The theory of Loewner chains provides a one-to-one correspondence between real valued continuous curves (called the Driver) and certain increasing family of compact sets in upper half plane. In this correspondence, one dimensional Brownian motion gives rise to Schramm-Loewner Evolution (SLE). One obstacle in this construction is that one has to additionally prove that the corresponding family of compact sets is given by a curve, called the Trace. We will address this existence of trace problem both in random and deterministic frameworks. We will also address the continuity of the map which maps the driver to the trace and present some new results in this direction.
