Multiple points of fractional stable processes
We consider a \(d\)-dimensional stochastic process \(X(t)=(X_ 1(t),\ldots,X_ d(t))\), where the components \(X_ j(t)\) are independent and each \(X_ j(t)\) is a linear fractional stable process with parameter \((\alpha_ j,H_ j)\), \(0<\alpha_ j<2\) and \(0<H_ j<1\), \(H_ j\neq 1/\alpha_ j\). \(X(t)\) is non-Gaussian and non-Markovian; yet it is self- similar and has stationary increments. We prove that: if \(1\leq\alpha_ j<2\), \(1/\alpha_ j<H_ j<1\) and we have chosen and fixed a continuous version of \(X(t)\), and if \(\theta:={k-1\over k}\sum^ d_{j=1}H_ j<1\), then almost surely the path \(X(\cdot,w)\) has \(k\)-multiple points. Furthermore, almost surely \(\dim\{(t_ 1,\ldots,t_ k): X(t_ 1)=\ldots=X(t_ k)\}\geq k(1-\theta)\). Here, \(k\) is a given positive integer \(\geq 2\). The basic idea is to prove that \(X(t)\) has jointly continuous \(k\)-fold intersection local times which is uniformly Hölder continuous in the set variable. Along the proof, a certain local nondeterminism property of linear fractional stable processes is established.
scientific article; zbMATH DE number 567074