Differentiability of the arrival time

TL;DR

This paper proves that the arrival time function for a monotonically advancing front is twice differentiable everywhere except at critical points, where it satisfies the equation classically, with implications for level set methods and game theory.

Contribution

It establishes the twice differentiability of the arrival time function and characterizes its behavior at critical points, connecting PDE theory, geometry, and game interpretations.

Findings

Arrival time is twice differentiable with bounded second derivatives.

Critical points form a set of finite codimensional two Hausdorff measure.

The arrival time satisfies the PDE classically at all points.

Abstract

For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite codimensional two Hausdorff measure. For a monotonically advancing front, the arrival time is equivalent to the level set method; a priori not even differentiable but only satisfies the equation in the viscosity sense. Using that it is twice differentiable and that we can identify the Hessian at critical points, we show that it satisfies the equation in the classical sense. The arrival time has a game theoretic interpretation. For the linear heat equation, there is a game theoretic interpretation that relates to Black-Scholes option pricing. From variations of the Sard and Lojasiewicz theorems, we relate differentiability to whether or not singularities all occur at only finitely many times for flows.