Local limit theorems for Markov chains with trend component of linear   growth
  We consider a sequence of Markov chains weakly convergent to a diffusion. We suppose that a drift term contains a linearly increasing component. The usual parametrix method fails because of this unbounded drift term. We show how to modify the parametrix method to obtain local limit theorems for this case. 
V. Konakov, A. Markova
math.PR