In this paper we study the second-order parabolic equation
formula here
in a domain [0,T]×ℝd ⊂ ℝd+1, where a = (aij)di,j=1 is matrix of bounded measurable
coefficients, b = (bj)dj=1, and bˆ = (bˆj)dj=1 are measurable (in general, singular) vector
fields, V is a measurable potential, T is a fixed positive number, and ∂tu = ∂u/∂t, and
we employ the notation
formula here
We introduce a new class of coefficients in the lower-order terms for which we prove
the existence and the uniqueness of the weak fundamental solution, and for this we
derive Gaussian upper and lower bounds.