Large deviations and renormalization for Riesz potentials of stable intersection measures

• Published in: Stochastic processes and their applications Vol. 120; no. 9; pp. 1837 - 1878
• Main Authors: Chen, Xia, Rosen, Jay
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.08.2010; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Exact sciences and technology; Large deviations; Large deviations Renormalization Riesz potentials Stable intersection measure; Limit theorems; Markov processes; Mathematics; Probability and statistics; Probability theory and stochastic processes; Renormalization; Riesz potentials; Sciences and techniques of general use; Stable intersection measure; Stochastic analysis; Stochastic processes; Stable intersection measure; Renormalization; Large deviations; Primary; Riesz potentials; 60G51; Law of iterated logarithm; Stable process; Stochastic process; Primary 60F10; Local time; Large deviation
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2010.05.006

We study the object formally defined as (0.1) γ ( [ 0 , t ] 2 ) = ∬ [ 0 , t ] 2 | X s − X r | − σ d r d s − E ∬ [ 0 , t ] 2 | X s − X r | − σ d r d s , where X t denotes the symmetric stable processes of index 0 < β ≤ 2 in R d . When β ≤ σ < min { 3 2 β , d } , this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for γ . This is applied to obtain results about stable processes in random potentials.