Fractional diffusion in Gaussian noisy environment

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic partial equations of the following form: $D_t^\alpha u(t, x)=\textit{B}u+u\cdot W^H$, where $D_t^\alpha$ is the fractional derivative of order $\alpha$ with respect to the time variable $t$, $\textit{B}$ is a second order elliptic operator with respect to the space variable $x\in\mathbb{R}^d$, and $W^H$ a fractional Gaussian noise of Hurst parameter $H=(H_1, \cdots, H_d)$. We obtain conditions satisfied by $\alpha$ and $H$ so that the square integrable solution $u$ exists uniquely

Stochastic integral representation of the L2L^{2} modulus of Brownian local time and a central limit theorem

The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in \cite{CLMR}, using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time