Isotropic Stable Process Research Articles

On Potential Theory of Markov Processes with Jump Kernels Decaying at the Boundary

Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the L\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\'evy process) in $\mathbb{R}^d$. The main novelty is that the term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\mathcal{B}(x,y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialise to the case of the half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, the $\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\alpha}$ and the killing function$\kappa(x)=c x_d^{-\alpha}$, $\alpha\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\alpha-1)_+$, there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid.

Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach

We compute explicitly the distribution of the point of closest reach to the origin in the path of any d-dimensional isotropic stable process, with d ≥ 2. Moreover, we develop a new radial excursion theory, from which we push the classical Blumenthal–Getoor–Ray identities for first entry/exit into a ball (cf. Blumenthal et al. Trans. Amer. Math. Soc., 99, 540–554 1961) into the more complex setting of n-tuple laws for overshoots and undershoots. We identify explicitly the stationary distribution of any d-dimensional isotropic stable process when reflected in its running radial supremum. Finally, for such processes, and as consequence of some of the analysis of the aforesaid, we provide a representation of the Wiener–Hopf factorisation of the MAP that underlies the stable process through the Lamperti–Kiu transform. Our analysis continues in the spirit of Kyprianou (Ann. Appl. Probab., 20(2), 522–564 2010) and Kyprianou et al. (2015) in that our methodology is largely based around treating stable processes as self-similar Markov processes and, accordingly, taking advantage of their Lamperti-Kiu decomposition.

Asymptotic properties of some space-time fractional stochastic equations

Consider non-linear time-fractional stochastic heat type equations of the following type, ∂tβut(x)=-ν(-Δ)α/2ut(x)+It1-β[λσ(u)F·(t,x)]\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial ^\\beta _tu_t(x)=-\\nu (-\\Delta )^{\\alpha /2} u_t(x)+I^{1-\\beta }_t[\\lambda \\sigma (u)\\mathop {F}\\limits ^{\\cdot }(t,x)] \\end{aligned}$$\\end{document}in (d+1) dimensions, where nu >0, beta in (0,1), alpha in (0,2]. The operator partial ^beta _t is the Caputo fractional derivative while -(-Delta )^{alpha /2} is the generator of an isotropic stable process and I^{1-beta }_t is the Riesz fractional integral operator. The forcing noise denoted by mathop {F}limits ^{cdot }(t,x) is a Gaussian noise. And the multiplicative non-linearity sigma :mathbb {R}rightarrow mathbb {R} is assumed to be globally Lipschitz continuous. Mijena and Nane (Stochastic Process Appl 125(9):3301–3326, 2015) have introduced these time fractional SPDEs. These types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter lambda . In particular, our results are significant extensions of those in Ann Probab (to appear), Foondun and Khoshnevisan (Electron J Probab 14(21): 548–568, 2009), Mijena and Nane (2015) and Mijena and Nane (Potential Anal 44:295–312, 2016). Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

Non-linear Noise Excitation for some Space-Time Fractional Stochastic Equations in Bounded Domains

Abstract In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: ∂ t β u t ( x ) = − ν ( − Δ ) α / 2 u t ( x ) + I t 1 − β [ λ σ ( u ) F ⋅ ( t , x ) ] $$\partial^\beta_tu_t(x) = -\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u) \stackrel{\cdot}{F}(t,x)] $$ in (d+1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator ∂ t β $\partial^\beta_t$ is the Caputo fractional derivative, −(−Δ) α/2 is the generator of an isotropic stable process and I 1−β t is the fractional integral operator. The forcing noise denoted by F ⋅ ( t , x ) $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. The multiplicative non-linearity σ : ℝ → ℝ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane [32]. We first study the existence and uniqueness of the solution of these equations and under suitable conditions on the initial function, we also study the asymptotic behavior of the solution with respect to the parameter λ. In particular, our results are significant extensions of those in [14], [16], [32], and [33].

Intermittency fronts for space-time fractional stochastic partial differential equations in [formula omitted] dimensions

We consider time fractional stochastic heat type equation ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[σ(u)W⋅(t,x)] in (d+1) dimensions, where ν>0, β∈(0,1), α∈(0,2], d<min{2,β−1}α, ∂tβ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, W⋅(t,x) is space-time white noise, and σ:R→R is Lipschitz continuous. Mijena and Nane proved in (Mijena and Nane, 2016) that: (i) absolute moments of the solutions of this equation grow exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. The last result was proved under the assumptions α=2 and d=1. In this paper we extend this result to the case α=2 and d∈{1,2,3}.

Fractional Laplacian with singular drift

For $\alpha \in (1,2)$ we consider the equation $\partial _t u = \varDelta ^{\alpha /2} u + b \cdot \nabla u$, where $b$ is a time-independent, divergence-free singular vector field of the Morrey class $M_1^{1-\alpha }$. We show that if the Morrey norm $

Intrinsic Ultracontractivity for Stable Semigroups on Unbounded Open Sets

We study the intrinsic ultracontractivity of the semigroup of the isotropic stable process on an unbounded open set. A two-sided estimate of the first eigenfunction of the semigroup is derived. We obtain general geometric conditions for intrinsic ultracontractivity which yield a necessary and sufficient condition for non-degenerate horn-shaped sets.

