  We show that some boundary conditions assumed at a thin membrane may result
in normal diffusion not being the stochastic Markov process. We consider
boundary conditions defined in terms of the Laplace transform in which there is
a linear combination of probabilities and probability fluxes defined on both
membrane surfaces. The coefficients of the combination may depend on the
Laplace transform parameter. Such boundary conditions are most commonly used
when considering diffusion in a membrane system unless collective or non-local
processes in particles diffusion occur. We find
Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation in terms of the
Laplace transform and we derive the criterion to check whether the boundary
conditions lead to fundamental solutions of diffusion equation satisfying this
equation. If the BSCK equation is not met, the Markov property is broken. When
a probability flux is continuous at the membrane, the general forms of the
boundary conditions for which the fundamental solutions meet the BSCK equation
are derived. A measure of broken of semi-group property is also proposed. The
relation of this measure to the non-Markovian property measure is discussed.

Statistical Mechanics