arXiv subjects

Pierre Patie

Publications and source records attributed to Pierre Patie.

At least 19 recordsLinked to original sources

arXiv2026

Non-local heat equations with moving boundary

In this paper we consider non-local (in time) heat equations on time-increasing parabolic sets whose boundary is determined by a suitable curve. We provide a notion of solution for these equations and we study well-posedness under Dirichlet conditions outside the domain. A maximum principle is proved and used to derive uniqueness and continuity with respect to the initial datum of the solutions of the Dirichlet problem. Existence is proved by showing a stochastic representation based on the delayed Brownian motion killed on the boundary. Several related distributional properties of the delayed Brownian motion and its crossing probabilities are also obtained. The asymptotic behaviour of the mean square displacement of the process is determined, showing that the diffusive behaviour is anomalous.

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arXiv2023

Jacobi processes with jumps as neuronal models: a first passage time analysis

To overcome some limits of classical neuronal models, we propose a Markovian generalization of the classical model based on Jacobi processes by introducing downwards jumps to describe the activity of a single neuron. The statistical analysis of inter-spike intervals is performed by studying the first-passage times of the proposed Markovian Jacobi process with jumps through a constant boundary. In particular, we characterize its Laplace transform which is expressed in terms of some generalization of hypergeometric functions that we introduce, and deduce a closed-form expression for its expectation. Our approach, which is original in the context of first passage time problems, relies on intertwining relations between the semigroups of the classical Jacobi process and its generalization, which have been recently established in [11]. A numerical investigation of the firing rate of the considered neuron is performed for some choices of the involved parameters and of the jumps distributions.

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arXiv2023

Weak Similarity Orbit of (log)-Self-Similar Markov Semigroups on the Euclidean Space

We start by identifying a class of pseudo-differential operators, generated by the set of continuous negative definite functions, that are in the weak similarity (WS) orbit of the self-adjoint log-Bessel operator on the Euclidean space. These WS relations turn out to be useful to first characterize a core for each operator in this class, which enables us to show that they generate a class, denoted by $\mathscr{P}$, of non-self-adjoint $\mathcal{C}_0$-contraction positive semigroups. Up to a homeomorphism, $\mathscr{P}$ includes, as fundamental objects in probability theory, the family of self-similar Markov semigroups on $\mathbb{R}_+^d$. Relying on the WS orbit, we characterize the nature of the spectrum of each element in $\mathscr{P}$ that is used in their spectral representation which depends on analytical properties of the Bernstein-gamma functions defined from the associated negative definite functions, and, it is either the point, residual, approximate or continuous spectrum. We proceed by providing a spectral representation of each element in $\ccP$ which is expressed in terms of Fourier multiplier operators and valid, at least, on a dense domain of a natural weighted $L^2$-space. Surprisingly, the domain is the full Hilbert space when the spectrum is the residual one, something which seems to be noticed for the first time in the literature. We end up the paper by presenting a series of examples for which all spectral components are computed explicitly in terms of special functions or recently introduced power series.

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arXiv2022

Discrete self-similar and ergodic Markov chains

The first aim of this paper is to introduce a class of Markov chains on $\mathbb{Z}_+$ 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 relation, a concept introduced in [26], between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on $\mathbb{R}_+$. 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 non-reversible. By means of intertwining and interweaving relations, where the latter was recently introduced in [27], we derive several deep analytical properties of such ergodic chains including the description of the spectrum, the spectral expansion of their semigroups, the study of their convergence to equilibrium in the $Φ$-entropy sense as well as their hypercontractivity property.

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arXiv2021

On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability measures and study their spectral and convergence properties. In particular, we derive a series expansion of the semigroup in terms of explicitly defined polynomials, which generalize the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators and semigroups, with the local objects serving as reference points for transferring properties from the local to the non-local case.

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arXiv2020

Non-classical Tauberian and Abelian type criteria for the moment problem

The aim of this paper is to provide some new criteria for the Stieltjes moment problem. We first give a Tauberian type criterion for moment indeterminacy that is expressed purely in terms of the asymptotic behavior of the moment sequence (and its extension to imaginary lines). Under an additional assumption this provides a converse to the classical Carleman's criterion, thus yielding an equivalent condition for moment determinacy. We also provide a criterion for moment determinacy that only involves the large asymptotic behavior of the distribution (or of the density if it exists), which can be thought of as an Abelian counterpart to the previous Tauberian type result. This latter criterion generalizes Hardy's condition for determinacy, and under some further assumptions yields a converse to the Pedersen's refinement of the celebrated Krein's theorem. The proofs utilize non-classical Tauberian results for moment sequences that are analogues to the ones developed in Feigin and Yashchin, and, Balkema et al. for the bi-lateral Laplace transforms in the context of asymptotically parabolic functions. We illustrate our results by studying the time-dependent moment problem for the law of log-Lévy processes viewed as a generalization of the log-normal distribution. Along the way, we derive the large asymptotic behavior of the density of spectrally-negative Lévy processes having a Gaussian component, which may be of independent interest.

