Schedule for: 16w5123 - Stable Processes

A welcome drink will be served at the hotel.

In this talk, we study the cut-off phenomenon for a family of d-dimensional Ornstein-Uhlenbeck processes driven by Lévy processes. Under some suitable conditions on the drift matrix and the Lévy measure of the driven processes present a profile cut-off with respect to the total variation distance. This is a joint work with Gerardo Barrera

As a generalisation of Lamperti’s transformation, it has been proved by Alili, Chaumont, Grackzyk and Zak that a stable process in dimension d can be seen as the exponential of a Markov additive process (MAP) time changed. In this talk, we aim at describing the so called upward, respectively downward, ladder height processes associated to this MAP. We will provide a precise description in the case where d=1, and, in general, how the characteristics of this process could be derived from known identities for stable processes. This is based on a ongoing collaboration with Kyprianou, Satitkanitkul and Sengul.

Boolean convolution is an opration between probaility measure related to Boolean Independence in Non Commutative Probability. As in probability, one can consider stable laws in relation to Boolean convolution. In this talk I will explain relation between classical, free and Boolean stable laws. Moreover I will also describe the parameters for infinite divisibility of the Boolean stable laws, both in the classical and the free setting.

In this talk, we study the process obtained from a Brownian bridge after excising all the excursions below the waterline level which reach zero. Three variables of interest are the maximum of this process, the value where this maximum is attained, and the total length of the excursions which are excised. Our analysis relies on some interesting transformations connecting Brownian path fragments and the 3-dimensional Bessel process.

This talk illustrates the approach that we, with M. Savov, have set up to develop the spectral theory in Hilbert space of some classes of non-self-adjoint Markov operators using intertwining techniques. We will focus on the spectral decomposition of the class, denoted by $\mathcal{K}$, of markovian self-similar $\rm{C_0}$-semigroup on $L^2(\mathbb{R}_+)$. We start by showing that any two elements in $\mathcal{K}$ are intertwined with a closed linear (non-necessarily positive) operator. Relying on these commutation relationships, we characterize, for each semigroup in $\mathcal{K}$, the spectrum including the residual part, the spectral operator, expressed in terms of (weak) Fourier kernels, and its domain. We conclude this talk by discussing a series of on-going projects and open problems.

In this talk, we analyze the strong solution of a particular family of stochastic differential equations. This result allows us to introduce continuous state branching process (CSBP) in a Lévy random environment. When the underlying CSBP is stable, we study the asymptotic behavior of the explosion and extinction probabilities. These probabilities are related with the exponential functional of a Lévy process.

In this talk, we present a necessary and sufficient condition for the existence of recurrent extensions of real self-similar Markov processes. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. To be precise, our main result ensures that a real self-similar Markov process with a finite hitting time of the point zero has a recurrent extension that leaves 0 continuously if and only if the MAP associated, via Lamperti transformation, satisfies the Cramér's condition. In doing so, we solve an old problem originally posed by Lamperti for positive self-similar Markov processes. We generalize Rivero’s (2005, 2007) and Fitzsimmons’s (2006) results to real-valued case. Finally, we describe the recurrent extension of a stable Lévy process.

We propose isomorphism type identities for nonlinear functionals of infinitely divisible (ID) processes. Such identities can be viewed as an analogy of the Cameron-Martin formula for Poissonian ID processes but with random translations, or perturbation identities. Their applicability relies on some knowledge of Lévy measures of stochastic processes and their representations. We will illustrate this approach on examples, including stable processes.

Extremal limit theorems for certain long memort stationary stable processes lead unexpectedly to a nonlinear time change in the usual extremal process. Extending the convergence to the context of random sup measures reveals that the nonlinear time change is a result of a certain optimization procedure over ranges of i.i.d. stable subordinators. When the memory in the original stationary stable process is particularly long, new random variables may appear in the limit, whose laws bridge the gap between Fréchet and stable random variables.

A classical result of H.J Brascamp and E.H. Lieb says that the ground state eigenfunction for the Laplacian in convex domains (and of Schrödinger operators with convex potentials) on $\mathbb{R}^n$, is log-concave. A proof can be given (interpreted) in terms of the finite dimensional distributions of Brownian motion. Some years ago the speaker raised similar questions, and made some conjectures, when the Brownian motion is replaced by other stochastic processes and in particular by the rotationally symmetric $\alpha$-stable processes. These problems, for the most part, remain open even for an interval in $\mathbb{R}$. In this talk we elaborate on this topic, discuss some known results, and outline the proof of one of these.

