Potential theory of one-dimensional geometric stable processes

Tomasz Grzywny; Michał Ryznar

Displaying similar documents to “Potential theory of one-dimensional geometric stable processes”

Boundary potential theory for stable Lévy processes

Paweł Sztonyk (2003)

Colloquium Mathematicae

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We investigate properties of harmonic functions of the symmetric stable Lévy process on $ℝ^{d}$ without the assumption that the process is rotation invariant. Our main goal is to prove the boundary Harnack principle for Lipschitz domains. To this end we improve the estimates for the Poisson kernel obtained in a previous work. We also investigate properties of harmonic functions of Feynman-Kac semigroups based on the stable process. In particular, we prove the continuity and the Harnack inequality...

A remarkable $σ$ -finite measure unifying supremum penalisations for a stable Lévy process

Yuko Yano (2013)

Annales de l'I.H.P. Probabilités et statistiques

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The $σ$ -finite measure $𝒫_{sup}$ which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s $h$ -transform processes with respect to these functions are utilized for the construction of $𝒫_{sup}$ .

Approximation of a symmetric α-stable Lévy process by a Lévy process with finite moments of all orders

Z. Michna (2007)

Studia Mathematica

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In this paper we consider a symmetric α-stable Lévy process Z. We use a series representation of Z to condition it on the largest jump. Under this condition, Z can be presented as a sum of two independent processes. One of them is a Lévy process $Y_{x}$ parametrized by x > 0 which has finite moments of all orders. We show that $Y_{x}$ converges to Z uniformly on compact sets with probability one as x↓ 0. The first term in the cumulant expansion of $Y_{x}$ corresponds to a Brownian motion which implies...

Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domains

Krzysztof Bogdan, Bartłomiej Dyda (2003)

Studia Mathematica

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For $C^{1, 1}$ domains we give exact asymptotics near the domain’s boundary for the Green function and Martin kernel of the rotation invariant α-stable Lévy process. We also obtain a relative Fatou theorem for harmonic functions of the stable process.

Universal stability of Banach spaces for ε -isometries

Lixin Cheng, Duanxu Dai, Yunbai Dong, Yu Zhou (2014)

Studia Mathematica

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Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T : L (f) \equiv \bar{s p a n} f (X) \to X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally...

Stochastic differential equations with Sobolev drifts and driven by $α$ -stable processes

Xicheng Zhang (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this article we prove the pathwise uniqueness for stochastic differential equations in $ℝ^{d}$ with time-dependent Sobolev drifts, and driven by symmetric $α$ -stable processes provided that $α \in (1, 2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $α \in (\frac{2 d}{d + 1}, 2)$ . Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

On stable currents in positively pinched curved hypersurfaces

Jintang Li (2003)

Colloquium Mathematicae

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Let Mⁿ (n ≥ 3) be an n-dimensional complete hypersurface in a real space form N(c) (c ≥ 0). We prove that if the sectional curvature $K_{M}$ of M satisfies the following pinching condition: $c + δ < K_{M} \leq c + 1$ , where δ = 1/5 for n ≥ 4 and δ = 1/4 for n = 3, then there are no stable currents (or stable varifolds) in M. This is a positive answer to the well-known conjecture of Lawson and Simons.

On Paszkiewicz-type criterion for a.e. continuity of processes in $L^{p}$ -spaces

Jakub Olejnik (2010)

Banach Center Publications

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In this paper we consider processes Xₜ with values in $L^{p}$ , p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type...

Stable invariant subspaces for operators on Hilbert space

John B. Conway, Don Hadwin (1997)

Annales Polonici Mathematici

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If T is a bounded operator on a separable complex Hilbert space ℋ, an invariant subspace ℳ for T is stable provided that whenever $T_{n}$ is a sequence of operators such that $‖ T_{n} - T ‖ \to 0$ , there is a sequence of subspaces $ℳ_{n}$ , with $ℳ_{n}$ in $L a t T_{n}$ for all n, such that $P_{ℳ_{n}} \to P_{ℳ}$ in the strong operator topology. If the projections converge in norm, ℳ is called a norm stable invariant subspace. This paper characterizes the stable invariant subspaces of the unilateral shift of finite multiplicity and normal operators. It also...

α-stable random walk has massive thorns

Alexander Bendikov, Wojciech Cygan (2015)

Colloquium Mathematicae

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We introduce and study a class of random walks defined on the integer lattice $ℤ^{d}$ -a discrete space and time counterpart of the symmetric α-stable process in $ℝ^{d}$ . When 0 < α <2 any coordinate axis in $ℤ^{d}$ , d ≥ 3, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for a thorn to be a massive set.

Convergence to stable laws and a local limit theorem for stochastic recursions

Mariusz Mirek (2010)

Colloquium Mathematicae

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We consider the random recursion $X ₙ^{x} = M ₙ X_{n - 1}^{x} + Q ₙ + N ₙ (X_{n - 1}^{x})$ , where x ∈ ℝ and (Mₙ,Qₙ,Nₙ) are i.i.d., Qₙ has a heavy tail with exponent α > 0, the tail of Mₙ is lighter and $N ₙ (X_{n - 1}^{x})$ is smaller at infinity, than $M ₙ X_{n - 1}^{x}$ . Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $S ₙ^{x} = \sum_{k = 0}^{n} X_{k}^{x}$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.

Stationary distributions for jump processes with memory

K. Burdzy, T. Kulczycki, R. L. Schilling (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$ . The state space of $(Z, S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z, S)$ is the product of the uniform probability measure and a Gaussian distribution.

