  Various gedanken experiments of quantum gravity phenomenology in search of a
complete theory of gravity near the Planck scale indicate a modification of the
Heisenberg uncertainty principle to the generalized uncertainty principle
(GUP). This modification leads to nontrivial contributions on the Hamiltonian
of a nonrelativistic particle moving in an arbitrary potential. In this paper
we study the path integral representation of a particle moving in an arbitrary
potential using the most general form of the GUP containing both the linear and
quadratic contributions in momentum. First we work out the action of the
particle in an arbitrary potential and hence find an upper bound to the
velocity of a free particle. This upper bound interestingly imposes
restrictions on the relation between the GUP parameters $\alpha$ and $\beta$.
Analysis shows that $ \beta > 4 \alpha^2$. We then deduce the mathematical
expressions of classical action and the quantum fluctuations for both free
particle and the harmonic oscillator systems.