\noindent Let $(\mathbf{X}_{t})_{t\geq 0}$ be a $d$--dimensional Lévy process, such that for each $t>0$ the distribution of $\mathbf{X}_{t}$ is full. We say that $\mathbf{X}_{t}\in \mathcal{F}$, the Feller class at zero, if there exist nonsingular $d\times d$ matrices $\mathbf{A}_{t}$ and centering vectors $\mathbf{b}_{t}$ such that for every sequence of positive constants $s_{m}\downarrow 0$, there is a further subsequence $ t_{n}\downarrow 0$, such that \[ \mathbf{A}_{t_{n}}(\mathbf{X}_{t_{n}}-\mathbf{b}_{t_{n}})\overset{\mathrm{D}} {\longrightarrow }\mathbf{Y,} \label{Y} \] where $\mathbf{Y}$ is a full $d$--dimensional random vector in $\mathbb{R} ^{d}$ depending on the subsequence $t_{n}.$\medskip We characterize when $\mathbf{X}_{t}\in\mathcal{F}$ and define for each $t>0$ a nonsingular $d\times d$ matrix $\mathbf{A}_{t}$ and centering $\mathbf{b} _{t}$ vector such that the above distributional convergence holds. We also study the asymptotic distribution of the $d\times\left( d+1\right) $ matrix valued Lévy process \[ (\mathbf{Y}_{t})_{t\geq0}=\left( \left( \mathbf{X}_{t},\mathbf{V}_{t}\right) \right) _{t\geq0}, \] where $(\mathbf{V}_{t})_{t\geq0}$ is the $d\times d$ matrix valued {\it quadratic variation process} corresponding to $(\mathbf{X}_{t})_{t\geq0}$. Examples, include Lévy processes in the domain of attraction at zero of a standard $d$--dimensional normal law, stable processes and $\alpha$--semistable Lévy processes. This talk is based on joint work in progress with Ross Maller.

It has been known for some time that extremal $\alpha$ stable variables ($S_{\alpha}$) are Generalized Gamma Convolutions. Recently, L. Bondesson has shown that the family of GGC distributions is stable by multiplicative convolution. I will exhibit a first consequence of this result with a short proof of the fact that $\alpha$-stable densities are hyperbolically completely monotone (HCM in short). This result was first obtained by T. Simon and P. Bosch. I will give a general representation of all positive $\alpha$-densities. This result can be used for numerical reasons and gives a new point of view on the famous Zolotarev result on unimodality problem. Also we obtain the optimal $\beta$ such that $S_{\alpha}^{\beta}$ has an HCM density.

Nonlocal minimal surfaces are introduced in [1] as boundary of sets that minimize the fractional perimeter in a bounded and open set $\Omega \subset \mathbb{R}^n$, among sets with fixed exterior data. It is a known result that when $\Omega$ has a smooth boundary and the exterior data is a half-space, the $s$-minimal set is the same half-space. On the other hand, if one removes (even from far away) some small set from the half-space, for $s$ small enough the $s$-minimal set completely sticks to the boundary, that is, the $s$-minimal set is empty inside $\Omega$. In the paper [3], it is proved indeed that fixing the first quadrant of the plane as boundary data, the $s$-minimal set in $B_1\subset\mathbb{R}^2$ is empty in $B_1$ for $s$ small enough. In this talk, we will present the behavior of $s$-minimal surfaces when the fractional parameter $s\in (0,1)$ is small, in a bounded and connected open set with $C^2$ boundary $\Omega$. We classify the behavior of $s$-minimal surfaces with respect to the fixed exterior data. So, for $s$ small and depending on the data at infinity, the $s$-minimal set can be either empty in $\Omega$, fill all $\Omega$, or possibly develop a wildly oscillating boundary. Also, we will present the asymptotic behavior of the fractional mean curvature (see [2]) when $s\to 0^+$. In particular, as $s$ gets smaller, the fractional mean curvature at any point on the boundary of a $C^{1,\gamma}$ set (for $\gamma\in(0,1)$) takes into account only the nonlocal contribution. The results in this talk are obtained in a preprint by myself, Luca Lombardini and Enrico Valdinoci. \[ \mbox{ } \] [1] L.Caffarelli, J.-M. Roquejoffre, and O.Savin. Nonlocal minimal surfaces. Comm. Pure Appl. Math., 63(9):1111--1144, 2010. \[ \mbox{ } \] [2] Nicola Abatangelo and Enrico Valdinoci. \newblock A notion of nonlocal curvature. \newblock {\em Numer. Funct. Anal. Optim.}, 35(7-9):793--815, 2014. \[ \mbox{ } \] [3] Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci. Boundary behavior of nonlocal minimal surfaces. arXiv preprint arXiv:1506.04282, 2015.

