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
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