Boundary Behavior of Harmonic Functions for Truncated Stable Processes

For any α∈(0,2), a truncated symmetric α-stable process in ℝ ^d is a symmetric Lévy process in ℝ ^d with no diffusion part and with a Lévy density given by c|x|^−d−α 1_{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in ℝ ^d called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κ-fat open sets. The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00037). The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167.     

Two-Sided Optimal Bounds for Green Functions of Half-Spaces for Relativistic <Emphasis Type="Italic">α</Emphasis>-Stable Process

The purpose of this paper is to find optimal estimates for the Green function of a half-space of the relativistic α -stable process with parameter m on ℝ ^d space. This process has an infinitesimal generator of the form mI–(m ^2/α I–Δ) ^α/2, where 0<α<2, m>0, and reduces to the isotropic α-stable process for m=0. Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets. Our approach combines the recent results obtained in Byczkowski et al. (Bessel Potentials, Hitting Distributions and Green Functions (2006) (preprint). ), where an explicit integral formula for the m-resolvent of a half-space was found, with estimates of the transition densities for the killed process on exiting a half-space. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic α-stable process. For example, for d≥3, it is comparable with the Green function for the isotropic α-stable process, provided that the points are close enough. Research supported by KBN Grants.     

Perturbation of symmetric Markov processes

We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator $\mathcal {L}We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator of a general symmetric Markov process. An illuminating concrete example for is , where D is a bounded Euclidean domain in is the Laplace operator in D with zero Dirichlet boundary condition and is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to is a Lévy process that is the sum of Brownian motion in and an independent symmetric (2s)-stable process in killed upon exiting the domain D. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is the use of an extension of Nakao’s stochastic integral for zero-energy additive functionals and the associated It? formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)]. The research of T.-S. Zhang is supported by the British EPSRC.     

Fine Properties of Sets of Finite Perimeter in Doubling Metric Measure Spaces

The aim of this paper is to study the properties of the perimeter measure in the quite general setting of metric measure spaces. In particular, defining the essential boundary ^* E of E as the set of points where neither the density of E nor the density of XE is 0, we show that the perimeter measure is concentrated on ^* E and is representable by an Hausdorff-type measure.     

Multiplizitäten “unendlich-ferner” Spitzen

LetX be the quotient of a bounded symmetric domainD by an arithmetically defined subgroup of all analytic automorphisms ofD and letX ^* be theSatake-compactification ofX. In the present note, the multiplicities of the local rings of the zero-dimensional boundary components ofX ^* will be computed in a completely elementary manner using reduction-theory in selfadjoint homogeneous cones.     

Asymptotic relation for the density of a multidimensional random evolution with rare poisson switchings

A symmetric random evolution X(t) = (X ₁ (t), …, X _m (t)) controlled by a homogeneous Poisson process with parameter λ > 0 is considered in the Euclidean space ℝ ^m , m ≥ 2. We obtain an asymptotic relation for the transition density p(x, t), t > 0, of the process X(t) as λ → 0 and describe the behavior of p(x, t) near the boundary of the diffusion domain in spaces of different dimensions. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1631 – 1641, December, 2008.     

The Smallest Parallelepiped of n Random Points and Peeling

Let n random points be given with uniform distribution in the d-dimensional unit cube [0,1]^d. The smallest parallelepiped A which includes all the n random points is dealt with. We investigate the asymptotic behavior of the volume of A as n tends to . Using a point process approach, we derive also the asymptotic behavior of the volumes of the k-th smallest parallelepipeds A _n ^(k) which are defined by iteration. Let A _n = A _n ⁽¹⁾ . Given A _n ^(k,-,1) delete the random points X _i which are on the boundary A _n ^(k,-,1) , and construct the smallest parallelepiped which includes the inner points of A _n ^(k,-,1) , this defines A _n ^(k) . This procedure is known as peeling of the parallelepiped A_n.     

Some Markov processes with Brownian exit distributions

Summary LetD be a bounded domain inR ^d with regular boundary. LetX=(X_t, P^x) be a standard Markov process inD with continuous paths up to its lifetime. IfX satisfies some weak conditions, then it is possible to add a non-local part to its generator, and construct the corresponding standard Markov process inD with Brownian exit distributions fromD.This work was done while the author was an Alexander von Humboldt fellow at the Universität des Saarlandes in Saarbrücken, Germany     

A note on small ball probability of a Gaussian process with stationary increments

Let {X(t), 0t1} be a Gaussian process with mean zero and stationary increments. Let ²(h) =EX ²(h) be nondecreasing and concave on (0,1). A sharp bound on the small ball probability ofX(·) is given in this paper.Research supported by Charles Phelps Taft Post-doctoral Fellowship of the University of Cincinnati and by the Fok Yingtung Education Foundation of China.     

Extension of stationary stochastic processes

Summary LetX be an arbitrary Hausdorff space, and consider a stationary stochastic process inX with time interval [0, 1], i.e. a tight probability onX ^{[0, 1]}, equipped with the Borel -field of the product space. We prove the existence of a stationary extension of this process to ₀ ⁺ . Furthermore, we show that the extended process may be chosen to have continuous paths if the original process has this property. Under stronger topological assumptions, we derive the corresponding results whenX ^{[0, 1]} is equipped with the product of the Borel -fields.Corporate Research and Development, SIEMENS AG, D-81730 Munich, Germany     

Potential theory of truncated stable processes

For any , a truncated symmetric α-stable process is a symmetric Lévy process in with a Lévy density given by for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a non-convex domain for which the boundary Harnack principle fails. The research of Panki Kim is supported by Research Settlement Fund for the new faculty of Seoul National University. The research of Renming Song is supported in part by a joint US-Croatia grant INT 0302167.     

