Potential kernels,probabilities of hitting a ball,harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process.In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space.     

Wiener’s criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries

This paper establishes a necessary and sufficient condition for the existence of a unique bounded solution to the classical Dirichlet problem in arbitrary open subset of R^N${\mathbb{R}}^N$ (N≥3$N\ge 3$) with a non-compact boundary. The criterion is the exact analogue of Wiener’s test for the boundary regularity of harmonic functions and characterizes the “thinness” of a complementary set at infinity. The Kelvin transformation counterpart of the result reveals that the classical Wiener criterion for the boundary point is a necessary and sufficient condition for the unique solvability of the Dirichlet problem in a bounded open set within the class of harmonic functions having a “fundamental solution” kind of singularity at the fixed boundary point. Another important outcome is that the classical Wiener’s test at the boundary point presents a necessary and sufficient condition for the “fundamental solution” kinds of singularities of the solution to the Dirichlet problem to be removable.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Potential theory of subordinate Brownian motions with Gaussian components

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C^1,1$C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C^1,1$C^{1,1}$ open set D$D$ and identify the Martin boundary of D$D$ with respect to the subordinate Brownian motion with the Euclidean boundary.     

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X$X$ in a bounded κ$\kappa$-fat open set; if u$u$ is a positive harmonic function with respect to X$X$ in a bounded κ$\kappa$-fat open set D$D$ and h$h$ is a positive harmonic function in D$D$ vanishing on D^c$D^c$, then the non-tangential limit of u/h$u/h$ exists almost everywhere with respect to the Martin-representing measure of h$h$.     

Hyperbolic Harmonic Functions: Weak Approach with Applications in Function Spaces

Harmonic functions with respect to the Poincare metric on the unit ball are called hyperbolic harmonic functions. We establish the weak formulation of hyperbolic harmonic functions and use it in the study of hyperbolic harmonic function spaces. In particular, we give the Carleson measure characterization for the whole spectrum of spaces, whose analytic counterparts include among else Bloch spaces, Bergman-spaces, Besov-spaces, and Q_p-spaces. The second author was supported by the Finnish Cultural Foundation.     

Construction of Markov processes and associated multiplicative functionals from given harmonic measures

Summary LetE be a noncompact locally compact second countable Hausdorff space. We consider the question when, given a family of finite nonzero measures onE that behave like harmonic measures associated with all relatively compact open sets inE (i.e. that satisfy a certain consistency condition), one can construct a Markov process onE and a multiplicative functional with values in [0, ) such that the hitting distributions of the process inflated by the multiplicative functional yield the given harmonic measures. We achieve this construction under weak continuity and local transience conditions on these measures that are natural in the theory of Markov processes, and a mild growth restriction on them. In particular, if the spaceE equipped with the measures satisfies the conditions of a harmonic space, such a Markov process and associated multiplicative functional exist. The result extends in a new direction the work of many authors, in probability and in axiomatic potential theory, on constructing Markov processes from given hitting distributions (i.e. from harmonic measures that have total mass no more than 1).     

Symmetric Skorohod topology onn-variable functions and hierarchical Markov properties ofn-parameter processes

Summary We introduce a new Skorohod topology for functions of several variables. Since ann-variable function may be viewed as a one-variable function with values in the set of (n–1)-variable functions, this topology is defined by induction from the classical Skorohod topology for one-variable functions. This allows us to define the notion of completen-parameter symmetric Markov processes: Such processes are, for any 1pn, rawp-parameter Markov processes (in the sense of our previous paper [17]) with values in the space of (n–p)-variable functions. We prove, for these processes and their Bochner subordinates, a maximal inequality which implies the continuity of additive functionals associated with finite energy measures. We finally present several important examples.     

Conditions for the existence of quasi-stationary distributions for birth–death processes with killing

We consider birth–death processes on the nonnegative integers, where {1,2,…}$\{1,2,\dots \}$ is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 (killing) may occur from any state. Assuming that absorption at 0 is certain we are interested in additional conditions on the transition rates for the existence of a quasi-stationary distribution. Inspired by results of Kolb and Steinsaltz [M. Kolb, D. Steinsaltz, Quasilimiting behavior for one-dimensional diffusions with killing, Ann. Probab. 40 (2012) 162–212] we show that a quasi-stationary distribution exists if the decay rate of the process is positive and exceeds at most finitely many killing rates. If the decay rate is positive and smaller than at most finitely many killing rates then a quasi-stationary distribution exists if and only if the process one obtains by setting all killing rates equal to zero is recurrent.     

