Renormalization constant for the local times of self-intersections of a diffusion process in the plane

We study the local times of self-intersection of a diffusion process in the plane. Our main result is connected with the investigation of the asymptotic behavior of the renormalization constant of this local time. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1489–1498, November, 2008.     

Asymptotics of Intersection Local Time for Diffusion Processes

In the paper, we investigate the intersection local time for two correlated Brownian motions on the plane that form a diffusion process in ?⁴ associated with a divergence-form generator. Using Gaussian heat kernel bounds, we prove the existence of intersection local time for these Brownian motions, obtain estimates of its moments, and establish the law of iterated logarithm for it.

     

Hilbert-valued self-intersection local times for planar Brownian motion

Dynkin's construction for self-intersection local time of a planar Wiener process is extended to Hilbert-valued weights. In Dynkin's construction, the weight is bounded and measurable. Since the weight function describes the properties of the medium in which the Brownian motion moves, relative to the external medium's properties, the weight function can be random and unbounded. In this article, we discuss the possibility to consider Hilbert-space-valued weights. It appears that the existence of Hilbert-valued Dynkin-renormalized self-intersection local time is equivalent to the embedding of the values of Hilbert-valued weight into a Hilbert–Schmidt brick. Using Dorogovtsev's sufficient condition for the embedding of compact sets into a Hilbert–Schmidt brick in terms of an isonormal process, we prove the existence of Hilbert-valued Dynkin-renormalized self-intersection local time. Also using Dynkin's construction we construct the self-intersection local time for the deterministic image of the planar Wiener process.     

Clark Representation for Local Times of Self-Intersection of Gaussian Integrators

Ukrainian Mathematical Journal - We prove the existence of multiple local times of self-intersection for a class of Gaussian integrators generated by operators with finite-dimensional kernels,,...