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We are interested in the kernel of one-dimensional diffusion equations withcontinuous coefficients as evaluated by means of explicit discretizationschemes of uniform step $h>0$ in the limit as $h\to0$. We consider bothsemidiscrete triangulations with continuous time and explicit Euler schemeswith time step small enough for the method to be stable. We find sharp uniformbounds for the convergence rate as a function of the degree of smoothness whichwe conjecture. The bounds also apply to the time derivative of the kernel andits first two space derivatives. Our proof is constructive and is based on anew technique of path conditioning for Markov chains and a renormalizationgroup argument. Convergence rates depend on the degree of smoothness andH\"older differentiability of the coefficients. We find that the fastestconvergence rate is of order $O(h^2)$ and is achieved if the coefficients havea bounded second derivative. Otherwise, explicit schemes still converge for anydegree of H\"older differentiability except that the convergence rate isslower. H\"older continuity itself is not strictly necessary and can be relaxedby an hypothesis of uniform continuity.

Kernel Convergence Estimates for Diffusions with Continuous Coefficients