Tomasz Grzywny
We prove the intrinsic ultracontractivity for the semigroup generated by a large class of symmetric Levy processes such that the Levy measure satisfies some conditions in the neighborhood of 0, killed on exiting a bounded and connected Lipschitz domain.
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We prove that a general (not necessarily symmetric) Lévy process killed on exiting a bounded open set (without regular condition on the boundary) is intrinsically ultracontractive, provided that $B(0,R_0)\subseteq \rm{supp}(ν)$ for some constant $R_0>0$, where $\rm{supp}(ν)$ denotes the support of the associated Lévy measure $ν$. For a symmetric Lévy process killed on exiting a bounded Hölder domain of order $0$, we also obtain the intrinsic ultracontractivity under much weaker assumption on the associated Lévy measure.
Recently we extended the concept of intrinsic ultracontractivity to non-symmetric semigroups and proved that for a large class of non-symmetric diffusions Z with measure-valued drift and potential, the semigroup of Z^D (the process obtained by killing Z upon exiting D) in a bounded domain is intrinsic ultracontractive under very mild assumptions. In this paper, we study the intrinsic ultracontractivity for non-symmetric discontinuous Levy processes. We prove that, for a large class of non-symmetric discontinuous Levy processes X such that the Lebesgue measure is absolutely continuous with resp…
We introduce a class of Lévy processes subject to specific regularity conditions, and consider their Feynman-Kac semigroups given under a Kato-class potential. Using new techniques, first we analyze the rate of decay of eigenfunctions at infinity. We prove bounds on $λ$-subaveraging functions, from which we derive two-sided sharp pointwise estimates on the ground state, and obtain upper bounds on all other eigenfunctions. Next, by using these results, we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asym…
In this paper we consider a large class of symmetric Markov processes $X=(X_t)_{t\ge0}$ on $\R^d$ generated by non-local Dirichlet forms, which include jump processes with small jumps of $α$-stable-like type and with large jumps of super-exponential decay. Let $D\subset \R^d$ be an open (not necessarily bounded and connected) set, and $X^D=(X_t^D)_{t\ge0}$ be the killed process of $X$ on exiting $D$. We obtain explicit criterion for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup $(P^{D}_t)_{t\ge0}$ of $X^D$. When $D$ is a horn-shaped region, we further o…
By establishing the intrinsic super-Poincaré inequality, some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, are sharp for some concrete examples.
We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for Lévy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a byproduct, we obtain for stable-like processes (in the sense of R.\ Bass) on $\R^d$ with smooth variable index $α(x)\in(0,2)$ a transience criterion in terms of the exponent $α(x)$; if $d=1$ and $\inf_{x\in\R} α(x)\in (1,2)$, then the stable-like proce…
We study the Feynman-Kac semigroup generated by the Schr{ö}dinger operator based on the fractional Laplacian $-(-Δ)^{α/2} - q$ in $\Rd$, for $q \ge 0$, $α\in (0,2)$. We obtain sharp estimates of the first eigenfunction $ϕ_1$ of the Schr{ö}dinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials $q$ such that $\lim_{|x| \to \infty} q(x) = \infty$ and comparable on unit balls we obtain that $ϕ_1(x)$ is comparable to $(|x| + 1)^{-d - α} (q(x) + 1)^{-1}$ and intrinsic ultracontractivity holds iff $\lim_{|x| \to \infty} q(x)/\log|x| = \i…
We study the long-time asymptotic behaviour of semigroups generated by non-local Schrödinger operators of the form $H = -L+V$; the free operator $L$ is the generator of a symmetric Lévy process in $\mathbb R^d$, $d > 1$ (with non-degenerate jump measure) and $V$ is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss vario…
For a symmetric $α$-stable process $X$ on $\RR^n$ with $0<α<2$, $n\geq 2$ and a domain $D \subset \RR^n$, let $L^D$ be the infinitesimal generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class function $q$, it is shown that $L^D+q$ is intrinsic ultracontractive on a Hölder domain $D$ of order 0. This is then used to establish the conditional gauge theorem for $X$ on bounded Lipschitz domains in $\RR^n$. It is also shown that the conditional lifetimes for symmetric stable process in a Hölder domain of order 0 are uniformly bounded.
Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process $Y$ is a diffusion process whose generator can be formally written as $L+μ\cdot\nabla-ν$ with Dirichlet boundary conditions, where $L$ is a uniformly elliptic second-order differential operator and $μ=(μ^1,...,μ^d)$ is such that each component $μ^i$, $i=1,...,d$, is a signed measure belonging to the Kato class…
Intrinsic ultracontractivity for Schrödinger semigroups based on cylindrical fractional Laplacian on the plane
研究平面上带柱状分数阶拉普拉斯算子的 Schrödinger 半群，给出内在超收缩性的充要条件及第一特征函数的尖锐估计。
Consider the symmetric non-local Dirichlet form $(D,\D(D))$ given by $$ D(f,f)=\int_{\R^d}\int_{\R^d}\big(f(x)-f(y)\big)^2 J(x,y)\,dx\,dy $$with $\D(D)$ the closure of the set of $C^1$ functions on $\R^d$ with compact support under the norm $\sqrt{D_1(f,f)}$, where $D_1(f,f):=D(f,f)+\int f^2(x)\,dx$ and $J(x,y)$ is a nonnegative symmetric measurable function on $\R^d\times \R^d$. Suppose that there is a Hunt process $(X_t)_{t\ge 0}$ on $\R^d$ corresponding to $(D,\D(D))$, and that $(L,\D(L))$ is its infinitesimal generator. We study the intrinsic ultracontractivity for the Feynman-Kac semigrou…
We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a Lévy process in small time. By intrinsic we mean that such estimates reflect the structure of the characteristic exponent of the process. The technique used in the paper relies on the asymptotic analysis of the inverse Fourier transform of the respective characteristic function. We provide several examples, in particular, with rather irregular Lévy measure, to illustrate our results.
Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials
本文研究被势灭杀的离散时间马尔可夫链，通过热核双边估计定义内在超压缩性，并刻画其均匀（拟）遍历性，适用于任意缓慢增长势。
In this paper we construct the fundamental solution to some integro-differential equation, as well as the intrinsic upper and lower estimates for this solution. As an application of constructed estimates we state a criterion when a given Borel measure belongs to the respective Kato and Dynkin classes.
We define and prove existence of fractional $P(ϕ)_1$-processes as random processes generated by fractional Schrödinger semigroups with Kato-decomposable potentials. Also, we show that the measure of such a process is a Gibbs measure with respect to the same potential. We give conditions of its uniqueness and characterize its support relating this with intrinsic ultracontractivity properties of the semigroup and the fall-off of the ground state. To achieve that we establish and analyze these properties first.
Let $X(t)$, $t\geq0$, be a Lévy process in $\mathbb{R}^d$ starting at the origin. We study the closed convex hull $Z_s$ of $\{X(t): 0\leq t\leq s\}$. In particular, we provide conditions for the integrability of the intrinsic volumes of the random set $Z_s$ and find explicit expressions for their means in the case of symmetric $α$-stable Lévy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of $Z_s$ for all $s>0$. Limit theorems for the convex hull of Lévy processes with normal and stable lim…
Let $(X_t)_{t\ge 0}$ be a symmetric strong Markov process generated by non-local regular Dirichlet form $(D,\D(D))$ as follows \begin{equation*} \begin{split} & D(f,g)=\int_{\R^d}\int_{\R^d}\big(f(x)-f(y)\big)\big(g(x)-g(y)\big) J(x,y)\,dx\,dy, \quad f,g\in \D(D) \end{split} \end{equation*} where $J(x,y)$ is a strictly positive and symmetric measurable function on $\R^d\times \R^d$. We study the intrinsic hypercontractivity, intrinsic supercontractivity and intrinsic ultracontractivity for the Feynman-Kac semigroup $$ T^V_t(f)(x)=\Ee^x\left(\exp\Big(-\int_0^tV(X_s)\,ds\Big)f(X_t)\right),\,\, x…
Intrinsic scales for high-dimensional LEVY-driven models with non-Markovian synchronizing updates
本文研究高维LEYV驱动模型的同步特性，证明对称模型在随机意义下达到同步，并探讨大系统中内在空间尺度的存在条件，讨论了非均匀矩阵基内在尺度的推广。
We study for a class of symmetric Lévy processes with state space $\rn$ the transition density $p_t(x)$ in terms of two one-parameter families of metrics, $(d_t)_{t>0}$ and $(δ_t)_{t>0}$. The first family of metrics describes the diagonal term $p_t(0)$; it is induced by the characteristic exponent $ψ$ of the Lévy process by $d_t(x,y)=\sqrt{tψ(x-y)}$. The second and new family of metrics $δ_t$ relates to $\sqrt{tψ}$ through the formula $$ \exp(-δ_t^2(x,y)) = \Ff[\frac{e^{-tψ}}{p_t(0)}](x-y) $$ where $\Ff$ denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of t…
This paper introduces Switching Processes, called SP. Their constructions are inspired by the PDMP's ones (PDMP stands for Piecewise Deterministic Markov Process). A Markov process, called the intrinsic process, replaces the PDMP's flow. Jumps are added ; they occur randomly as their locations ; their distributions depend on the process's trajectory between them. When the intrinsic process is a Levy process, thanks to its L{é}vy-It{ô} decomposition as a semi-martingale, we obtain the expected Kolmogorov equations for the SP. The results are extended to It{ô}-L{é}vy processes, in particular to …
The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable Lévy processes, which are then shown to arise as the scaling limit of the perimeter proce…
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