Self-intersection local time of (α,d,β)(α,d,β)-superprocess☆

Renormalized Self - Intersection Local Time for Fractional Brownian Motion

Let B H t be a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). Assume d ≥ 2. We prove that the renor-malized self-intersection local time ℓ = T 0 t 0 δ(B H t − B H s) ds dt − E T 0 t 0 δ(B H t − B H s) ds dt exists in L 2 if and only if H < 3/(2d), which generalizes the Varadhan renormalization theorem to any dimension and with any Hurst parameter. Motivated by a resul...

Derivatives of self-intersection local times

We show that the renormalized self-intersection local time γt(x) for both the Brownian motion and symmetric stable process in R is differentiable in the spatial variable and that γ′ t(0) can be characterized as the continuous process of zero quadratic variation in the decomposition of a natural Dirichlet process. This Dirichlet process is the potential of a random Schwartz distribution. Analogo...

Self-intersection Local Time of the Fluctuation Limit of a Particle System with Mutation

Continuous local time of a purely atomic immigration superprocess with dependent spatial motion

A purely atomic immigration superprocess with dependent spatial motion in the space of tempered measures is constructed as the unique strong solution of a stochastic integral equation driven by Poisson processes based on the excursion law of a Feller branching diffusion, which generalizes the work of Dawson and Li [3]. As an application of the stochastic equation, it is proved that the superpro...

Regularity of Renormalized Self-intersection Local Time for Fractional Brownian Motion

Let B H t be a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). We study the regularity, in the sense of the Malliavin calculus, of the renormalized self-intersection local time ℓ = T 0 t 0 δ 0 (B H t − B H s)dsdt − E T 0 t 0 δ 0 (B H t − B H s)dsdt , where δ 0 is the Dirac delta function.