Stochastic formulations of the parametrix method

In this manuscript, we consider stochastic expressions of the parametrix method for solutions ofd-dimensional stochastic differential equations (SDEs) with drift coefficients which belong toLp(Rd),p>d. We prove the existence and Hölder continuity of probability density functions for distributions of solutions at fixed points and obtain an explicit expansionvia(stochastic) parametrix methods. We also obtain Gaussian type upper and lower bounds for these probability density functions.