Stochastic harmonic morphisms: functions mapping the paths of one diffusion into the paths of another
Annales de l'Institut Fourier, Volume 33 (1983) no. 2, p. 219-240

We give several necessary and sufficient conditions that a function ϕ maps the paths of one diffusion into the paths of another. One of these conditions is that ϕ is a harmonic morphism between the associated harmonic spaces. Another condition constitutes an extension of a result of P. Lévy about conformal invariance of Brownian motion. The third condition implies that two diffusions with the same hitting distributions differ only by a chance of time scale. We also obtain a converse of the above theorem of Lévy.

Nous donnons plusieurs conditions nécessaires et suffisantes pour qu’une fonction ϕ transforme les trajectoires d’une diffusion dans les trajectoires d’une autre. Une de ces conditions est que ϕ est un morphisme harmonique entre les espaces harmoniques associés. Une autre condition constitue une extension d’un résultat de P. Lévy sur l’invariance conforme du mouvement brownien. De la troisième condition on déduit que deux diffusions avec la même distribution de sortie d’ensembles ouverts ne diffère que par un changement d’horloge. Nous obtenons aussi un renversement du théorème de Lévy ci-dessus.

@article{AIF_1983__33_2_219_0,
     author = {Oksendal, Bernt and Csink, L.},
     title = {Stochastic harmonic morphisms: functions mapping the paths of one diffusion into the paths of another},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {33},
     number = {2},
     year = {1983},
     pages = {219-240},
     doi = {10.5802/aif.925},
     zbl = {0498.60083},
     mrnumber = {85i:60069},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1983__33_2_219_0}
}
Oksendal, Bernt; Csink, L. Stochastic harmonic morphisms: functions mapping the paths of one diffusion into the paths of another. Annales de l'Institut Fourier, Volume 33 (1983) no. 2, pp. 219-240. doi : 10.5802/aif.925. http://www.numdam.org/item/AIF_1983__33_2_219_0/

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