Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

In this paper we survey some results on the Dirichlet problem ( Lu = f in u = g in R n n for nonlocal operators of the form Lu(x) = PV Z Rn u(x) u(x + y) K(y)dy: We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators L in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the eld. 2010 Mathematics Subject Classication: 47G20, 60G52, 35B65.

http://mat.uab.cat/pubmat/fitxers/download/FileType:pdf/FolderName:v60(1)/FileName:60116_01.pdf

Nonlocal elliptic equations in bounded domains: a survey

distribution, queuing theory, random walks, and so on. On many topical issues he is prepared to admit that there are no definitive answers, considering, inter alia, the following questions: how convinced are we that the trends in climate change over the past thirty years are an indication of global warming rather than just random fluctuations? how much belief can there be in miracles? is the movement of share prices better explained by chaos theory than by statistics? He also emphasizes that issues such as psychology and economic efficiency sometimes have as much of a bearing on eventual decisions as purely statistical considerations.

Foundations of modern probability (2nd edn), by Olav Kallenberg. Pp. 638. £49 (hbk). 2002. ISBN 0 387 95313 2 (Springer-Verlag).

Density and tails of unimodal convolution semigroups

Regularity theory for general stable operators

In this paper, we consider subordinate Brownian motion  $X$ in $\mathbb{R}^d$, $d \ge 1$,  where the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions. The scale invariant Harnack inequality  is proved for $X$.   We first give new forms of asymptotical properties of the Levy and potential density of the subordinator near zero. Using these results we find asymptotics of the Levy density and potential density of $X$ near the origin, which is essential to our approach. The examples which are covered byour results include geometric stable processes and relativistic geometric stable processes, i.e. the cases when the subordinator has the Laplace exponent\[\phi(\lambda)=\log(1+\lambda^{\alpha/2})\  (0 0)\,.\]

/pdf/harnack-inequalities-for-subordinate-brownian-motions-4i08qt6sil.pdf

Harnack inequalities for subordinate Brownian motions

We study integrodifferential operators and regularity estimates for solutions to integrodifferential equations. Our emphasis is on kernels with a critically low singularity which does not allow for standard scaling. For example, we treat operators that have a logarithmic order of differentiability. For corresponding equations we prove a growth lemma and derive a priori estimates. We derive these estimates by classical methods developed for partial differential operators. Since the integrodifferential operators under consideration generate Markov jump processes, we are able to offer an alternative approach using probabilistic techniques.

Intrinsic scaling properties for nonlocal operators

A subordinate Brownian motion $X$ is a L\'evy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $\phi$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates of the Green functions of these subordinate Brownian motions in any bounded $C^{; ; ; 1, 1}; ; ; $ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{; ; ; 1, 1}; ; ; $ open set with explicit decay rate. Unlike \cite{; ; ; KSV2, KSV4}; ; ; , our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent \[ \phi(\lambda)=\log(1+\lambda^{; ; ; \alpha/2}; ; ; )\ \ \ \ (0 \alpha)\] and \[ \phi(\lambda)=\log(1+(\lambda+m^{; ; ; \alpha/2}; ; ; )^{; ; ; 2/\alpha}; ; ; -m)\ \ \ \ (0 0, d >2)\, . \]

/pdf/green-function-estimates-for-subordinate-brownian-motions-2vnbdmug7c.pdf

Green function estimates for subordinate Brownian motions : stable and beyond

In this article we study transition probabilities of a class of subordinate Brownian motions. Under mild assumptions on the Laplace exponent of the corresponding subordinator, sharp two sided estimates of the transition probability are established. This approach, in particular, covers subordinators with Laplace exponents that vary regularly at infinity with index one, e.g. φ(λ) = λ log(1 + λ) − 1 or φ(λ) = λ log(1 + λβ/2) , β ∈ (0, 2) that correspond to subordinate Brownian motions with scaling order that is not necessarily stricty between 0 and 2. These estimates are applied to estimate Green function (potential) of subordinate Brownian motion. We also prove the equivalence of the lower scaling condition of the Laplace exponent and the near diagonal upper estimate of the transition estimate.

/pdf/heat-kernel-estimates-for-subordinate-brownian-motions-4bb1ff5tu9.pdf

Heat kernel estimates for subordinate Brownian motions

We consider the following non-local operator 

$$ \mathcal{A}f(x)=\lim\limits_{\varepsilon\to 0}\int_{\{y\in \mathbb{R}^d\colon |x-y|>\varepsilon\}}(f(y)-f(x))n(x,y)\,dh. $$

where 

$$ n(x,y)\asymp \frac{1}{|x-y|^{d+2}\left(\ln\frac{2}{|x-y|}\right)^{1+\beta}}\ \textrm{ for }\ |x-y|\leq 1 $$

and β ∈ (0, 1]. We prove upper estimates for the transition density of the associated symmetric Markov jump process X. Examples of Levy processes with generator of the type above are studied.

/pdf/heat-kernel-upper-estimates-for-symmetric-jump-processes-27muwomc6c.pdf

Ante Mimica

Papers

Harnack inequalities for subordinate Brownian motions

Intrinsic scaling properties for nonlocal operators

Green function estimates for subordinate Brownian motions : stable and beyond

Heat kernel estimates for subordinate Brownian motions

Heat Kernel Upper Estimates for Symmetric Jump Processes with Small Jumps of High Intensity