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Ouyang, Cheng; Roberson-Vickery, William

Quasi-sure non-self-intersection for rough differential equations driven by fractional Brownian motion. (English) Zbl 1492.60302

Electron. Commun. Probab. 27, Paper No. 15, 12 p. (2022).
Summary: In this paper we study the self-intersection of paths solving elliptic stochastic differential equations driven by fractional Brownian motion. We show that such a path has no self-intersection – except for paths forming a set of zero \((r,q)\)-capacity in the sample space – provided the dimension \(d\) of the space and the Hurst parameter \(H\) satisfy the inequality \(d> rq+2/ H\). This inequality is sharp in the case of brownian motion and fractional brownian motion according to existing results. Various results exist for the critical case where \(d=rq+4\) for Brownian motion.

Cited in 1 Document

MSC:

60L20 Rough paths
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus

Keywords:

rough differential equations; stochastic differential equations; fractional Brownian motion; Malliavin calculus; capacity
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