Stable solutions of $- Δ u = f (u)$ in $ℝ^{N}$

Louis Dupaigne, Alberto Farina (2010)

Journal of the European Mathematical Society

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Several Liouville-type theorems are presented for stable solutions of the equation $- Δ u = f (u)$ in $ℝ^{N}$ , where $f > 0$ is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $α$ with drift and diffusion coefficients $b$ , $σ$ . When $α \in (1, 2)$ , we investigate pathwise uniqueness for this equation. When $α \in (0, 1)$ , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $α \in (0, 1)$ or $α \in (1, 2)$ and on whether the driving stable process is symmetric or not. Our...

Large scale behaviour of the spatial $𝛬$ -Fleming–Viot process

N. Berestycki, A. M. Etheridge, A. Véber (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the spatial $𝛬$ -Fleming–Viot process model ( (2010) 162–216) for frequencies of genetic types in a population living in $ℝ^{d}$ , in the special case in which there are just two types of individuals, labelled $0$ and $1$ . At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$ . We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the...

Theorem-proving systems

Ewa Orłowska

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CONTENTSIntroduction.................................................................................................................... 6Chapter I. Theorem-proving system§ 1. Theory...................................................................................................................... 7§ 2. Fundamental theory $T_{ƒ}$ ................................................................................ 8§ 3. Theorem-proving system.......................................................................................

Heat kernel estimates for the Dirichlet fractional Laplacian

Zhen-Qing Chen, Panki Kim, Renming Song (2010)

Journal of the European Mathematical Society

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We consider the fractional Laplacian $- {(- Δ)}^{α / 2}$ on an open subset in $ℝ^{d}$ with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in $C^{1, 1}$ open sets. This heat kernel is also the transition density of a rotationally symmetric $α$ -stable process killed upon leaving a $C^{1, 1}$ open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

On the strong Brillinger-mixing property of $α$ -determinantal point processes and some applications

Lothar Heinrich (2016)

Applications of Mathematics

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First, we derive a representation formula for all cumulant density functions in terms of the non-negative definite kernel function $C (x, y)$ defining an $α$ -determinantal point process (DPP). Assuming absolute integrability of the function $C_{0} (x) = C (o, x)$ , we show that a stationary $α$ -DPP with kernel function $C_{0} (x)$ is “strongly” Brillinger-mixing, implying, among others, that its tail- $σ$ -field is trivial. Second, we use this mixing property to prove rates of normal convergence for shot-noise processes and sketch...

On smoothing properties of transition semigroups associated to a class of SDEs with jumps

Seiichiro Kusuoka, Carlo Marinelli (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in $ℝ^{d}$ driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process $Y$ (i.e. $Y = W \circ T$ , where $T$ is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process $W$ ) and of an arbitrary Lévy process independent of $Y$ , that the drift coefficient is continuous...

Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

Leonid Berlyand, Volodymyr Rybalko (2010)

Journal of the European Mathematical Society

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We study solutions of the 2D Ginzburg–Landau equation $- Δ u + ε^{- 2} {u (| u |}^{2} - 1) = 0$ subject to “semi-stiff” boundary conditions: Dirichlet conditions for the modulus, $| u | = 1$ , and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small $ε$ . For the Dirichlet boundary condition (“stiff” problem), the existence of stable solutions with vortices,...

Stable cohomotopy groups of compact spaces

Sławomir Nowak (2003)

Fundamenta Mathematicae

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We show that one can reduce the study of global (in particular cohomological) properties of a compact Hausdorff space X to the study of its stable cohomotopy groups $π_{s}^{k} (X)$ . Any cohomology functor on the homotopy category of compact spaces factorizes via the stable shape category ShStab. This is the main reason why the language and technique of stable shape theory can be used to describe and analyze the global structure of compact spaces. For a given Hausdorff compact space X, there exists...

On vertex stability with regard to complete bipartite subgraphs

Aneta Dudek, Andrzej Żak (2010)

Discussiones Mathematicae Graph Theory

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A graph G is called (H;k)-vertex stable if G contains a subgraph isomorphic to H ever after removing any of its k vertices. Q(H;k) denotes the minimum size among the sizes of all (H;k)-vertex stable graphs. In this paper we complete the characterization of $(K_{m, n}; 1)$ -vertex stable graphs with minimum size. Namely, we prove that for m ≥ 2 and n ≥ m+2, $Q (K_{m, n}; 1) = m n + m + n$ and $K_{m, n} * K ₁$ as well as $K_{m + 1, n + 1} - e$ are the only $(K_{m, n}; 1)$ -vertex stable graphs with minimum size, confirming the conjecture of Dudek and Zwonek.

Two stationarity conditions in the $G I / r_{0} / M / r_{1}$ dam

M. Jankiewicz (1988)

Applicationes Mathematicae

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A Weak-Type Inequality for Submartingales and Itô Processes

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $∥ Y ∥_{W} \leq (2 (α + 1) ²) / (2 α + 1) ∥ X ∥_{L^{\infty}}$ . Here W is the weak- $L^{\infty}$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Lévy processes conditioned on having a large height process

Mathieu Richard (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In the present work, we consider spectrally positive Lévy processes $(X_{t}, t \geq 0)$ not drifting to $+ \infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$ ) before hitting $0$ . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law $ℙ_{x}^{☆}$ of this conditioned process (starting at $x g t; 0$ ) is defined as a Doob $h$ -transform via a martingale. For Lévy processes with infinite variation paths,...