  We introduce and study a new model consisting of a single classical random
walker undergoing continuous monitoring at rate $\gamma$ on a discrete lattice.
Although such a continuous measurement cannot affect physical observables, it
has a non-trivial effect on the probability distribution of the random walker.
At small $\gamma$, we show analytically that the time-evolution of the latter
can be mapped to the Stochastic Heat Equation (SHE). In this limit, the width
of the log probability thus follows a Family-Vicsek scaling law,
$N^{\alpha}f(t/N^{\alpha/\beta})$, with roughness and growth exponents
corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e
$\alpha^{\rm{1D}}_{\rm{KPZ}}=1/2$ and $\beta^{\rm{1D}}_{\rm{KPZ}}=1/3$
respectively. When $\gamma$ is increased outside this regime, we find
numerically in 1D a crossover from the KPZ class to a new universality class
characterized by exponents $\alpha^{1\rm{D}}_{\text{M}}\approx 1$ and
$\beta^{1\rm{D}}_{\text{M}}\approx 1.4$. In 3D, varying $\gamma$ beyond a
critical value $\gamma^c_{\rm{M}}$ leads to a phase transition from a smooth
phase that we identify as the Edwards-Wilkinson (EW) class to a new
universality class with $\alpha^{3\rm{D}}_{\text{M}}\approx1$.