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arXiv2019

Analysis of non-reversible Markov chains via similarity orbit

In this paper, we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as the one of birth-death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth-death one. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and ${\rm{L}}^2$-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper, we investigate a particular similarity orbit of reversible Markov kernels, that we call the pure birth orbit, and analyze various possibly non-reversible variants of classical birth-death processes in this orbit.

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arXiv2019

On interweaving relations

Interweaving relations are introduced and studied here in a general Markovian setting as a strengthening of usual intertwining relations between semigroups, obtained by adding a randomized delay feature. They provide a new classification scheme of the set of Markovian semigroups which enables to transfer from a reference semigroup and up to an independent warm-up time, some ergodic, analytical and mixing properties including the $φ$-entropy convergence to equilibrium, the hyperboundedness and when the warm-up time is deterministic the cut-off phenomena. We also present several useful transformations that preserve interweaving relations. We provide a variety of examples of interweaving relations ranging from classical, discrete, and non-local Laguerre and Jacobi semigroups to degenerate hypoelliptic Ornstein-Uhlenbeck semigroups and some non-colliding particle systems

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arXiv2019

On Doney's striking factorization of the arc-sine law

R. Doney identifies a striking factorization of the arc-sine law in terms of the suprema of two independent stable processes of the same index by an elegant random walks approximation. In this paper, we provide an alternative proof and a generalization of this factorization based on the theory recently developed for the exponential functional of Lévy processes. As a by-product, we provide some interesting distributional properties for these variables and also some new examples of the factorization of the arc-sine law.

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arXiv2019

Spectral projections correlation structure for short-to-long range dependent processes

Let $\mathbf{X}=(\mathbf{X}_t)_{t \geq 0}$ be a stochastic process issued from $x \in \mathbb R$ that admits a marginal stationary measure $ν$, i.e. $ν\mathbf{P}_t f = νf$ for all $t \geq 0$, where $\mathbf{P}_t f(x)= \mathbb{E}_x[f(\mathbf{X}_t)]$. In this paper, we introduce the (resp. biorthogonal) spectral projections correlation functions which are expressed in terms of projections into the eigenspaces of $\mathbf{P}_t$ (resp. and of its adjoint in the weighted Hilbert space $L^2(ν)$). We obtain closed-form expressions involving eigenvalues, the condition number and/or the angle between the projections in the following different situations: when $\mathbf{X}=X$ with $X=(X_t)_{t \geq 0}$ is a Markov process, $\mathbf{X}$ is the subordination of $X$ in the sense of Bochner, and $\mathbf X$ is a non-Markovian process which is obtained by time-changing $X$ with an inverse of a subordinator. It turns out that these spectral projections correlation functions have different expressions with respect to these classes of processes which enables to identify substantial and deep properties about their dynamics. This interesting fact can be used to design original statistical tests to make inferences, for example, about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to-long-range dependence. To reveal the usefulness of our results, we apply them to a class of non-self-adjoint Markov semigroups studied in Patie and Savov [28], and then time-change by subordinators and their inverses.

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arXiv2019

A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators

The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup $\tilde{P} = (e^{-t\tilde{\mathbf A}})_{t \geq 0}$ to a target semigroup $P$ which is the object of study. This allows us to obtain conditions under which $P$ satisfies a hypocoercive estimate with exponential decay rate given by the spectral gap of $\tilde{\mathbf A}$. Along the way we also develop a functional calculus involving the non-self-adjoint resolution of identity induced by the intertwining relations. We apply these results in a general Hilbert space setting to two cases: degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on $\mathbb R^d$, and non-local Jacobi semigroups on $[0,1]^d$, which have been recently introduced and studied for $d=1$. In both cases we obtain hypocoercive estimates and are able to explicitly identify the hypocoercive constants

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arXiv2019

The log-Lévy moment problem via Berg-Urbanik semigroups

We consider the Stieltjes moment problem for the Berg-Urbanik semigroups which form a class of multiplicative convolution semigroups on $\mathbb{R}_+$ that is in bijection with the set of Bernstein functions. Berg and Durán proved that the law of such semigroups is moment determinate (at least) up to time $t=2$, and, for the Bernstein function $ϕ(u)=u$, Berg made the striking observation that for time $t>2$ the law of this semigroup is moment indeterminate. We extend these works by estimating the threshold time $\scr{T}_ϕ\in [2,\infty]$ that it takes for the law of such Berg-Urbanik semigroups to transition from moment determinacy to moment indeterminacy in terms of simple properties of the underlying Bernstein function $ϕ$, such as its Blumenthal-Getoor index. One of the several strategies we implement to deal with the different cases relies on a non-classical Abelian type criterion for the moment problem, recently proved by the authors. To implement this approach we provide detailed information regarding distributional properties of the semigroup such as existence and smoothness of a density, and, the large asymptotic behavior for all $t > 0$ of this density along with its successive derivatives. In particular, these results, which are original in the Lévy processes literature, may be of independent interests.

