arXiv subjects

René L. Schilling

Publications and source records attributed to René L. Schilling.

At least 37 records · Page 2Linked to original sources

arXiv2021

Homogenization of Symmetric Lévy Processes on $\mathbb{R}^d$

In this short note we study homogenization of symmetric $d$-dimensional Lévy processes. Homogenization of one-dimensional pure jump Markov processes has been investigated by Tanaka \emph{et al.} in 1992; their motivation was the work by Benssousan \emph{et al.}\ from 1975 on the homogenization of diffusion processes in $\mathbb{R}^d$. We investigate a similar problem for a class of symmetric pure-jump Lévy processes on $\mathbb{R}^d$ and we identify -- using Mosco convergence -- the limit process.

math.PR↗

arXiv2020

Asymptotic behaviour and functional limit theorems for a time changed Wiener process

We study the asymptotic behaviour of a properly normalized time changed Wiener processes. The time change reflects the fact that we consider the Laplace operator (which generates a Wiener process) multiplied by a possibly degenerate state-space dependent intensity $λ(x)$. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time changed Wiener process. The normalization depends on the asymptotic behaviour of the intensity function $λ$. One of the possible limits is a skew Brownian motion.

math.PR↗

arXiv2020

Strong convergence of the Euler--Maruyama approximation for a class of Lévy-driven SDEs

Consider the following stochastic differential equation (SDE) $$dX_t = b(t,X_{t-}) \, dt+ dL_t, \quad X_0 = x,$$ driven by a $d$-dimensional Lévy process $(L_t)_{t \geq 0}$. We establish conditions on the Lévy process and the drift coefficient $b$ such that the Euler--Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of $b$ and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples of Lévy processes, e.g. isotropic stable, relativistic stable, tempered stable and layered stable.

math.PR↗

arXiv2019

Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the $n$-dimensional fractional heat equations in both time and space \begin{gather*} \frac{\partial^β}{\partial t^β}u(t,x) = -(-Δ_x)^γu(t,x), \quad β,γ\in(0,1). \end{gather*}

math.PR↗

arXiv2019

Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators

We study the long-time asymptotic behaviour of semigroups generated by non-local Schrödinger operators of the form $H = -L+V$; the free operator $L$ is the generator of a symmetric Lévy process in $\mathbb R^d$, $d > 1$ (with non-degenerate jump measure) and $V$ is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat trace and heat content. Our examples cover a wide range of processes and we have to assume only mild restrictions on the growth, resp.\ decay, of the potential and the jump intensity of the free process. Our approach is based on a combination of probabilistic and analytic methods; our examples include fractional and quasi-relativistic Schrödinger operators.

math.PR↗

arXiv2018

On the domain of fractional Laplacians and related generators of Feller processes

In this paper we study the domain of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder-Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process $(X_t)_{t \geq 0}$, \begin{equation*} \lim_{t \to 0} \frac 1t\left(\mathbb{E}^x f(X_t)-f(x)\right), \quad x\in\mathbb{R}^d, \end{equation*} for functions $f$ which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x \in \mathbb{R}^d$ if $f$ satisfies a Hölder condition at $x$. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of $(X_t)_{t \geq 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.

math.PR↗

arXiv2018

A Unified Approach to Coupling SDEs driven by Lévy Noise and Some Applications

We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As an application, we establish regularity results for the transition semigroups of the solutions to stochastic differential equations driven by additive Lévy noise.

math.PR↗

arXiv2018

Detecting independence of random vectors: generalized distance covariance and Gaussian covariance

Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric Lévy measures and the corresponding real-valued continuous negative definite functions. The Lévy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework. From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance Multivariance: New dependence measures for random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the present article.

math.PR↗

arXiv2018

Distance multivariance: New dependence measures for random vectors

We introduce two new measures for the dependence of $n \ge 2$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted $L^2$-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to $n$-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of $n$ random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.

math.PR↗

arXiv2018

A probabilistic proof of Schoenberg's theorem

Assume that $g(|ξ|^2)$, $ξ\in\mathbb{R}^k$, is for every dimension $k\in\mathbb{N}$ the characteristic function of an infinitely divisible random variable $X^k$. By a classical result of Schoenberg $f:=-\log g$ is a Bernstein function. We give a simple probabilistic proof of this result starting from the observation that $X^k = X_1^k$ can be embedded into a Lévy process $(X_t^k)_{t\geq 0}$ and that Schoenberg's theorem says that $(X_t^k)_{t\geq 0}$ is subordinate to a Brownian motion. A key ingredient of our proof are concrete formulae which connect the transition densities, resp., Lévy measures of subordinated Brownian motions across different dimensions. As a by-product of our proof we obtain a gradient estimate for the transition semigroup of a subordinated Brownian motion.

