Let , where is an open domain in . Consider equation

where

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

**Assumption 1 (Local regularity)** * *

*
*
- ,
- ,
- , or if .

* *

We should note that by Lemma 6.1.1 of [Friedman 1976] ([Fri76]), under Assumption 4, , (or ) exists iff (or ). To avoid the explosion of from (3), we assume

**Assumption 2 (Uniform linear growth)** * *

**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).

**Assumption 3** * . *

Next, we present sufficient condition, under which a classical solution is a stochastic solution.

**Theorem 2** * Suppose is a classical solution, and Assumptions 1 and 3 hold. In addition, if either of two followings are ture, *

*
*
- is bounded;
- , Assumption 2 hold, and satisfies

* then is a stochastic solution. *

*Proof:* Assumption 1 is needed to have strong solution of (3). Ito’s formula shows that is a local martingale. Also, one can show , hence

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)

**Assumption 4 (Positive definite)** * . *

The next is Theorem 2.7 of [JT06].

**Theorem 4** * Suppose Assumptions 1, 2, 3, and 4 hold. If is a continuous stochastic solution, then is a classical solution. *

*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, .

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