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arXiv2019

On intertwining relations between Ehrenfest, Yule and Ornstein-Uhlenbeck processes

Markovian intertwining relations between two Markov semigroups are related to the partial inclusion of the spectra of their generators, at least for finite ergodic processes. We check the limitations of this observation by investigating the Markov intertwining relations between the Ehrenfest, Yule and Ornstein-Uhlenbeck processes, whose spectra are all included into -$\mathbb{Z}_+$. As a by-product, we offer a clarification of an intertwining relation found in Biane [2] between the Yule and the Ornstein-Uhlenbeck processes.

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arXiv2019

Skip-free Markov chains

The aim of this paper is to develop a general theory for the class of skip-free Markov chains on denumerable state space. This encompasses their potential theory via an explicit characterization of their potential kernel expressed in terms of family of fundamental excessive functions, which are defined by means of the theory of Martin boundary. We also describe their fluctuation theory generalizing the celebrated fluctuations identities that were obtained by using the Wiener-Hopf factorization for the specific skip-free random walks. We proceed by resorting to the concept of similarity to identify the class of skip-free Markov chains whose transition operator has only real and simple eigenvalues. We manage to find a set of sufficient and easy-to-check conditions on the one-step transition probability for a Markov chain to belong to this class. We also study several properties of this class including their spectral expansions given in terms of Riesz basis, derive a necessary and sufficient condition for this class to exhibit a separation cutoff, and give a tighter bound on its convergence rate to stationarity than existing results.

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arXiv2018

Extinction time of non-Markovian self-similar processes, persistence, annihilation of jumps and the Fréchet distribution

We start by providing an explicit characterization and analytical properties, including the persistence phenomena, of the distribution of the extinction time $\mathbb{T}$ of a class of non-Markovian self-similar stochastic processes with two-sided jumps that we introduce as a stochastic time-change of Markovian self-similar processes. For a suitably chosen time-changed, we observe, for classes with two-sided jumps, the following surprising facts. On the one hand, all the $\mathbb{T}$'s within a class have the same law which we identify in a simple form for all classes and reduces, in the spectrally positive case, to the Fréchet distribution. On the other hand, each of its distribution corresponds to the law of an extinction time of a single Markov process without positive jumps, leaving the interpretation that the time-changed has annihilated the effect of positive jumps. The example of the non-Markovian processes associated to Lévy stable processes is detailed.

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arXiv2018

On a gateway between continuous and discrete Bessel and Laguerre processes

By providing instances of approximation of linear diffusions by birth-death processes, Feller [13], has offered an original path from the discrete world to the continuous one. In this paper, by identifying an intertwining relationship between squared Bessel processes and some linear birth-death processes, we show that this connection is in fact more intimate and goes in the two directions. As by-products, we identify some properties enjoyed by the birth-death family that are inherited from squared Bessel processes. For instance, these include a discrete self-similarity property and a discrete analogue of the beta-gamma algebra. We proceed by explaining that the same gateway identity also holds for the corresponding ergodic Laguerre semi-groups. It follows again that the continuous and discrete versions are more closely related than thought before, and this enables to pass information from one semi-group to the other one.

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arXiv2018

Bernstein-gamma functions and exponential functionals of Levy Processes

We study the equation $M_Ψ(z+1)=\frac{-z}{Ψ(-z)}M_Ψ(z), M_Ψ(1)=1$ defined on a subset of the imaginary line and where $Ψ$ is a negative definite functions. Using the Wiener-Hopf method we solve this equation in a two terms product which consists of functions that extend the classical gamma function. These functions are in a bijection with Bernstein functions and for this reason we call them Bernstein-gamma functions. Via a couple of computable parameters we characterize of these functions as meromorphic functions on a complex strip. We also establish explicit and universal Stirling type asymptotic in terms of the constituting Bernstein function. The decay of $|M_Ψ(z)|$ along imaginary lines is computed. Important quantities for theoretical and applied studies are rendered accessible. As an application we investigate the exponential functionals of Levy Processes whose Mellin transform satisfies the recurrent equation above. Although these variables have been intensively studied, our new perspective, based on a combination of probabilistic and complex analytical techniques, enables us to derive comprehensive and substantial properties and strengthen several results on the law of these random variables. These include smoothness, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions, bounds, and Mellin-Barnes representations for the density and its successive derivatives. We also study the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. As a result of new factorizations of the law of the exponential functional we deliver important intertwining relation between members of the class of positive self-similar semigroups. The derivation of our results relies on a mixture of complex-analytical and probabilistic techniques.

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arXiv2018

Smoothness of continuous state branching with immigration semigroups

In this work we develop an original and thorough analysis of the (non)-smoothness properties of the semigroups, and their heat kernels, associated to a large class of continuous state branching processes with immigration. Our approach is based on an in-depth analysis of the regularity of the absolutely continuous part of the invariant measure combined with a substantial refinement of Ogura's spectral expansion of the transition kernels. In particular, we find new representations for the eigenfunctions and eigenmeasures that allow us to derive delicate uniform bounds that are useful for establishing the uniform convergence of the spectral representation of the semigroup acting on linear spaces that we identify. We detail several examples which illustrate the variety of smoothness properties that CBI transition kernels may enjoy and also reveal that our results are sharp. Finally, our technique enables us to provide the (eventually) strong Feller property as well as the rate of convergence to equilibrium in the total variation norm.

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