We introduce a class of continuous-state branching processes with immigration, predation and competition, which can be viewed as a combination of the classical Lotka-Volterra model and continuous-state branching processes with competition that were introduced by Berestycki, Fittipaldi, and Fontbona (Probab. Theory Relat. Fields, 2018). This model can be constructed as a unique strong solution to a class of two-dimensional stochastic differential equations with jumps. We establish sharp conditions for the uniform ergodicity in the total variation of this model. Our proof relies on a novel, localized Markovian coupling approach, which is of its own interest in the ergodicity theory of Markov processes with interactions.

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.

We establish the exponential ergodicity in a weighted total variation distance of continuous-state branching processes with immigration in random environments with competition and catastrophes, under a Lyapunov-type condition and other mild assumptions. The proof is based on a Markov coupling process along with some delicate estimates for the associated coupling generator. In particular, the main result indicates whether and how the competition mechanism, the random environment and the catastrophe could balance the branching mechanism respectively to guarantee the exponential ergodicity of the processes.

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 study long-time behaviors for branching-diffusion process corresponding to the drifted Schrödinger operator ℒ = 1/2 Δ+ ⟨∇ V,∇⟩ - K, where K represents the reduction rate of a population dynamics and ∇ V is a given drift term. In particular, we establish exponential convergence rates for the total mass of this process and characterize its quasi-stationary distribution. The proof is based on a novel transformation in spectral analysis, and heat kernel estimates for Schrödinger operators with unbounded potentials. The result is new even in the one-dimensional setting, which especially improves the recent work .

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.

We consider fractional stochastic heat equations with space-time Lévy white noise of the form ∂ X/∂ t(t,x)=ℒ_αX(t,x)+σ (X(t,x))Λ̇(t,x). Here, the principal part ℒ_α=-(-Δ )^α /2 is the d-dimensional fractional Laplacian with α∈ (0,2) , the noise term Λ̇(t,x) denotes the space-time Lévy white noise, and the function σ : ℝ→ℝ is Lipschitz continuous. Under suitable assumptions, we obtain bounds for the Lyapunov exponents and the growth indices of exponential type on pth moments of the mild solutions, which are connected with the weakly intermittency properties and the characterizations of the high peaks propagate away from the origin. Unlike the case of the Gaussian noise, the proofs heavily depend on the heavy tail property of heat kernel estimates for the fractional Laplacian. The results complement these in [8, 12] for fractional stochastic heat equations driven by space-time white noise and stochastic heat equations with Lévy noise, respectively.

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.

This paper investigates the essential spectrum and compactness property of Markov semigroups generated by multi-dimensional Brox diffusion processes under two types of random media: stationary Gaussian random fields and semi-selfsimilar Lévy random fields. For Gaussian environments satisfying general stationary covariance growth conditions, we prove that the associated semigroup is almost surely noncompact; if the covariance function grows sublinearly at infinity, then the bottom of the essential spectrum vanishes almost surely, with fractional Brownian fields as a concrete example. For multi-dimensional semi-selfsimilar Lévy environments with scaling index α∈(1,2), we show that the essential spectral bottom equals zero almost surely for arbitrary space dimension d≥1, and the same conclusion holds for one-dimensional symmetric α-stable Lévy processes with any α∈(0,2). Furthermore, we study one-dimensional diffusion operators perturbed by random Lévy drift. When 0<δ≤ 1/α, random environmental fluctuations can destroy the compactness of semigroups induced by deterministic power-law potentials ±|x|^δ, even if the unperturbed semigroup is compact. The analysis relies on almost sure volume growth estimates for the random reference measure induced by environmental potentials, sample path asymptotics of random fields, ergodic theory of scaling transforms, and spectral criteria for regular Dirichlet forms. A unified framework linking sample path behaviors of random potentials to spectral characteristics of diffusion semigroups is established, and several open problems for large potential exponents are stated.

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 are interested in the issues on existence, uniqueness, and multiplicity of stationary distributions for McKean-Vlasov SDEs with jumps. In detail, with regarding to McKean-Vlasov SDEs driven by pure jump Lévy processes, we principally (i) explore the existence of stationary distributions via Schauder's fixed point theorem under an appropriate Lyapunov condition; (ii) tackle the uniqueness of stationary distributions and the convergence to the equilibria as long as the underlying drifts are continuous with respect to the measure variables under the weighted total variation distance and the L^1-Wasserstein distance, respectively; (iii) demonstrate the multiplicity of stationary distributions under a locally dissipative condition. In addition, some illustrative examples are provided to show that the associated McKean-Vlasov SDEs possess a unique, two and three stationary distributions, respectively.

We study the ergodic property of a continuous-state branching process with immigration and competition. The exponential ergodicity in a weighted total variation distance is proved under natural assumptions. The main theorem applies to subcritical, critical and supercritical branching mechanisms, including all those of stable types. The proof is based on the construction of a Markov coupling process and the choice of a nonsymmetric control function for the distance. Those are designed to identify and to take the advantage of the dominating factor from the branching, immigration and competition mechanisms in different parts of the state space. The approach provides a way of finding a lower bound of the ergodicity rate.

