Rough Paths above Weierstrass Functions

Cellarosi, Francesco; Selk, Zachary

Comptes Rendus. Mathématique, Tome 362 (2024) no. G12, pp. 1555-1570

Rough paths theory allows for a pathwise theory of solutions to differential equations driven by highly irregular signals. The fundamental observation of rough paths theory is that if one can define “iterated integrals” above a signal, then one can construct solutions to differential equations driven by the signal.

The typical examples of the signals of interest are stochastic processes such as (fractional) Brownian motion. However, rough paths theory is not inherently random and therefore can treat irregular deterministic driving signals such as a (vector-valued) Weierstrass function. This note supplies a construction of a rough path above vector-valued Weierstrass functions.

La théorie des chemins rugueux fournit une théorie trajectorielle des solutions aux équations différentielles gouvernées par des signaux très irréguliers. L’observation fondamentale de la théorie des chemins rugueux est que si l’on peut définir des “intégrales itérées” au-dessus d’un signal donné, alors on peut construire des solutions aux équations différentielles gouvernées par ce signal. Les exemples typiques de signaux d’intérêt sont des processus stochastiques tels le mouvement brownien (fractionnaire). Ceci dit, la théorie des chemins rugueux n’est pas intrinsèquement probabiliste et peut donc traiter des signaux déterministes irréguliers tels les fonctions de Weierstrass (à valeurs vectorielles). Cette note fournit une construction d’un chemin rugueux au-dessus de fonctions de Weierstrass (à valeurs vectorielles).

Classification : 60L20
Keywords: Rough paths, Weierstrass functions
Mots-clés : Chemins rugueux, fonctions de Weierstrass

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