Related papers: Hypercontractivity for Stochastic Hamiltonian Syst…
In a separable Hilbert space, we study supercontractivity and ultracontractivity properties for a transition semigroups associated with a stochastic partial differential equations. This is done in terms of exponential integrability of…
We show that the Ornstein-Uhlenbeck semigroup associated with a general Poisson random measure is hypercontractive, whenever it is restricted to non-increasing mappings on configuration spaces. We deduce from this result some versions of…
$\mu$ being a nonnegative measure satisfying some log-Sobolev inequality, we give conditions on F for the measure $\nu=e^{-2F} \mu$ to also satisfy some log-Sobolev inequality. Explicit examples are studied.
We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic functions. We introduce a new large class of measures, Euclidean regular and…
We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we identify a convenient auxiliary measure and then we prove conditions equivalent to the existence of an invariant…
Given $p,N>1,$ we prove the sharp $L^p$-log-Sobolev inequality on noncompact metric measure spaces satisfying the ${\sf CD}(0,N)$ condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based…
By proving an $L^2$-gradient estimate for the corresponding Galerkin approximations, the log-Harnack inequality is established for the semigroup associated to a class of stochastic Burgers equations. As applications, we derive the strong…
We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity…
In this paper, we provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. We will illustrate our method with free groups, triangular groups and finite cyclic groups, for which we…
We establish convergence to an invariant measure as time tends to infinity, for a large class of (possibly non-Markovian) stochastic volatility models. Our arguments are based on a novel coupling idea for Markov chains which also extends to…
By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of this proof is…
We establish that, for a Markov semi-group, $L^2$ hypocoercivity, i.e. contractivity for a modified $L^2$ norm, implies quantitative deviation bounds for additive functionals of the associated Markov process and exponential integrability of…
We analyse certain degenerate infinite dimensional sub-elliptic generators, and obtain estimates on the long-time behaviour of the corresponding Markov semigroups that describe a certain model of heat conduction. In particular, we establish…
We study the connection between the $p$--Talagrand inequality and the $q$--logarithmic Sololev inequality for conjugate exponents $p\geq 2$, $q\leq 2$ in proper geodesic metric spaces. By means of a general Hamilton--Jacobi semigroup we…
We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or…
By the method of coupling and Girsanov transformation, Harnack inequalities [F.-Y. Wang, 1997] and strong Feller property are proved for the transition semigroup associated with the multivalued stochastic evolution equation on a Gelfand…
We present a class of potentials $q \colon \mathbb{R}^{n} \to (0,\infty)$ that implies the weighted Schr\"odinger semigroup $\varphi^{-1}\mathrm{e}^{-tH}\varphi$ to map a weighted Lebesgue function space…
Necessary and sufficient conditions are given for a substochastic semigroup on $L^1$ obtained through the Kato--Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise…
Let $\gamma_{d}$ be the $d$-dimensional standard Gaussian measure and $\{Q_{t}\}_{t\ge 0}$ the Ornstein-Uhlenbeck semigroup acting on $L^{1}(\gamma_{d})$. We show that the hypercontractivity of $\{Q_{t}\}_{t\ge 0}$ is equivalent to the…
In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise from two perspectives. Firstly, by adopting the analytic semigroup theory, we prove the upper…