Pseudo Jordan domains and reflecting Brownian motions

Summary The manifold metric between two points in a planar domain is the minimum of the lengths of piecewiseC ¹ curves in the domain connecting these two points. We define a bounded simply connected planar region to be a pseudo Jordan domain if its boundary under the manifold metric is topologically homeomorphic to the unit circle. It is shown that reflecting Brownian motionX on a pseudo Jordan domain can be constructed starting at all points except those in a boundary subset of capacity zero.X has the expected Skorokhod decomposition under a condition which is satisfied when G has finite 1-dimensional lower Minkowski content.     

Pettis integrability of Stone transforms

Letf be a bounded Pettis integrable function ranging in a Banach spaceX (the range of the indefinite Pettis integral is separable). We consider Pettis integrability conditions for the Stone transform off and relate this problem to the regular oscillation condition for the family of functions {x ^* fx^*B(X^*)}, whereB(X^*) is the unit ball inX ^*.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 238–253, August, 1996.     

The spectral projections and the resolvent for scattering metrics

In this paper, we consider a compact manifold with boundaryX equipped with a scattering metricg as defined by Melrose [9]. That is,g is a Riemannian metric in the interior ofX that can be brought to the formg=x ⁻⁴ dx²+x^{−2 h’} near the boundary, wherex is a boundary defining function andh’ is a smooth symmetric 2-cotensor which restricts to a metrich on ϖX. LetH=Δ+V, whereV∈x ²C^∞ (X) is real, soV is a ‘short-range’ perturbation of Δ. Melrose and Zworski started a detailed analysis of various operators associated toH in [11] and showed that the scattering matrix ofH is a Fourier integral operator associated to the geodesic flow ofh on ϖX at distance π and that the kernel of the Poisson operator is a Legendre distribution onX×ϖX associated to an intersecting pair with conic points. In this paper, we describe the kernel of the spectral projections and the resolvent,R(σ±i0), on the positive real axis. We define a class of Legendre distributions on certain types of manifolds with corners and show that the kernel of the spectral projection is a Legendre distribution associated to a conic pair on the b-stretched productX _b ² (the blowup ofX ² about the corner, (ϖX)²). The structure of the resolvent is only slightly more complicated. As applications of our results, we show that there are ‘distorted Fourier transforms’ forH, i.e., unitary operators which intertwineH with a multiplication operator and determine the scattering matrix; we also give a scattering wavefront set estimate for the resolventR(σ±i0) applied to a distributionf.     

Hausdorff Measure for a Stable-Like Process over an Infinite Extension of a Local Field

We consider an infinite extension K of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. K is equipped with an inductive limit topology; its conjugate K is a completion of K with respect to a topology given by certain explicitly written seminorms. The semigroup of measures, which defines a stable-like process X(t) on K, is concentrated on a compact subgroup S K. We study properties of the process X _S (t), a part of X(t) in S. It is shown that the Hausdorff and packing dimensions of the image of an interval equal 0 almost surely. In the case of tamely ramified extensions a correct Hausdorff measure for this set is found.     

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

Schrödinger Equations with Fractional Laplacians

It is shown that the unique solution of } can be represented as { } where X=(X _t , t≥ 0) is a stable process whose generator is (-Δ) ^α/2 with X ₀ =0 . Accepted 24 July 2000. Online publication 13 November 2000.     

Walsh's Brownian Motion-Type of Extensions

Let (X _t ) be a rotation invariant Feller process on the state space F R²{} consisting of finite number of rays, meeting at 0. We study a certain class of possible strong Markov extensions of (X _t ) to F ,{} given the corresponding radial extension to [0, ). A well-known example is the class of Walsh's Brownian motions, in the case where (X _t ) is the Brownian motion on F. It turns out that while the symmetric extension of Walsh's Brownian motion-type always exists, the non-symmetric extension exists iff (X _t ), roughly speaking, does not jump from one ray to another before hitting 0.     

A Class of Infinitesimal Generators and Time Reversal for the Corresponding One-Dimensional Markov Processes

A class of infinitesimal generators A of strongly continuous nonnegative contraction semigroups in a subspace of C[0, 1] is introduced. It contains the class of generators of regular gap diffusions. A construction of the Markov process X generated by A gives some stochastic interpretations of the integral term which appears in A. The infinitesimal generator of the time reversal of X (with respect to its life time) is explicitly given. It belongs to the introduced class of generators too. Thus, the considered class is invariant under this transformation. Two examples, the time reversal of gap diffusions with nonlocal boundary conditions and the time reversal of processes with Levy-measure, complete the note.     

Precise Asymptotics for Lévy Processes

Let {X(t), t ≥ 0} be a Lévy process with EX(1) = 0 and EX ²(1) < ∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t ≥ 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables. This work is supported by the National Natural Science Foundation (Grant No. 10671188) and Special Foundation of USTC