math.PR↗

arXiv2017

A Probabilistic proof of the breakdown of Besov regularity in $L$-shaped domains

{We provide a probabilistic approach in order to investigate the smoothness of the solution to the Poisson and Dirichlet problems in $L$-shaped domains. In particular, we obtain (probabilistic) integral representations for the solution. We also recover Grisvard's classic result on the angle-dependent breakdown of the regularity of the solution measured in a Besov scale.

math.PR↗

arXiv2017

Harnack Inequalities for SDEs Driven by Time-Changed Fractional Brownian Motions

We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition, otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.

math.PR↗

arXiv2017

Subgeometric rates of convergence for Markov processes under subordination

We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent) we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moments (sub-exponential, algebraic and logarithmic) for subordinators as time $t$ tends to infinity. At the end we discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.

math.PR↗

arXiv2016

Reflected Spectrally Negative Stable Processes and their Governing Equations

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.

math.PR↗

arXiv2016

Complete Bernstein functions and subordinators with nested ranges. A note on a paper by P. Marchal

Let $α:[0,1]\to [0,1]$ be a measurable function. It was proved by P. Marchal \cite{Mar15} that the function $$ ϕ^{(α)}(λ):=\exp\left[ \int_0^1\frac{λ-1}{1+(λ-1)x}\,α(x)\,d x \right],\quad λ>0 $$ is a special Bernstein function. Marchal used this to construct, on a single probability space, a family of regenerative sets $\mathcal R^{(α)}$ such that $\mathcal{R}^{(α)} \stackrel{\text{law}}{=} \overline{\{S^{(α)}_t:t\geq 0\}}$ ($S^{(α)}$ is the subordinator with Laplace exponent $ϕ^{(α)}$) and $\mathcal R^{(α)}\subset \mathcal R^{(β)}$ whenever $α\leqβ$. We give two simple proofs showing that $ϕ^{(α)}$ is a complete Bernstein function and extend Marchal's construction to all complete Bernstein functions.

math.PR↗

arXiv2016

Markov Chain Approximation of Pure Jump Processes

In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains.

math.PR↗

arXiv2016

Continuity Assumptions in Cake-Cutting

In important papers on cake-cutting -- one of the key areas in fair division and resource allocation -- the measure-theoretical fundamentals are not fully correctly given. It is not clear (i) which family of sets should be taken for the pieces of cake, (ii) which set-functions should be used for evaluating the pieces, and (iii) which is the relationship between various continuity properties appearing in cake-cutting. We show that probably the best choice for the familiy of subsets of $[0,1]$ is the Borel $σ$-algebra and for the set-function any `sliceable' Borel measure. At least in dimension one it does not make sense to work with only finitely additive contents on finite unions of intervals. For the continuity property we see two possibilities. The weaker is the traditional divisibility property, which is equivalent to being atom-free. The stronger is simply absolute continuity with respect to Lebesgue measure. We also consider the case of a base set (cake or pie) more general than $[0,1]$.

cs.DS↗

arXiv2016

An Introduction to Lévy and Feller Processes. Advanced Courses in Mathematics - CRM Barcelona 2014

These lecture notes are an extended version of my lectures on Lévy and Lévy-type (Feller) processes given at the "Second Barcelona Summer School on Stochastic Analysis" 2014 organized by the Centre de Recerca Matemaatica (CRM). The lectures are aimed at advanced graduate and PhD students. In order to read these notes, one should have sound knowledge of measure theoretic probability theory and some background in stochastic processes, as it is covered in my books "Measures, Integals and Martingales" (Cambridge University Press) and "Brownian Motion" (de Gruyter). My purpose in these lectures is to give an introduction to Levy processes, and to show how one can extend this approach to space inhomogeneous processes which behave locally like Lévy processes: Lévy-type or Feller processes. These course notes will be published, together Davar Khoshnevisan's notes on "Invariance and Comparison Principles for Parabolic SPDEs" as "From Lévy-Type Processes to Parabolic SPDEs" by the CRM, Barcelona and Birkäuser, Cham 2017 (ISBN: 978-3-319-34119-4). The arXiv-version and the published version may differ in layout, pagination and wording, but not in content

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