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Processes with prescribed local regularity. (Processus à régularité locale prescrite.) (French. Abridged English version) Zbl 0988.60028
Let \(X=\{X(t):t\in [0,1]\}\) be a continuous and nowhere differentiable stochastic process. The Hölder process \(\alpha_X\) of \(X\) is defined by \[ \alpha_X(t):=\sup\Bigl\{\alpha:\limsup_{h\to 0} |X(t+h)-X(t)|/|h|^{\alpha}=0\Bigr\} \] for each \(t\in [0,1]\). The authors construct a continuous random process \(W\) extending the Weierstrass function and whose Hölder process may be, with probability \(1\), any lower limit of continuous functions with values in \([0,1]\). The construction allows to build stochastic processes with an arbitrary singularities spectrum. These processes are not obtained via a multiplicative cascade and yet they can be multifractal.

MSC:
60G17 Sample path properties
26A16 Lipschitz (Hölder) classes
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