Chaotic expansion and smoothness of some functionals of the fractional Brownian motion
Let the \(N\)-parameter fractional Brownian motion be defined as \[ B^H_t= \int_{[0,1]^n} \prod^N_{j=1} K_{H_j}(t_j, s_j)\,dW_s,\quad \in ]0,1[^n. \] This paper concerns the local time associated to \(T^H\) (i.e. \(l(t, x)= \int^t_0\delta_x(B^H_s)\,ds)\). Using the expansion in Wiener chaos of fractional derivative and Sobolev spaces, the main result is that for any \(\gamma\in ]0\), \({1\over 2}\sup_j H_j-{1\over 2}[\), \(D\) being some singular operators (fractional derivatives), then \(Dl(t, x)=\sum_{n\geq 0} I_n(Df_n(.,t,x))\), where \(f_n\) is defined on \((\mathbb{R}^+)^{n+1}\times\mathbb{R}\) as an integral. Some applications are given: the expansion in Wiener chaos of additive functionals (operators applied to the local time), for instance \[ A^{-\beta}_H:(t, x)\mapsto \int_{\mathbb{R}}(y- x)^{\beta-1}_+ l(t,y)\,dy,\qquad 0< \beta< 1, \] is developed as \(\Gamma(\beta)\sum_{n\geq 0} I_n(I^\beta(f_n(.,t,.)(x))\).
scientific article; zbMATH DE number 2124041