Hölder conditions for the local times of multiscale fractional Brownian motion
For \(K\in \mathbb{N}\) an \(M_K\)-multiscale fractional Brownian motion is defined: it is the Gaussian process with stationary increments defined by \[ X_\rho(t)=\int_{\mathbb{R}} \frac{e^{it\lambda}-1}{\rho(\lambda)}\,dW(\lambda), \] where \(dW(\lambda)\) is the Brownian measure on \(L_2(\mathbb{R})\), and for \(i=0,\dots,K\) there exist \((w_i,\sigma_i,H_i)\in (\mathbb{R}_+,\mathbb{R}_+,]0,1[)\) such that \(\rho(x)=\sigma_i^{-1}| x| ^{H_i+1/2}\) for \(| x| \in D_i=[w_i,w_{i+1}[\) with \(w_0=0<w_1<\dots <w_K<w_{K+1}=\infty\). The law of the iterated logarithm for \(X_\rho\) is proved. Further, \(X_\rho\) admits a.s. jointly continuous local time \(L(t,x)\), and the estimates on the modulus of continuity of \(L(x,t)\) are found. Also, some corollaries from these two results are derived.
scientific article; zbMATH DE number 5080200