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When is it possible to interpret a given Markov process as a L\'evy-likeprocess? Since the class of L\'evy processes can be defined by the relationbetween transition probabilities and convolutions, the answer to this questionlies in the existence of a convolution-like operator satisfying the samerelation with the transition probabilities of the process. It is known that theso-called Sturm-Liouville convolutions have the desired properties andtherefore the question above has a positive answer for a certain class ofone-dimensional diffusions. However, more general processes have never beensystematically treated in the literature. This study addresses this gap byconsidering the general problem of constructing a convolution-like operator fora given strong Feller process on a general locally compact metric space. Bothnecessary and sufficient conditions for the existence of such convolution-likestructures are determined, which reveal a connection between the answer to theabove question and certain analytical and geometrical properties of theeigenfunctions of the transition semigroup. The case of reflected Brownianmotions on bounded domains of R d and compact Riemannian manifolds isconsidered in greater detail: various special cases are analysed, and a generaldiscussion on the existence of appropriate convolution-like structures ispresented.

On the construction of convolution-like operators associated with multidimensional diffusion processes