Non-local Dirichlet forms and symmetric jump processes

• Published in: Transactions of the American Mathematical Society Vol. 361; no. 4; pp. 1963 - 1999
• Main Authors: Barlow, Martin T., Bass, Richard F., Chen, Zhen-Qing, Kassmann, Moritz
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-04-2009
• Subjects: Continuous functions; Exact sciences and technology; Harmonic functions; Inference from stochastic processes; time series analysis; Lebesgue measures; Markov processes; Martingales; Mathematical analysis; Mathematical functions; Mathematical inequalities; Mathematics; Measure and integration; Partial differential equations; Perceptron convergence procedure; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Semigroups; Statistics; Harnack inequality; parabolic; symmetric processes; Jump processes; heat kernel; integro-differential operators; Dirichlet forms; harmonic; Markov process; Heat equation; Closure; Position; Heat kernel; Dirichlet process; Statistical estimation; Continuous function; Direction; Harmonic function; Kernel method; Kernels; Dirichlet form; Jump process
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-08-04544-3

We consider the non-local symmetric Dirichlet form (\mathcal {E}, \mathcal {F}) given by \[\mathcal {E} (f,f)=\int \limits _{\mathbb {R}^d} \int \limits _{\mathbb {R}^d} (f(y)-f(x))^2 J(x,y) dx dy ] with \mathcal {F} the closure with respect to \mathcal {E}_1 of the set of C^1 functions on \mathbb {R}^d with compact support, where \mathcal {E}_1 (f, f):=\mathcal {E} (f, f)+\int _{\mathbb {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 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 (\mathcal {E}, \mathcal {F}). We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to \mathcal {E}. Finally we construct an example where the corresponding harmonic functions need not be continuous.