Plane brownian motion in a region with holes

We consider a fixed family of balls with decreasing radii in the plane. We establish a relationship between a Dirichlet problem in a region without the balls and the solution of a Schroedinger equation in the complete region. Then we find upper bounds for the probability that a brownian motion exits the region without touching these balls. This is used to study harmonic measure and entire functions.     

Hölder estimates of p-harmonic extension operators

It is now a well-known fact that for 1<p<∞ the p-harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a (1,p)-Poincaré inequality are locally Hölder continuous. In this note we provide a characterization of domains in such metric spaces for which p-harmonic extensions of Hölder continuous boundary data are globally Hölder continuous. We also provide a link between this regularity property of the domain and the uniform p-fatness of the complement of the domain.     

Liouville theorems for non-local operators

The paper characterizes some classes of pseudo-differential operators for which there are (or there are not) non-constant bounded harmonic functions. Non-local perturbations of Ornstein-Uhlenbeck operators and operators with dissipative coefficients are considered. The methods used are probabilistic and based on the concept of absorption function and on a new extension of the Bismut-Elworthy-Li formula. The probabilistic interpretation of the Liouville theorem by means of absorption functions for general Markov processes is given as well.     

Gradient estimates for positive harmonic functions by stochastic analysis

We prove Cheng–Yau type inequalities for positive harmonic functions on Riemannian manifolds by using methods of Stochastic Analysis. Rather than evaluating an exact Bismut formula for the differential of a harmonic function, our method relies on a Bismut type inequality which is derived by an elementary integration by parts argument from an underlying submartingale. It is the monotonicity inherited in this submartingale which allows us to establish the pointwise estimates.     

Enumerative meaning of mirror maps for toric Calabi–Yau manifolds

We prove that the inverse of a mirror map for a toric Calabi–Yau manifold of the form _KY$K_Y$, where Y$Y$ is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov–Witten invariants defined by Fukaya–Oh–Ohta–Ono (2010)  [15]. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert (2011)  [24, Conjecture 0.2 and Remark 5.1] as part of their program, and was later formulated in terms of Fukaya–Oh–Ohta–Ono’s invariants in the toric Calabi–Yau case in Chan et al. (2012)  [8, Conjecture 1.1].     

Multifractal spectra and precise rates of decay in homogeneous fragmentations

We consider a mass-conservative fragmentation of the unit interval. Motivated by a result of Berestycki [J. Berestycki, Multifractal spectra of fragmentation processes, J. Statist. Phys. 113 (3–4) (2003) 411–430], the main purpose of this work is to specify the Hausdorff dimension of the set of locations having exactly an exponential decay. The study relies on an additive martingale which arises naturally in this setting, and a class of Lévy processes constrained to stay in a finite interval.     

Nonlinear potential theory and PDEs

We consider equations like -div(|u| ^p–2u)=, where is a nonnegative Radon measure and 1u and the measure are reviewed. A link between potential estimates and the boundary regularity of the Dirichlet problem is established.     

FBSDEs with time delayed generators: L-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6] and [7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general L^p-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in L^p. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L²-path regularity to delay FBSDEs.     

Green function estimates for relativistic stable processes in half-space-like open sets

In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m−(m^2/α−Δ)^α/2) in half-space-like C^1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M∈(0,∞). When m↓0, our estimates reduce to the sharp Green function estimates for −(−Δ)^α/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for X^m, which is uniform for all m∈(0,∞), holds for a large class of non-smooth open sets.     

On a Poincaré-type Inequality for Energy Forms in L p

We consider Dirichlet spaces ( ) in L ² and more general energy forms in L ^p , . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that , resp. , are compactly embedded in L ², resp. L ^p , we prove a Poincaré inequality for transient (Dirichlet) forms. If both and its adjoint are sub-Markovian semigroups, we show that the transience of T _t is independent of ) and that it is implied by the transience of the energy form of and the form belonging to .     

Space and time inversions of stochastic processes and Kelvin transform

Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.