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A probabilistic interpretation is constructed for the symmetry group G of the finite difference-differential equation ðtη(x, t) = η(x, t) − η(x + 1, t) using the Doob transform for Markov (jump) processes. While the first three generators of G correspond to the identity and to space and time shifts, we show that in this interpretation the fourth generator of G is associated to time dilations and is linked to a creation operator on the Poisson space. [ABSTRACT FROM AUTHOR]