Sungularity as $L_{2}$-regularity and vice versa

T. Pogany

Displaying similar documents to “Sungularity as $L_{2}$ -regularity and vice versa”

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...

On reduction of two-parameter prediction problems

J. Friedrich, L. Klotz, M. Riedel (1995)

Studia Mathematica

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We present a general method for the extension of results about linear prediction for q-variate weakly stationary processes on a separable locally compact abelian group $G_{2}$ (whose dual is a Polish space) with known values of the processes on a separable subset $S_{2} \subseteq G_{2}$ to results for weakly stationary processes on $G_{1} \times G_{2}$ with observed values on $G_{1} \times S_{2}$ . In particular, the method is applied to obtain new proofs of some well-known results of Ze Pei Jiang.

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.

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 $𝒫_{sup}$ .

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.

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.......................................................................................

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...

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...

Linearized plasticity is the evolutionary $Γ$ -limit of finite plasticity

Alexander Mielke, Ulisse Stefanelli (2013)

Journal of the European Mathematical Society

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We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via $Γ$ -convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

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...

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

M. Jankiewicz (1988)

Applicationes Mathematicae

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Small positive values for supercritical branching processes in random environment

Vincent Bansaye, Christian Böinghoff (2014)

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

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Branching Processes in Random Environment (BPREs) $(Z_{n} : n \geq 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $ℙ (1 \leq Z_{n} \leq k | Z_{0} = i)$ , $k, i \in ℕ$ as $n \to \infty$ . More precisely, we characterize...

The weak convergence of regenerative processes using some excursion path decompositions

Amaury Lambert, Florian Simatos (2014)

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

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We consider regenerative processes with values in some general Polish space. We define their $ε$ -big excursions as excursions $e$ such that $ϕ (e) g t; ε$ , where $ϕ$ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of $e$ . We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of $ε$ -big excursions and of their endpoints, for all $ε$ in a set whose closure contains $0$ . Finally,...

Some properties of stationary sets

C. A. Di Prisco, W. Marek

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CONTENTSIntroduction..................................................................51. Derivative of a stationary set...................................72. Stationary degrees ...............................................133. Subsets of $P_{ϰ} (λ)$ ..............................................194. Stationary subsets of $P_{ϰ} (λ)$ .............................255. Superstationary substes of $P_{ϰ} (λ)$ ....................326. End-stationary subsets of $P_{ϰ} (λ)$ ......................34References................................................................37 ...

Banach-space-valued stationary processes and their linear prediction

S. A. Chobanyan, A. Weron

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Contents0. Introduction............................................................................................................................................. 51. Linear operators generated by random elements.......................................................................... 62. Covariance operator of generalized random elements................................................................. 93. The space of generalized random elements of the second-order as an LVH-space.................

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,...

Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times

Anton Bovier, Michael Eckhoff, Véronique Gayrard, Markus Klein (2004)

Journal of the European Mathematical Society

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We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ on $ℝ^{d}$ or subsets of $ℝ^{d}$ , where $F$ is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be related, up to multiplicative errors that tend to one as $ϵ ↓ 0$ , to the capacities of suitably constructed sets. We show that...

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 Stochastic Differential Equations with Reflecting Boundary Condition in Convex Domains

Weronika Łaukajtys (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let D be an open convex set in $ℝ^{d}$ and let F be a Lipschitz operator defined on the space of adapted càdlàg processes. We show that for any adapted process H and any semimartingale Z there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of D: $X_{t} = H_{t} + \int_{0}^{t} ⟨ F {(X)}_{s -}, d Z_{s} ⟩ + K_{t}$ , t ∈ ℝ⁺. Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.

Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

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A notion of a wide-sense Markov process $X_{t}$ of order k ≥ 1, $X_{t} \sim W M (k)$ , is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of $X_{t}$ is the k-dimensional process $x_{t} = (X_{t - k + 1}, . . ., X_{t})$ . The covariance structure of $X_{t} \sim W M (k)$ is considered in the general case and in the periodic case. In the general case it is shown that $X_{t} \sim W M (k)$ iff $x_{t}$ is a k-dimensional WM(1) process and iff the covariance function of $x_{t}$ has the triangular...

Displaying similar documents to “Sungularity as L 2 -regularity and vice versa”

Displaying similar documents to “Sungularity as $L_{2}$ -regularity and vice versa”