Renormalized self-intersection local time for sub-bifractional Brownian motion

Abstract

Let $S^{H,K}=\{S^{H,K}(t),t\geq 0\}$ be a $d-$dimensional sub-bifractional Brownian
motion with indices $H\in (0,1)$ and $K\in (0,1].$ Assuming $d\geq 2,$ as $HKd<1,$ we mainly prove that the renormalized self-intersection local time
$$
\int_0^t\int_0^s\delta(S^{H,K}(s)-S^{H,K}(r))drds-\textbf{E}\left(\int_0^t\int_0^s\delta(S^{H,K}(s)-S^{H,K}(r))drds\right)
$$
exists in $L^2,$ where $\delta(x)$ is the Dirac delta function for $x\in \textbf{R}^d$.