LAUSR

In Borowiecka et al. (Bernoulli 21(4):2513–2551, 2015) the authors show that every generalized convolution can be used to define a Markov process, which can be treated as a Lévy process in the sense of this convolution. The Bessel process is the best known example here. In this paper we present new classes of regular generalized convolutions enlarging the class of such Markov processes. We give here a full characterization of such generalized convolutions $$\diamond $$⋄ for which $$\delta _x \diamond \delta _1$$δx⋄δ1, $$x \in [0,1]$$x∈[0,1], is a convex linear combination of $$n=3$$n=3 fixed measures and only the coefficients of the linear combination depend on x. For $$n=2$$n=2 it was shown in Jasiulis-Goldyn and Misiewicz (J Theor Probab 24(3):746–755, 2011) that such a convolution is unique (up to the scale and power parameters). We show also that characterizing such convolutions for $$n \geqslant 3$$n⩾3 is equivalent to solving the Levi-Civita functional equation in the class of continuous generalized characteristic functions.