Skorohod metric

PDF: here

Reference: [Bil99] ([Patrick Billingsley 1999])

Goal: RCLL space is the collection of all right continuous processes with left limit exists. In this below, we will give a definition of Skorohod metric on this space. More precisely, for a ${t\in (0, \infty)}$, we denote by ${\mathbb D^{d}_{t}}$ the collection of RCLL processes on ${[0, t]}$ taking values in ${\mathbb R^{d}}$. We also use ${\mathbb D^{d}_{\infty}}$ to denote the collection of RCLL processes on ${[0, \infty)}$.

The definition of the viscosity solution of Dirichlet problem

We have discussed the definition of viscosity property (here). In this note, we will discuss the definition of the viscosity solution of Dirichlet problem. (PDF) Continue reading

A discussion on the solvability of Dirichlet problem

Last Friday, I’ve had a pleasant discussion with Professor Lions (here) on my recent paper (here). PDF is also available (here).

Supremum of uncountably many measurable functions may not be measurable

Given a family of measurable functions ${f_{\alpha}(x)}$ with ${\alpha \in \Lambda}$, shall we have ${g(x) = \sup_{\alpha\in \Lambda} f_{\alpha}(x)}$ measurable? If each ${f_{\alpha}}$ is continuous, then ${g(x)}$ is lower semicontinuous, so is measurable. However, it may not be true in general.

A motivating example for a generalized Dirichlet problem

This is an example of Dirichlet problem whose unique solution only meets its boundary in the generalized sense. In other words, there is no solution in classical Dirichlet sense. It is taken from Example 7.8 of [Crandall, Ishii, Lion 1992].

Uniqueness of a generalized Dirichlet problem: An Example

Recalling that, strong comparison principle is able to identify a unique viscosity solution in ${C(O)}$. In this below, we have an example having its interior being discontinuous. Is there any theory to justify its uniqueness? Continue reading