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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
##### Keywords:
stochastic process; multifractal; Weierstrass function
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