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We give several necessary and sufficient conditions that a function maps the paths of one diffusion into the paths of another. One of these conditions is that 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...
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