We are going to identify sufficient condition, under which continuous stochastic solution is indeed a classical solution. The results are from ([JT06]).
First, we define stochastic solution of (1). Consider -valued stochastic process solving
where . Note that, (3) has unique strong solution if
- , or if .
Definition 1 (Stochastic solution) A stochastic solution to (1) is a function defined on satisfying,
where is the first exit time from , i.e.
We need one more assumption to avoid explosion of stochastic solution from (4).
Next, we present sufficient condition, under which a classical solution is a stochastic solution.
- is bounded;
- , Assumption 2 hold, and satisfies
then is a stochastic solution.
is a uniformly integrable martingale in , and so is a martingale.
Next, we consider converse of Theorem 2: when do we have stochatics solution being a classical solution.
Lemma 3 Consider a bounded cylinder being for some . Suppose is nondegenerate with constant of ellipticity . Also, assume . Let be a continuous function on the parabolic boundary . Then, there is a unique classical solution to (1) in with boundary data .
If boundary data satisfies smooth and consistency conditions of (10.4.2-3) of [Kry96], then it is the result of Theorem 10.4.1 of [Kry96]. However, is just assumed to be continuous without any regularity. In this case, Shauder’s interior estimate will be useful. A proof of similar result of Lemma 3 is provided in [JT06]. (The proof will be completed later)
The next is Theorem 2.7 of [JT06].
Proof: Consider arbitrary with with . By strong Markov property, for any , we have
where is the first hitting time of to . This implies, is stochastic solution in with continuous boundary data . Thus, . Therefore, .