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.

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},
     pages = {219--240},
     publisher = {Institut Fourier},
     volume = {33},
     number = {2},
     year = {1983},
     doi = {10.5802/aif.925},
     zbl = {0498.60083},
     mrnumber = {85i:60069},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.925/}
}
TY  - JOUR
AU  - Oksendal, Bernt
AU  - Csink, L.
TI  - Stochastic harmonic morphisms: functions mapping the paths of one diffusion into the paths of another
JO  - Annales de l'Institut Fourier
PY  - 1983
DA  - 1983///
SP  - 219
EP  - 240
VL  - 33
IS  - 2
PB  - Institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.925/
UR  - https://zbmath.org/?q=an%3A0498.60083
UR  - https://www.ams.org/mathscinet-getitem?mr=85i:60069
UR  - https://doi.org/10.5802/aif.925
DO  - 10.5802/aif.925
LA  - en
ID  - AIF_1983__33_2_219_0
ER  - 
%0 Journal Article
%A Oksendal, Bernt
%A Csink, L.
%T Stochastic harmonic morphisms: functions mapping the paths of one diffusion into the paths of another
%J Annales de l'Institut Fourier
%D 1983
%P 219-240
%V 33
%N 2
%I Institut Fourier
%U https://doi.org/10.5802/aif.925
%R 10.5802/aif.925
%G en
%F 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/articles/10.5802/aif.925/

[1] A. Bernard, E. A. Campbell and A. M. Davie, Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier, 29-1 (1979), 207-228. | Numdam | MR | Zbl

[2] H. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, 1968. | Zbl

[3] H. M. Blumenthal, R. K. Getoor and H. P. Mckean Jr., Markov processes with identical hitting distributions, Illinois Journal of Math., 6 (1962), 402-420. | MR | Zbl

[4] H. M. Blumenthal, R. K. Getoor and H. P. Mckean Jr., A supplement to "Markov processes with identical hitting distributions", Illinois Journal of Math., 7 (1963), 540-542. | MR | Zbl

[5] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J., 25 (1965), 1-57. | MR | Zbl

[6] C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, 1972. | MR | Zbl

[7] R. W. Darling, Thesis, University of Warwick, 1982.

[8] B. Davis, Brownian motion and analytic functions, The Annals of Probability, 7 (1979), 913-932. | MR | Zbl

[9] E. B. Dynkin, Markov Processes I, Springer-Verlag, 1965. | Zbl

[10] E. B. Dynkin, Markov Processes II, Springer-Verlag, 1965. | Zbl

[11] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier, 28-2 (1978), 107-144. | EuDML | Numdam | MR | Zbl

[12] B. Fuglede, Harmonic morphisms, In Proc. Coll. on Complex Analysis, Springer Lecture Notes in Math., 747 (1979). | MR | Zbl

[13] T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19-2 (1979), 215-229. | MR | Zbl

[14] H. P. Mckean, Stochastic Integrals, Academic Press, 1969. | MR | Zbl

[15] P. A. Meyer, Géométrie stochastique sans larmes. Sem. de Probabilités XV, Springer Lecture Notes in Math., 850, Springer-Verlag, 1981. | Numdam | Zbl

[16] B. Øksendal and D. W. Stroock, A characterization of harmonic measure and Markov processes whose hitting distributions are preserved by rotations, translations and dilatations, Ann. Inst. Fourier, 32-4 (1982). | EuDML | Numdam | MR | Zbl

[17] D. Sibony, Allure à la frontière minimale d'une classe de transformations. Théorème de Doob généralisé, Ann. Inst. Fourier, 18 (1968), 91-120. | EuDML | Numdam | MR | Zbl

[18] D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, 1979. | MR | Zbl

Cited by Sources: