  We are interested in the kernel of one-dimensional diffusion equations with
continuous coefficients as evaluated by means of explicit discretization
schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both
semidiscrete triangulations with continuous time and explicit Euler schemes
with time step small enough for the method to be stable. We find sharp uniform
bounds for the convergence rate as a function of the degree of smoothness which
we conjecture. The bounds also apply to the time derivative of the kernel and
its first two space derivatives. Our proof is constructive and is based on a
new technique of path conditioning for Markov chains and a renormalization
group argument. Convergence rates depend on the degree of smoothness and
H\"older differentiability of the coefficients. We find that the fastest
convergence rate is of order $O(h^2)$ and is achieved if the coefficients have
a bounded second derivative. Otherwise, explicit schemes still converge for any
degree of H\"older differentiability except that the convergence rate is
slower. H\"older continuity itself is not strictly necessary and can be relaxed
by an hypothesis of uniform continuity.

Numerical Analysis;Probability