

We establish global two-sided heat kernel estimates (for full time and space) of the Schrödinger operator -1/2Δ+V on ^d, where the potential V(x) is locally bounded and behaves like c|x|^-α near infinity with α∈ (0,2) and c> 0, or with α>0 and c<0.Our results improve all known results in the literature, and it seems that the current paper is the first one where consistent two-sided heat kernel bounds for the long range potentials are established.

Homogenization for non-local operators in periodic environments has been studied intensively. So far, these works are mainly devoted to the qualitative results, that is, to determine explicitly the operators in the limit. To the best of authors' knowledge, there is no result concerning the convergence rates of the homogenization for stable-like operators in periodic environments. In this paper, we establish a quantitative homogenization result for symmetric α-stable-like operators on ^d with periodic coefficients. In particular, we show that the convergence rate for the solutions of associated Dirichlet problems on a bounded domain D is of order ε^(2-α)/2_{α∈ (1,2)}+ε^α/2_{α∈ (0,1)}+ε^1/2|log|^2_{α=1}, while, when the solution to the equation in the limit is in C^2_c(D), the convergence rate becomes ε^2-α_{α∈ (1,2)}+ε^α_{α∈ (0,1)}+ε |log|^2_{α=1}. This indicates that the boundary decay behaviors of the solution to the equation in the limit affects the convergence rate in the homogenization.

We consider random conductance models with long range jumps on $\Z^d$, where the one-step transition probability from $x$ to $y$ is proportional to $w_{x,y}|x-y|^{-d-\alpha}$ with $\alpha\in (0,2)$. Assume that $\{w_{x,y}\}_{(x,y)\in E}$ are independent, identically distributed and uniformly bounded non-negative random variables with $\Ee w_{x,y}=1$, where $E$ is the set of all unordered pairs on $\Z^d$. We obtain a quantitative version of stochastic homogenization for these random walks, with explicit polynomial rates up to logarithmic corrections.

Consider the Schr\"odinger operator $ \mathcal L^V=-\Delta+V $ on $\R^d$, where $V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$ with some constants $c_1,c_2>0$ and a nondecreasing and strictly positive function $g:[0,\infty)\to [1,+\infty)$ that satisfies $g(2r)\le c_0 g(r)$ for all $r>0$ and $\lim_{r\to \infty} g(r)=\infty.$ We establish global in time and qualitatively sharp bounds for the heat kernel of the associated Schrödinger semigroup by the probabilistic method. In particular, we can present global in space and time two-sided bounds of heat kernel even when the Schrödinger semigroup is not intrinsically ultracontractive. Furthermore, two-sided estimates for the corresponding Green's functions are also obtained.

Abstract In 2020, the lockdown of Wuhan due to the outbreak of COVID‐19 impacted various aspects of local college students' life and may further negatively affect their psychological state. This study was conducted among 652 Wuhan local college students during the quarantine of this city. We assessed their psychological state using Depression‐Anxiety‐Stress Scale 21 and evaluated their living condition including diet, schedule, recreational activities, social contact, academic life, and attention paid to pandemic news. Results showed that 16.87% of the students reported stress, 28.68% with anxiety, and 35.12% had depression. According to multivariate logistic regression analysis, having a medical background was associated with higher stress levels; students who had an irregular diet and schedule were more likely to develop stress, anxiety, and depression; students with their academic life affected had a higher prevalence of anxiety and depression. By studying local students in the hardest‐hit area during the pandemic, our findings can provide references for the improvement of college students' mental health in the long term.