Stable-like Processes Research Articles

Homogenization of non-symmetric jump processes

AbstractWe study homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times.

Maximal inequalities and some applications

A maximal inequality is an inequality which involves the (absolute) supremum sups⩽t|Xs| or the running maximum sups⩽tXs of a stochastic process (Xt)t⩾0. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type – including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy process –, strong Markov processes and Gaussian processes. Using the Burkholder–Davis–Gundy inequalities we also discuss some relations between maximal estimates in probability and the Hardy–Littlewood maximal functions from analysis.

Upper functions for sample paths of Lévy(-type) processes

We study the small-time asymptotics of sample paths of Lévy processes and Lévy-type processes. Namely, we investigate under which conditions the limit lim supt→0 1 f(t)|Xt−X0| is finite resp. infinite with probability 1. We establish integral criteria in terms of the infinitesimal characteristics and the symbol of the process. Our results apply to a wide class of processes, including solutions to Lévy-driven SDEs and stable-like processes. For the particular case of Lévy processes, we recover and extend earlier results from the literature. Moreover, we present a new maximal inequality for Lévy-type processes, which is of independent interest.

Operator-stable-like processes

In the present article, we introduce so-called operator-stable-like processes. Roughly speaking, they behave locally like operator-stable processes, but they need not to be homogenous in space. Having shown existence for this class of processes, we analyze maximal estimates, the existence of moments, the short- and long-time behavior of the sample paths and p-variation. The class introduced, here, includes stable-like processes as special case.

Homogenization of Symmetric Stable-like Processes in Stationary Ergodic Media

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 23 March 2020Accepted: 22 February 2021Published online: 20 May 2021Keywordshomogenization, symmetric nonlocal Dirichlet form, ergodic random medium, $\alpha$-stable-like operatorAMS Subject Headings60G51, 60G52, 60J25, 60J75Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation Af=g associated with the (extended) infinitesimal generator of a Feller process. The regularity of f is described in terms of Hölder–Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equations. Most prominently, we establish Schauder estimates for the Poisson equation associated with the fractional Laplacian of variable order. As a by-product, we obtain new regularity estimates for semigroups associated with stable-like processes.

Uniqueness in Law for Stable-Like Processes of Variable Order

Let $$d\ge 1$$ . Consider a stable-like operator of variable order $$\begin{aligned} {\mathcal {A}}f(x)&=\int _{{\mathbb {R}}^{d}\backslash \{0\}} \left[ f(x+h)-f(x)-\mathbf {1}_{\{|h|\le 1\}}h\cdot \nabla f(x)\right] n(x,h)|h|^{-d-\alpha (x)}\mathrm{d}h, \end{aligned}$$ where $$0<\inf _{x}\alpha (x)\le \sup _{x}\alpha (x)<2$$ and n(x, h) satisfies $$\begin{aligned} n(x,h)=n(x,-h),\quad 0<\kappa _{1}\le n(x,h)\le \kappa _{2}, \quad \forall x,h\in {\mathbb {R}}^{d}, \end{aligned}$$ with $$\kappa _{1}$$ and $$\kappa _{2}$$ being some positive constants. Under some further mild conditions on the functions n(x, h) and $$\alpha (x)$$ , we show the uniqueness of solutions to the martingale problem for $${\mathcal {A}}$$ .

Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators

Let $A$ be a pseudo-differential operator with symbol $q(x,\xi)$. In this paper we derive sufficient conditions which ensure the existence of a solution to the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem. If the symbol $q$ depends continuously on the space variable $x$, then the existence of solutions is well understood, and therefore the focus lies on martingale problems for pseudo-differential operators with discontinuous coefficients. We prove an existence result which allows us, in particular, to obtain new insights on the existence of weak solutions to a class of L\'evy-driven SDEs with Borel measurable coefficients and on the the existence of stable-like processes with discontinuous coefficients. Moreover, we establish a Markovian selection theorem which shows that - under mild assumptions - the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem gives rise to a strong Markov process. The result applies, in particular, to L\'evy-driven SDEs. We illustrate the Markovian selection theorem with applications in the theory of non-local operators and equations; in particular, we establish under weak regularity assumptions a Harnack inequality for non-local operators of variable order.

