On perturbed substochastic semigroups in abstract state spaces

无穷维凸区域上扰动Ornstein-Uhlenbeck半群的梯度估计

Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains

本文研究了无穷维凸区域上扰动Ornstein-Uhlenbeck半群的逐点梯度估计，推导出对数Sobolev不等式、Poincaré不等式及可和性改进性质，并分析了半群的渐近行为。

Sharp growth rates for semigroups using resolvent bounds

We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroup is positive and the underlying space is an $L^{p}$-space or a space of continuous functions. We also prove variations of the main results on fractional domains; these are valid on more general Banach spaces. In the second part of the article we apply our main theorem to prove optimality in a classical exam…

Fourth-order Schrödinger type operator with singular potentials

In this paper we study the biharmonic operator perturbed by an inverse fourth-order potential. In particular, we consider the operator $A=Δ^2-V=Δ^2-c|x|^{-4}$ where $c$ is any constant such that $c<\left(\frac{N(N-4)}{4}\right)^2$. The semigroup generated by $-A$ in $L^2(\mathbb{R}^N)$, $N\geq5$, extrapolates to a bounded holomorphic $C_0$-semigroup on $L^p(\mathbb{R}^N)$ for $p\in [p^{'}_0,p_0]$ where $p_0=\frac{2N}{N-4}$ and $p_0^{'}$ is its dual exponent. Furthermore, we study the boundedness of the Riesz transform $ΔA^{-1/2}$ on $L^p(\mathbb{R}^N)$ for all $p\in(p_0^{'},2]$.

On some new results on anisotropic singular perturbations of second order elliptic operators

In this article, we deal with some problems involving a class of singularly perturbed elliptic operator. We prove the asymptotic preserving of a general Galerkin method associated to a semilinear problem. We use a particular Galerkin approximation to estimate the convergence rate on the whole domain, for the linear problem. Finally, we study the asymptotic behavior of the semigroup generated.

Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$-spaces

In this paper we give sufficient conditions on $α\geq 0$ and $c\in \mathbb{R}$ ensuring that the space of test functions $C_c^\infty(\mathbb{R}^N)$ is a core for the operator $$L_0u=(1+|x|^α)Δu+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u,$$ and $L_0$ with suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(\mathbb{R}^N),\,1<p<\infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.

Rescaling approach for a stochastic population dynamics equation perturbed by a linear multiplicative Gaussian noise

We are concerned with a nonlinear nonautonomous model represented by an equation describing the dynamics of an age-structured population diffusing in a space habitat $O,$ governed by local Lipschitz vital factors and by a stochastic behavior of the demographic rates possibly representing emigration, immigration and fortuitous mortality. The model is completed by a random initial condition, a flux type boundary conditions on $\partial O$ with a random jump in the population density and a nonlocal nonlinear boundary condition given at age zero. The stochastic influence is expressed by a linear m…

An integrated version of Varadhan's asymptotics for lower-order perturbations of strong local Dirichlet forms

The studies of Ramírez, Hino-Ramírez, and Ariyoshi-Hino showed that an integrated version of Varadhan's asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type.

Regularity of solutions to Kolmogorov equations with perturbed drifts

We prove that a probability solution of the stationary Kolmogorov equation generated by a first order perturbation $v$ of the Ornstein--Uhlenbeck operator $L$ possesses a highly integrable density with respect to the Gaussian measure satisfying the non-perturbed equation provided that $v$ is sufficiently integrable. More generally, a similar estimate is proved for solutions to inequalities connected with Markov semigroup generators under the curvature condition $CD(θ,\infty)$. For perturbations from $L^p$ an analog of the Log-Sobolev inequality is obtained. It is also proved in the Gaussian ca…

Maximal $L^2$-regularity in nonlinear gradient systems and perturbations of sublinear growth

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $φ$ has a smoothing effect, discovered by H. Brézis, which implies maximal regularity for the evolution equation. We use this and Schaefer's fixed point theorem to solve the evolution equation perturbed by a Nemytskii-operator of sublinear growth. For this, we need that the sublevel sets of $φ$ are not only closed but even compact. We apply our results to the $p$-Laplacian and also to the Dirichlet-to-Neumann operator with respect to $p$-harmonic functions.

Physiologically structured populations with diffusion and dynamic boundary conditions

We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. The model is equipped with generalized Wentzell-Robin (or dynamic) boundary conditions. This allows modelling of "adhesion" at extremely small or large sizes. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. This is carried out via establishing dissipativity of a suitably perturbed semigroup generator…

Spectral Analysis and Long-Time Behaviour of a Fokker-Planck Equation with a Non-Local Perturbation

In this article we consider a Fokker-Planck equation with a non-local, mass preserving perturbation. We show that the perturbed Fokker-Planck operator generates a $C_0$-semigroup on an exponentially weighted $L^2$-space. Surprisingly, the spectrum of the Fokker-Planck operator is not affected by the perturbation. In particular there still exists a unique (normalized) stationary solution of the perturbed equation. And we have convergence towards the stationary state with exponential rate -1, the same rate as for the unperturbed Fokker-Planck equation. Moreover, for any $k\in\mathbb N$ there exi…

由带粗糙输运噪声的三维Navier-Stokes方程生成的随机动力系统

Random dynamical system generated by the 3D Navier-Stokes equation with rough transport noise

针对带粗糙输运噪声的三维Navier-Stokes系统解不唯一、无普遍半群性质的问题，研究者选取满足半群性质且符合能量耗散最大化准则的解，首次构造出该类非唯一SPDE的可测单值随机动力系统。

与对流 Brinkman-Forchheimer 扩展 Darcy 方程相关的转移半群的近似能控性与不可约性

Approximate controllability and Irreducibility of the transition semigroup associated with Convective Brinkman-Forchheimer extended Darcy Equations

本文研究受控 CBFeD 方程在超临界与临界条件下的近似能控性，并据此证明非退化 Gaussian 噪声驱动的随机 CBFeD 系统转移半群在能量空间中不可约。

Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation

The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances. Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I. This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators pertur…