We look at models of fragmentation with growth. In such a model, one has a number of independent cells, each of which grows continuously in time until a fragmentation event occurs, at which point the cell splits into two child cells of a smaller mass. Each of the children is independent and behaves in the same way as its parent. The rate of fragmentation is self-similar, that is, the rate at which each cell splits is a power of the mass. This is a random model; looking at its mean-field behaviour gives the growth-fragmentation equation, which is a deterministic PDE. We describe probabilistic solutions to the equation, using growth-fragmentations and positive self-similar Markov processes. In certain cases, we see spontaneous generation of positive solutions from zero initial mass. Based on joint work with Jean Bertoin (University of Zurich).

Growth-fragmentation processes are stochastic processes, recently introduced by Bertoin, which describe the evolution of the sizes of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. We will identify a distinguished one-parameter family of self-similar growth-fragmentations closely related to stable Lévy processes, which arise in scaling limits of Markovian explorations of certain random planar maps with large degrees. Based on joint work with Jean Bertoin, Timothy Budd and Nicolas Curien.

In this talk, we consider symmetric jump processes of mixed-type on metric measure spaces and establish stability of two-sided heat kernel estimates and parabolic Harnack inequalities. We obtain their stable equivalent characterizations in terms of the jump kernels, some cut-off Sobolev inequalities, and the Poincar\'e inequalities (or the Faber-Krahn inequalities). In particular, we establish stability of heat kernel estimates for $\alpha$-stable-like processes even with $\alpha\ge 2$ when the underlying spaces have walk dimensions larger than $2$, which has been one of the major open problems in this area.

We study regularity properties of the super-Brownian motion with stable branching mechanism. The spectrum of singularities is established in different dimensions. This is based on collaboration with V. Wachtel, K. Fleischmann, and P. Balança.

I will report on joint work with Victoria Knopova and Pawel Sztonyk. We construct and estimate Markovian semigroups in $\mathbb{R}^d$ with the jump kernel comparable to that of an anisotropic symmetric $\alpha$-stable Lévy process. The order $\gamma$ of Haussdorff regularity of the L\'evy measure is a critical ingredient in our development: we require $\alpha+\gamma>d$.

For a hyperbolic $\alpha$-stable process in the hyperbolic space $\mathbb{H}^d, d\ge 2$, we prove that the mean exit time from a halfspace $H(a)=\{x_d>a\}\subset \mathbb{H}^d $ is equal to $E^x \tau_{H(a)} = c(\alpha, d)\delta^{\alpha/2}_{H(a)}(x),$ where $\delta_D(x)$ is the (hyperbolic) distance of $x$ to $D^c$. Based on this exact result we provide a sharp estimate of the mean exit time from a hyperbolic ball $B(x_0,R)$ of radius $R$ and center $x_0$: $E^x\tau_{B(x_0,R)}\approx (\delta_{B(x_0,R)}(x) \tanh R)^{\alpha/2},\ x\in \mathbb{H}^d.$ By usual isomorphism argument the same estimate holds in any other model of real hyperbolic space. Joint work with Michal Ryznar.

I will explain how Meijer G-functions appear naturally in the study of the fractional Laplace operator. Our main result states the following: if we apply the fractional Laplace operator to a product of a solid harmonic polynomial and a radial function (the latter being expressed in terms of the Meijer G-function) we would obtain a function in the same class. This result has a number of important consequences. First of all, one can re-derive by a simple and unified approach many known expressions, such as the Green function for the unit ball. Second, one can also obtain many new and potentially useful results. For example, one of these new results gives a complete set of eigenfunctions for an operator $(1-|x|^2)_+^{\alpha/2} (-\Delta)^{\alpha/2}$ with the Dirichlet boundary conditions outside of the unit ball, and this result has already proved important for studying the eigenvalues of the fractional Laplace operator in the unit ball. This talk is based on joint work with Bartłomiej Dyda and Mateusz Kwasnicki.

We investigate certain analytical properties of free $\alpha-$stable laws on the line. In the one-sided case, we show that their densities are whale-shaped (that is, their successive derivatives vanish once), perfectly skew, and infinitely divisible in the classical sense. The latter property conveys to the two-sided case when $\alpha \le 1,$ and we also investigate the structure of the Lévy measure. Our method could be useful to the fine study of hitting densities for points or half-lines of real classical stable processes, and we will present several natural open questions in this respect.