Based on the hypocoercivity approach due to Villani (2009), Dolbeault, Mouhot and Schmeiser (2015) established a new and simple framework to investigate directly the L 2 {L<^>{2}} -exponential convergence to the equilibrium for the solution to the kinetic Fokker-Planck equation. Nowadays, the general framework advanced by Dolbeault, Mouhot and Schmeiser (2015) is named as the DMS framework for hypocoercivity. Subsequently, Grothaus and Stilgenbauer (2014) built a dual version of the DMS framework in the kinetic Fokker-Planck setting. No matter what the abstract DMS framework by Dolbeault, Mouhot and Schmeiser (2015) or the dual counterpart by Grothaus and Stilgenbauer (2014), the densely defined linear operator involved is assumed to be decomposed into two parts, where one part is symmetric and the other part is anti-symmetric. Thus, the existing DMS framework is not applicable to investigate the L 2 {L<^>{2}} -exponential ergodicity for stochastic Hamiltonian systems with alpha-stable L & eacute;vy noises, where one part of the associated infinitesimal generators is anti-symmetric whereas the other one is not symmetric at all. In this paper, we shall develop a dual version of the DMS framework in the fractional kinetic Fokker-Planck setup, where one part of the densely defined linear operator under consideration need not to be symmetric. As a direct application, we explore the L 2 {L<^>{2}} -exponential ergodicity of stochastic Hamiltonian systems with alpha-stable L & eacute;vy noises. The proof is also based on Poincar & eacute; inequalities for non-local stable-like Dirichlet forms and the potential theory for fractional Riesz potentials.

In this note, under weak monotonicity and weak coercivity, we address strong well-posedness of McKean–Vlasov stochastic differential equations (SDEs) driven by Lévy jump processes, where the coefficients are Lipschitz continuous (with respect to the measure variable) under the L^β -Wasserstein distance for β∈ [1,2]. Moreover, the issues of on the weak propagation of chaos (i.e., convergence in distribution via convergence of the empirical measure) and strong propagation of chaos (i.e., at the level paths by coupling) are explored simultaneously. To treat the strong well-posedness of McKean–Vlasov SDEs we are interested in, we investigate strong well-posedness of classical time-inhomogeneous SDEs with jumps under a local weak monotonicity and a global weak coercivity. Such a result is of independent interest, and, most importantly, can provide an available reference on strong well-posedness of Lévy-driven SDEs under the monotonicity condition, which has been is missing for a long time. Based on the theory derived, along with the interlacing technique and the Banach fixed point theorem, the strong well-posedness of McKean–Vlasov SDEs driven by Lévy jump processes can be established. Additionally, as a potential extension, strong well-posedness and conditional propagation of chaos are treated for Lévy-driven McKean–Vlasov SDEs with common noise under a weak monotonicity.

We establish explicit integral tests for spatial asymptotic behaviors of fractional stochastic heat equations driven by additive L & eacute;vy white noise. Our results indicate that fractional stochastic heat equations enjoy the so-called additive physical intermittent property in all dimensions when the driven L & eacute;vy white noise is sufficiently light-tailed. The proofs are based on heat kernel estimates for the fractional Laplacian and exact tail behaviors for Poissonian functionals associated with the driven L & eacute;vy white noise.

Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations. Our findings give the explicit decay rates for the energy, but do not address reflection/transmission of waves at the interface of the damping. Still for a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.

Characterizing the differential privacy (DP) of learning algorithms has become a major challenge in recent years. In parallel, many studies suggested investigating the behavior of stochastic gradient descent (SGD) with heavy-tailed noise, both as a model for modern deep learning models and to improve their performance. However, most DP bounds focus on light-tailed noise, where satisfactory guarantees have been obtained but the proposed techniques do not directly extend to the heavy-tailed setting. Recently, the first DP guarantees for heavy-tailed SGD were obtained. These results provide (0,δ)-DP guarantees without requiring gradient clipping. Despite casting new light on the link between DP and heavy-tailed algorithms, these results have a strong dependence on the number of parameters and cannot be extended to other DP notions like the well-established Rényi differential privacy (RDP). In this work, we propose to address these limitations by deriving the first RDP guarantees for heavy-tailed SDEs, as well as their discretized counterparts. Our framework is based on new Rényi flow computations and the use of well-established fractional Poincaré inequalities. Under the assumption that such inequalities are satisfied, we obtain DP guarantees that have a much weaker dependence on the dimension compared to prior art.

In this paper, we are interested in conditional McKean-Vlasov jump diffusions, which are also termed as McKean-Vlasov stochastic differential equations with jump idiosyncratic noise and jump common noise. As far as conditional McKean-Vlasov jump diffusions are concerned, the corresponding conditional distribution flow is a measure-valued process, which indeed satisfies a stochastic partial integral differential equation driven by a Poisson random measure. Via a novel construction of the asymptotic coupling by reflection, we explore the ergodicity of the underlying measure-valued process corresponding to a one-dimensional conditional McKean-Vlasov jump diffusion when the associated drift term fulfils a partially dissipative condition with respect to the spatial variable. In addition, the theory derived demonstrates that the intensity of the jump common noise and the jump idiosyncratic noise can simultaneously enhance the convergence rate of the exponential ergodicity.

Let X:=(X_t)_t≥ 0 be an ergodic Markov process on ^d, and p>0. We derive upper bounds of the p-Wasserstein distance between the invariant measure and the empirical measures of the Markov process X. For this we assume, e.g. that the transition semigroup of X is exponentially contractive in terms of the 1-Wasserstein distance, or that the iterated Poincaré inequality holds together with certain moment conditions on the invariant measure. Typical examples include diffusions and underdamped Langevin dynamics.