  Positive self-similar Markov processes (pssMp) are positive Markov processes
that satisfy the scaling property and it is known that they can be represented
as the exponential of a time-changed L\'evy process via Lamperti
representation. In this work, we are interested in the following problem: what
happens if we consider Markov processes in dimension $1$ or $2$ that satisfy
self-similarity properties of a more general form than a scaling property ? Can
they all be represented as a function of a time-changed L\'evy process ? If
not, how can Lamperti representation be generalized ? We show that, not
surprisingly, a Markovian process in dimension $1$ that satisfies
self-similarity properties of a general form can indeed be represented as a
function of a time-changed L\'evy process, which shows some kind of
universality for the classical Lamperti representation in dimension $1$.
However, and this is our main result, we show that a Markovian process in
dimension $2$ that satisfies self-similarity properties of a general form is
represented as a function of a time-changed exponential functional of a
bivariate L\'evy process, and processes that can be represented as a function
of a time-changed L\'evy process form a strict subclass. This shows that the
classical Lamperti representation is not universal in dimension $2$. We briefly
discuss the complications that occur in higher dimensions. In dimension $2$ we
present an example, built from a self-similar fragmentation process, where our
representation in term of an exponential functional of a bivariate L\'evy
process appears naturally and has a nice interpretation in term of the
self-similar fragmentation process.
