Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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

We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump Levy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C 1,� regularity for general fully nonlinear integro- differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.

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Regularity theory for fully nonlinear integro‐differential equations

A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in $\mbRn$. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficie...

Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

We consider the non-local symmetric Dirichlet form (E,F) given by with F the closure with respect to E 1 of the set of C 1 functions on R d with compact support, where E 1 (f, f):= E(f, f) + f Rd f(x) 2 dx, and where the jump kernel J satisfies for 0 < α < β < 2, |x - y| < 1. This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E,F). We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.

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Non-local dirichlet forms and symmetric jump processes

The aim of this work is to develop a localization technique and to establish a regularity result for non-local integro-differential operators ${\fancyscript{L}}$ of order ${\alpha\in (0,2)}$ . Thereby we extend the De Giorgi–Nash–Moser theory to non-local integro-differential operators. The operators ${\fancyscript{L}}$ under consideration generate strong Markov processes via the theory of Dirichlet forms. As is well known, regularity properties of the resolvents are important for many aspects of the corresponding stochastic process. Therefore, this work is related to probability theory and analysis, especially partial differential equations, at the same time.

A priori estimates for integro-differential operators with measurable kernels

In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the “boundary data” are prescribed on the complement of a given bounded set. We formulate the problem in the classical framework of Hilbert spaces and prove unique solvability using standard techniques like the Fredholm alternative.

The Dirichlet problem for nonlocal operators

We consider the symmetric non-local Dirichlet form $(E, F)$ given by \[ E (f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$ the closure of the set of $C^1$ functions on $R^d$ with compact support with respect to $E_1$, where $E_1 (f, f):=E (f, f)+\int_{R^d} f(x)^2 dx$, and where the jump kernel $J$ satisfies \[ \kappa_1|y-x|^{-d-\alpha} \leq J(x,y) \leq \kappa_2|y-x|^{-d-\beta} \] for $0<\alpha< \beta <2, |x-y|<1$. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to $(E, F)$. We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to $E$. Finally we construct an example where the corresponding harmonic functions need not be continuous.

Non-local Dirichlet Forms and Symmetric Jump Processes

We consider harmonic functions with respect to the operator formula math Under suitable conditions on n(x,h) we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator L is allowed to be anisotropic and of variable order.

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Moritz Kassmann

Papers

Non-local dirichlet forms and symmetric jump processes

A priori estimates for integro-differential operators with measurable kernels

The Dirichlet problem for nonlocal operators

Non-local Dirichlet Forms and Symmetric Jump Processes

Harnack inequalities for non-local operators of variable order