We will discuss probabilistic representation of the smallest solution of linear parabolic equation under the absence of uniqueness of the solution.

Consider a stochastic differential equation with filtered probability space of the form

*Assumption1:* SDE (1) has a unique strong solution satisfying

Let be a given function, and be

Then, satisfies following PDE in some sense:

where

*Problem:* If Assumption 1 holds, and furhter assume , then is the smallest lower-bounded classical solution of (3).

*Proof:* For simplicity, we denote by in this below. Note that

is a martingale process. Suppose is an aribtrary lower bounded solution of (3), then Ito’s formula applying to leads to

and is a lower bounded local martingale, hence is a super martingale.

Therefore, we have

and this implies