Lévy processes: Capacity and Hausdorff dimension

We use the recently-developed multiparameter theory of additive Lévy processes to establish novel connections between an arbitrary Lévy process X in Rd, and a new class of energy forms and their corresponding capacities. We then apply these connections to solve two long-standing problems in the folklore of the theory of Lévy processes. First, we compute the Hausdorff dimension of the image X(G) of a nonrandom linear Borel set G⊂R+, where X is an arbitrary Lévy process in Rd. Our work completes the various earlier efforts of Taylor [Proc. Cambridge Phil. Soc. 49 (1953) 31–39], McKean [Duke Math. J. 22 (1955) 229–234], Blumenthal and Getoor [Illinois J. Math. 4 (1960) 370–375, J. Math. Mech. 10 (1961) 493–516], Millar [Z. Wahrsch. verw. Gebiete 17 (1971) 53–73], Pruitt [J. Math. Mech. 19 (1969) 371–378], Pruitt and Taylor [Z. Wahrsch. Verw. Gebiete 12 (1969) 267–289], Hawkes [Z. Wahrsch. verw. Gebiete 19 (1971) 90–102, J. London Math. Soc. (2) 17 (1978) 567–576, Probab. Theory Related Fields 112 (1998) 1–11], Hendricks [Ann. Math. Stat. 43 (1972) 690–694, Ann. Probab. 1 (1973) 849–853], Kahane [Publ. Math. Orsay (83-02) (1983) 74–105, Recent Progress in Fourier Analysis (1985b) 65–121], Becker-Kern, Meerschaert and Scheffler [Monatsh. Math. 14 (2003) 91–101] and Khoshnevisan, Xiao and Zhong [Ann. Probab. 31 (2003a) 1097–1141], where dim X(G) is computed under various conditions on G, X or both. We next solve the following problem [Kahane (1983) Publ. Math. Orsay (83-02) 74–105]: When X is an isotropic stable process, what is a necessary and sufficient analytic condition on any two disjoint Borel sets F,G⊂R+ such that with positive probability, X(F)∩X(G) is nonempty? Prior to this article, this was understood only in the case that X is a Brownian motion [Khoshnevisan (1999) Trans. Amer. Math. Soc. 351 2607–2622]. Here, we present a solution to Kahane’s problem for an arbitrary Lévy process X, provided the distribution of X(t) is mutually absolutely continuous with respect to the Lebesgue measure on Rd for all t>0. As a third application of these methods, we compute the Hausdorff dimension and capacity of the preimage X−1(F) of a nonrandom Borel set F⊂Rd under very mild conditions on the process X. This completes the work of Hawkes [Probab. Theory Related Fields 112 (1998) 1–11] that covers the special case where X is a subordinator.

Advanced acoustic signal detection and localization with lower-order statistics

The importance of extending the statistical signal processing methodology to the so-called alpha-stable farmework is apparent. First, scientists and engineers have started to appreciate alpha-spectra and the elegant scaling and self-similarity properties of stable distributions. Additionally, real life sonar applications exist in which impulsive ocean channels tend to produce large-amplitude, short-duration interferences more frequently than Gaussian channels do. The stable law has been shown to successfully model noise over certain impulsive channels. In this lecture, new robust techniques are proposed for source detection and localization in the presence of noise modeled as a complex isotropic stable process. First, optimal, in the maximum likelihood sense, approaches are presented and the Cauchy beamformer is introduced. Also, subspace methods based on fractional lower-order statistics are developed for sonar applications where reduced computational cost is a crucial design parameter. Finally, simulation experiments demonstrating the performance of the proposed methods are presented.

Remarks on transmission, antitransmission and antilocal properties for sums of stable generators

The antilocal property (to all directions)of an operator A states that, if f=Af=0 in a domain U, then / is identically zero on the whole space. (For the precise definition see §2). This property is known to hold for the class of fractional powers of the Laplacian: A―Ax, where i is a non-integral real number, by Goodman-Segal [6] when the space dimension is odd and by Murata [14] in general. Liess [13] showed this property for the fractional powers of the ellipticdifferentialoperator with analytic coefficients using the theory of pseudodifferential operators with analytic symbol. As is well known, the fractional power Aa/2is the generator of the isotropic stable process in case 0 0, then /=0 on (0, + oo) but not necessarily f=Q in (-co, 0). The result has been shown for aG(0, 1) and will be extended to fl£(0,1)W(1, 2) in §4, b), 2). [It is stillopen if the asymmetric Cauchy generators (the generalized case for a=l) have the similar property.] The proof is carried out on the similar line as in Liess [13], and so Sato-Kashiwara-Kawai theory is essentialin the proof. The firstaim of this note is to show what type of antilocal property holds for a finitesum of one-dimensional stable generators, for example, the sum o1 a one-sided a-stable generator to the right and a one-sided /3-stablegeneratoi to the left. The solution will be given in §4, b), 3). In fact the example citec