Self-similar Markov Processes Research Articles

Positive self-similar Markov processes obtained by resurrection

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly α-stable process at its first exit time from (0,∞). We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes.

On the local times of noise reinforced Bessel processes

We investigate the effects of noise reinforcement on a Bessel process of dimension $d\in(0,2)$, and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.

Discrete self-similar and ergodic Markov chains

The first aim of this paper is to introduce a class of Markov chains on Z+ which are discrete self-similar in the sense that their semigroups satisfy an invariance property expressed in terms of a discrete random dilation operator. After showing that this latter property requires the chains to be upward skip-free, we first establish a gateway intertwining relation between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on R+. As a by-product, we prove that each of these Markov chains, after an appropriate scaling, converge in the Skorohod metric to the associated self-similar Markov process. By a linear perturbation of the generator of these Markov chains, we obtain a class of ergodic Markov chains which are nonreversible. By means of intertwining relations and their strengthened interweaving versions, we derive several deep analytical properties of such ergodic chains, including the description of the spectrum, the spectral expansion of their semigroups and the study of their convergence to equilibrium in the Φ-entropy sense as well as their hypercontractivity property.

Fragmentations with self-similar branching speeds

AbstractWe consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincaré Prob. Statist.38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy–Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.

General Self-Similarity Properties for Markov Processes and Exponential Functionals of Lévy Processes

Positive self-similar Markov processes are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a time-changed Lévy process via Lamperti representation. In this work, we are interested in what happens if we consider Markov processes in dimension 1 or 2 that satisfy self-similarity properties of a more general form than a scaling property. We characterize them by proving a generalized Lamperti representation. Our results show that, in dimension 1, the classical Lamperti representation only needs to be slightly generalized. However, in dimension 2, our generalized Lamperti representation is much more different and involves the exponential functional of a bivariate Lévy process. We briefly discuss the complications that occur in higher dimensions. We present examples in dimensions 1, 2 and 3 that are built from growth-fragmentation, self-similar fragmentation and Continuous-state Branching processes in Random Environment. Some of our arguments apply in the context of a general state space and show that we can exhibit a topological group structure on the state space of a Markov process that satisfies general self-similarity properties, which allows to write a Lamperti-type representation for this process in terms of a Lévy process on the group.

Attraction to and repulsion from a subset of the unit sphere for isotropic stable Lévy processes

Taking account of recent developments in the representation of d-dimensional isotropic stable Lévy processes as self-similar Markov processes, we consider a number of new ways to condition its path. Suppose that S is a region of the unit sphere Sd−1={x∈Rd:|x|=1}. We construct the aforesaid stable Lévy process conditioned to approach S continuously from either inside or outside of the sphere. Additionally, we show that these processes are in duality with the stable process conditioned to remain inside the sphere and absorb continuously at the origin and to remain outside of the sphere, respectively. Our results extend the recent contributions of Döring and Weissman (2020), where similar conditioning is considered, albeit in one dimension as well as providing analogues of the same classical results for Brownian motion, cf. Doob (1957). As in Döring and Weissman (2020), we appeal to recent fluctuation identities related to the deep factorisation of stable processes, cf. Kyprianou (2016), Kyprianou et al. (2020) and Kyprianou et al. (2017).

Discretization of the Lamperti representation of a positive self-similar Markov process

This paper considers discretization of the Lévy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given Lévy process. Additionally, the scaling limit of a positive self-similar Markov process at small times is provided. Finally, we present an application to simulation of self-similar Lévy processes conditioned to stay positive.

Self-Similar Cauchy Problems and Generalized Mittag-Leffler Functions

Abstract By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. This work could be seen as an-in depth analysis of a specific class, the one with the self-similarity property, of the general inverse of increasing Markov processes introduced in [15].

Double hypergeometric Lévy processes and self-similarity

AbstractMotivated by a recent paper (Budd (2018)), where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener–Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

A temporal factorization at the maximum for certain positive self-similar Markov processes

AbstractFor a spectrally negative self-similar Markov process on $[0,\infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.

Entrance laws at the origin of self-similar Markov processes in high dimensions

In this paper we consider the problem of finding entrance laws at the origin for self-similar Markov processes in R d \mathbb {R}^d , killed upon hitting the origin. Under suitable assumptions, we show the existence of an entrance law and the convergence to this law when the process is started close to the origin. We obtain an explicit description of the process started from the origin as the time reversal of the original self-similar Markov process conditioned to hit the origin.

Stable processes conditioned to hit an interval continuously from the outside

Conditioning stable Lévy processes on zero probability events recently became a tractable subject since several explicit formulas emerged from a deep analysis using the Lamperti transformations for self-similar Markov processes. In this article, we derive new harmonic functions and use them to explain how to condition stable processes to hit continuously a compact interval from the outside.

Entrance and exit at infinity for stable jump diffusions

In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on $-\infty \leq a<b\leq \infty $ in terms of their ability to access the boundary (Feller’s test for explosions) and to enter the interior from the boundary. Feller’s technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille–Yosida theory. In the present article, we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form \begin{equation*}dZ_{t}=\sigma (Z_{t-})\,dX_{t},\end{equation*} driven by stable Lévy processes for $\alpha \in (0,2)$. Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller’s work but exit and entrance from infinite boundaries has long remained open. We show that the presence of jumps implies features not seen in the diffusive setting without drift. Finite time explosion is possible for $\alpha \in (0,1)$, whereas entrance from different kinds of infinity is possible for $\alpha \in [1,2)$. Accordingly, we derive necessary and sufficient conditions on $\sigma $. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti–Kiu representation and new Wiener–Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz–Bogdan–Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt–Nagasawa duality and Getoor’s characterisation of transience and recurrence.

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.

Recurrent Extensions of Real-Valued Self-Similar Markov Processes

Let X = (Xt,t ≥ 0) be a self-similar Markov process taking values in \(\mathbb {R}\) such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (Electron. Commun. Probab. 11, 230–241 2006) and Rivero (Bernoulli 11, 471–509 2005, 13, 1053–1070 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable Lévy process which to the best of our knowledge has not been studied before.

First passage upwards for state-dependent-killed spectrally negative Lévy processes

AbstractFor a spectrally negative Lévy process X, killed according to a rate that is a function ω of its position, we complement the recent findings of [12] by analysing (in greater generality) the exit probability of the one-sided upwards passage problem. When ω is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for X that has been time-changed by the inverse of the additive functional $$\int_0^ \cdot \omega ({X_u}){\kern 1pt} {\rm{d}}u$$. In particular, our findings thus shed extra light on related results concerning first passage times downwards (resp. upwards) of continuous-state branching processes (resp. spectrally negative positive self-similar Markov processes).

Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes

This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process (X(t))t≥0 which drifts to ∞, namely U(t)≔e−tX(et−1). We point out that U is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponential functional Iˆ≔∫0∞exp(ξˆs)ds, where ξˆ is the dual of the real-valued Lévy process ξ related to X by the Lamperti transformation. This invariant measure is infinite (i.e. U is null-recurrent) if and only if ξ1∉L1(P). In that case, we determine the family of Lévy processes ξ for which U fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process ξ, and properties of time-substitutions based on additive functionals.

Conditioned real self-similar Markov processes

In recent work, Chaumont et al. (2013) showed that is possible to condition a stable process with index α∈(1,2) to avoid the origin. Specifically, they describe a new Markov process which is the Doob h-transform of a stable process and which arises from a limiting procedure in which the stable process is conditioned to have avoided the origin at later and later times. A stable process is a particular example of a real self-similar Markov process (rssMp) and we develop the idea of such conditionings further to the class of rssMp. Under appropriate conditions, we show that the specific case of conditioning to avoid the origin corresponds to a classical Cramér–Esscher-type transform to the Markov Additive Process (MAP) that underlies the Lamperti–Kiu representation of a rssMp. In the same spirit, we show that the notion of conditioning a rssMp to continuously absorb at the origin also fits the same mathematical framework. In particular, we characterise the stable process conditioned to continuously absorb at the origin when α∈(0,1). Our results also complement related work for positive self-similar Markov processes in Chaumont and Rivero (2007).

Stable windings at the origin

In 1996, Bertoin and Werner demonstrated a functional limit theorem, characterising the windings of planar isotropic stable processes around the origin for large times, thereby complementing known results for planar Brownian motion. The question of windings at small times can be handled using scaling. Nonetheless we examine the case of windings at the origin using new techniques from the theory of self-similar Markov processes. This allows us to understand upcrossings of (not necessarily symmetric) stable processes over the origin for large and small times in the one-dimensional setting.

Bivariate Markov chains converging to Lamperti transform Markov additive processes

Motivated by various applications, we describe the scaling limits of bivariate Markov chains (X,J) on Z+×{1,…,κ} where X can be viewed as a position marginal and {1,…,κ} is a set of κ types. The chain starts from an initial value (n,i)∈Z+×{1,…,κ}, with i fixed and n→∞, and typically we will assume that the macroscopic jumps of the marginal X are rare, i.e. arrive with a probability proportional to a negative power of the current state. We also assume that X is non-increasing. We then observe different asymptotic regimes according to whether the rate of type change is proportional to, faster than, or slower than the macroscopic jump rate. In these different situations, we obtain in the scaling limit Lamperti transforms of Markov additive processes, that sometimes reduce to standard positive self-similar Markov processes. As first examples of applications, we study the number of collisions in coalescents in varying environment and the scaling limits of Markov random walks with a barrier. This completes previous results obtained by Haas and Miermont (2011) and Bertoin and Kortchemski (2016) in the monotype setting. In a companion paper, we will use these results as a building block to study the scaling limits of multi-type Markov branching trees, with applications to growing models of random trees and multi-type Galton–Watson trees.