The following inversion property (IP) of rotationally invariant $\alpha$-stable processes on $\mathbb{R}^n$ was shown by Bogdan and Żak (2006). Let $I(x)=x/\|x\|^2$ and $h(x)=\|x\|^{\alpha-n}$, $n\ge 1$. Then $(I(X_{\gamma_t}), t>0)\stackrel{d}{=}(X^h_t,\,\, t>0)$, where the time change $\gamma_t$ is the inverse function of $A(t)=\int_0^t \|X_s\|^{-2\alpha}\, ds $. In the pointwise recurrent case $\alpha > n$ one must consider process $X_t^0$ killed at zero. Bogdan and Żak also showed that a Kelvin transformation of $\alpha$-harmonic functions exists. The IP property was extended, in a dual version, by Kyprianou(2016) and Alili, Chaumont, Graczyk and Żak (2016) to real valued stable Lévy processes. We will present our recent results, in which we obtain such inversion properties, often involving dual processes, for diffusions on $\mathbb{R}$ and large classes of Markov processes on $\mathbb{R}^n$, $n\geq 1$. We show that the Kelvin transform of harmonic functions exists for processes satisfying IP. \[ \mbox{ } \] [1] L. Alili, P. Graczyk and T. Żak: On inversions and Doob $h$-transforms of linear diffusions. Lecture Notes in Math, 2137, Séminaire de Probabilités. In Memoriam Marc Yor, 2015. \[ \mbox{ } \] [2] L. Alili, L. Chaumont, P. Graczyk and T. Żak: Inversion, duality and Doob $h$-transforms for self-similar Markov processes. To appear in Electron. J. Probab.(2016) \[ \mbox{ } \] [3] L. Alili, L. Chaumont, P. Graczyk and T. Żak: Space and time inversions of stochastic processes and Kelvin transform, preprint(2016) \[ \mbox{ } \] [4] K. Bogdan and T. Żak: On Kelvin Transformation. Journal of Theoretical Probability, Vol. 19, No. 1, 89--120, (2006). \[ \mbox{ } \] [5] A. E. Kyprianou, Deep factorisation of the stable process, Electron. J. Probab. 21(2016), 28 pp.

In this talk, I will describe a pathwise spine decomposition for superprocesses with both local and non-local branching mechanisms under a martingale change of measure. This result complements the related results obtained in Evans (1993), Kyprianou et al. (2012) and Liu, Ren and Song (2009) for superprocesses with purely local branching mechanisms and in Chen, Ren and Song (2016) and Kyprianou and Palau (2016) for multitype superprocesses. As an application of this decomposition, we obtain necessary/sufficient conditions for the limit of the fundamental martingale to be non-degenerate. In particular, we obtain extinction properties of superprocesses with non-local branching mechanisms as well as a Kesten-Stigum $LlogL$ theorem for the fundamental martingale.

We discuss a new strategy to solve the Skorokhod problem which is generic in the sense that it can be applied to many Markov processes. For the special case of Lévy processes we derive necessary and sufficient conditions (extending earlier work of Bertoin and Le Jan) for the existence of a Skorodhod embedding with finite mean.

Let $J$ be the Lévy density of a symmetric Lévy process in $\bf{R}^d$ with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator \[ {\cal L}^{\kappa}f(x):= \lim_{\epsilon \downarrow 0} \int_{\{z \in \bf{R}^d: |z|>\epsilon\}}(f(x+z)-f(z))\kappa(x,z)J(z)\, dz\, , \] where $\kappa(x,z)$ is a Borel measurable function on $\bf{R}^d\times \bf{R}^d$ satisfying $0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and $|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1)$. We construct the heat kernel $p^\kappa(t, x, y)$ of ${\cal L}^\kappa$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^\kappa$. This talk is based on a joint paper with Panki Kim and Zoran Vondracek.

We present a construction of a coupling of solutions to a certain class of SDEs with jumps, which includes SDEs driven by symmetric $\alpha$-stable processes with $\alpha \in (1,2)$. As an application, we quantify the speed of convergence of solutions to such equations to their invariant measures, both in the standard $L^1$-Wasserstein and the total variation distances. As a second application, we obtain some transportation inequalities, which characterize concentration of the distributions of these solutions, and which were previously known only under the global dissipativity assumption on the drift.

