Properties of trajectories of a multifractional Rosenblatt process (Q2890730)

The multifractional Rosenblatt process with Hurst function \(H\) is defined by NEWLINE\[NEWLINE X_t = \iint_{\mathbb{R}^2} \int_0^t (s-x)_+^{H(t)/2-1} (s-y)_+^{H(t)/2-1} ds \, W(dx) W(dy), \quad 0 \leq t \leq T, NEWLINE\]NEWLINE where \(\iint f(x,y) W(dx) W(dy)\) denotes double Wiener integral. It is assumed that \(H: [0, T] \to (\frac12, 1)\) is a Hölder continuous function with exponent \(\gamma > \max_{0 \leq t \leq T} H(t)\). This process \((X_t)\) has long range dependence, with non-Gaussian, ``moderate'' tail distribution. (A ``moderate'' tail distribution is heavier than normal, but lighter than other stable distributions.) The Hurst function is useful to model local properties that vary with time.NEWLINENEWLINEThe first main result of the paper is that \((X_t)\) is a.s. continuous, in fact, Hölder continuous with exponent \(\gamma_X < \min_{a \leq t \leq b}H(t)\) on any interval \([a,b] \subset [0,T]\). The second main result is that \((X_t)\) is strongly localizable at every point \(t_0 \in [0,T]\), that is, the distribution of the process \(\left\{\delta^{-H(t_0)} \left(X_{t_0+\delta t} - X_{t_0}\right), t \geq 0 \right\}\) converges to the distribution of a Rosenblatt process with constant Hurst parameter \(H(t_0)\) in \(C[0,T]\) as \(\delta \to 0^+\). The third main result is that \((X_t)\) has a square integrable local time.