On Mean Field Games with Common Noise based on Stable-Like Processes

In this paper, we study Mean Field games with common noise based on nonlinear stable-like processes. The MFG limit is specified by a single quasi-linear deterministic infinite-dimensional partial differential second order backward equation. The main result is that any its solution provides an 1/N-Nash equilibrium for the initial game of N agents. Our approach is based on interpreting the common noise as a kind of binary interaction of agents and our previous results on regularity and sensitivity with respect to the initial conditions of the solution to the nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes.

Semimartingale Decomposition and Heat Kernel Estimates of Reflected Stable-Like Processes with Variable Order

We investigate symmetric reflected stable-like processes on a compact set $ \overline{E} \subset \mathbf{R}^d$ associated to nonlocal Dirichlet forms with variable order $\alpha{(\,\cdot\,,\cdot\,)...

Semi-martingale decomposition and heat kernel estimates of reflected stable-like processes with variable order

Мы изучаем отраженные симметричные процессы устойчивого типа (на компактном множестве $\overline{E}\subset\mathbf{R}^d$), ассоциированные с нелокальными формами Дирихле, в случае переменного порядка $\alpha{( \cdot ,\cdot )}$ в ядре интенсивности скачков. Сначала, предполагая выполненными двусторонние оценки непрерывной переходной плотности отраженного процесса устойчивого типа $(X_t)_{t \ge 0}$, мы получаем, аналогично [10], семимартингальное разложение процесса $(X_t)_{t \ge 0}$. Затем, при некоторых дополнительных условиях на $\alpha{( \cdot ,\cdot )}$, мы выводим в явном виде верхние и нижние оценки для непрерывной по Гeльдеру переходной плотности процесса $(X_t)_{t \ge 0}$.

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

In this paper we study the domain of the generator 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 (Xt)t≥0,limt→0⁡1t(Exf(Xt)−f(x)),x∈Rd, for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed x∈Rd 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 (Xt)t≥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.

Regularity and Sensitivity for McKean-Vlasov Type SPDEs Generated by Stable-like Processes

In this paper we study the sensitivity of nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes. By using the method of stochastic characteristics, we transfer these equations to the non-stochastic equations with random coefficients thus making it possible to use the results obtained for nonlinear PDE of McKean-Vlasov type generated by stable-like processes in the previous works. The motivation for studying sensitivity of nonlinear McKean-Vlasov SPDEs arises naturally from the analysis of the mean-field games with common noise.

Multifractality of jump diffusion processes

We study the local regularity and multifractal nature of the sample paths of jump diffusion processes, which are solutions to a class of stochastic differential equations with jumps. This article extends the recent work of Barral {\it et al.} who constructed a pure jump monotone Markov process with random multifractal spectrum. The class of processes studied here is much larger and exhibits novel features on the extreme values of the spectrum. This class includes Bass' stable-like processes and non-degenerate stable-driven SDEs.

On martingale problems and Feller processes

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty }(\mathbb{R} ^d))$-martingale problem implies that the unique solution to the martingale problem is a Feller process. This provides a proof of a former claim by van Casteren. As an application we prove new existence and uniqueness results for Lévy-driven stochastic differential equations and stable-like processes with unbounded coefficients.

Laws of the iterated logarithm for symmetric jump processes

Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs) for sample paths, local times and ranges are established. In particular, the LILs are obtained for $\beta$-stable-like processes on $\alpha$-sets with $\beta>0$.

Hausdorff Dimension of the Range and the Graph of Stable-Like Processes

We determine the Hausdorff dimension for the range of a class of pure jump Markov processes in $\mathbb{R}^d$, which turns out to be random and depends on the trajectories of these processes. The key argument is carried out through the SDE representation of these processes. The method developed here also allows to compute the Hausdorff dimension for the graph.

Bottom crossing probability for symmetric jump processes

We determine the decay rate of the bottom crossing probability for symmetric jump processes under the condition on heat kernel estimates. Our results are applicable to symmetric stable-like processes and stable-subordinated diffusion processes on a class of (unbounded) fractals and fractal-like spaces.

Martin Kernels for Markov Processes with Jumps

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in Rd with positive continuous density of the Lévy measure; stable-like processes in Rd and in domains; and stable-like subordinate diffusions in metric measure spaces.

Convergence of measures penalized by generalized Feynman-Kac transforms

We prove the existence of limiting laws for symmetric stable-like processes penalized by generalized Feynman-Kac functionals and characterize them by the gauge functions and the ground states of Schrodinger type operators.