The eigenvalues $\lambda_n$ of the fractional Laplace operator $(-\Delta)^{\alpha/2}$ in the unit ball are not known explicitly, and many apparently simple questions concerning $\lambda_n$ remain unanswered. In my recent joint work with Bartłomiej Dyda and Alexey Kuznetsov we address two examples of such questions. Until recently, evaluating $\lambda_n$ was difficult. We provide two efficient numerical methods for finding lower and upper numerical estimates for $\lambda_n$. For the upper bounds, we use standard Rayleigh–Ritz variational method, while lower bounds involve Weinstein–Aronszajn method of intermediate problems. Both require closed-form expressions for the fractional Laplace operator. We use explicit formulae for the eigenvalues and eigenvectors of the operator $(1 - |x|^2)^{\alpha/2}_+ (-\Delta)^{\alpha/2}$, a topic that will be discussed in detail by Alexey Kuznetsov. The second problem that we address is a conjecture due to Tadeusz Kulczycki, which asserts that all eigenfunctions corresponding to $\lambda_2$ are antisymmetric. This was known to be true only in dimension $1$ when $\alpha \ge 1$. We are able to extend this result to arbitrary $\alpha$ in dimensions $1$ and $2$, and to $\alpha = 1$ in dimensions up to $9$. We prove this result by applying our estimates analytically. This is practically doable only when $2 \times 2$ matrices are involved. Larger matrices can be treated numerically, and such experiments strongly support the conjecture in full generality. At the end of my talk I will present an intriguing open problem, which originates in the following observation: the operators $A = -(1 - |x|^2)^{\alpha/2}_+ (-\Delta)^{\alpha/2}$ and $B = (1 - |x|^2) \Delta - (2 + \alpha) x \cdot \nabla$ have identical eigenfunctions! In dimension $1$ this can be used to prove that the time-changed isotropic $\alpha$-stable Lévy process generated by $A$ can be constructed by subordinating the Jacobi diffusion generated by $B$ using an appropriate subordinator. This result, however, does not extend to higher dimensions!

In this talk, we consider self-similar Markov processes defined on $R^d$ without the origin, which are killed upon hitting the origin. The goal is to try to take a weak limit as $x\rightarrow0$ under mild assumptions. The process started at the origin is obtained in a unique way by conditioning the process to be continuously absorbed at the origin and then reversing time from the absorption time. The proof uses recent techniques in Markov additive process and the Lamperti-Kiu tranformation. This is joint work with Loïc Chaumont, Andreas Kyprianou and Victor Rivero.

Growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. In the self-similar case, it is known that a simple Malthusian condition ensures that the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. We shall present here the converse: when this Malthusian condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes. Based on a joint work with Robin Stephenson.

We consider the diameter of Lévy trees that are random compact metric spaces obtained as the scaling limits of Galton-Watson trees. Lévy trees have been introduced by Le Gall and Le Jan (1998) and they generalise Aldous' Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of Lévy trees and we prove that it is realized by a unique pair of points. We prove that the law of Lévy trees conditioned to have a fixed diameter r is obtained by glueing at their respective roots two independent size-biased Lévy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. This decomposition relies on a similar one for Lévy trees along the geodesic realizing their height that has been obtained by Abraham and Delmas (2009). The law obtained by glueing two trees with height r/2 can be viewed as a natural law for unrooted Lévytrees: this can be justified thanks to a limit theorem for unrooted unlabelled planar trees conditioned by their total height that has been obtained in the recent preprint Wang (2016). As an application of this decomposition of Lévy trees according to their diameter, we characterize the joint law of the height and the diameter of stable Lévy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalized stable trees, which generalizes the identity due to Szekeres (1983) in the Brownian case. Note that Szekeres' result has been proved in a simple way and extended by Wang (2015). This is a joint work with Minmin Wang.

Let $W^D$ be a killed Brownian motion in a domain $D\subset {\mathbb R}^d$ and $S$ an independent subordinator with Laplace exponent $\phi$. The process $Y^D$ defined by $Y^D_t=W^D_{S_t}$ is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator $\phi(-\Delta|_D)$, where $\Delta|_D$ is the Dirichlet Laplacian. In this talk I will present several potential-theoretic results for $Y^D$ under a weak scaling condition on the derivative of $\phi$. These results include the scale invariant Harnack inequality for non-negative harmonic functions of $Y^D$, and two types of scale invariant boundary Harnack principles with explicit decay rates. The first boundary Harnack principle deals with a $C^{1,1}$ domain $D$ and non-negative functions which are harmonic near the boundary of $D$, while the second one is for a more general domain $D$ and non-negative functions which are harmonic near the boundary of an interior open subset of $D$. The obtained decay rates are not the same, reflecting different boundary and interior behavior of $Y^D$. The results are new even in the case of the stable subordinator. Joint work with P.Kim and R.Song