Hamilton-Jacobi方程式に対する有限差分法の数学解析―確率論と変分法を用いた枠組―

抄録

This study explains a stochastic and variational approach to the finite difference method applied to first order Hamilton-Jacobi equations generated by Hamiltonians of the Tonelli class. In Section 1, the connection between Hamilton-Jacobi equations and Hamiltonian dynamics is stated, which implies the necessity to introduce weak solutions. In Section 2, the representation formula of a local classical solution obtained via the method of characteristics is extended to a globally defined function employing Tonelli’s calculus of variations. This function is called a value function in optimal control theory, and it is a weak solution of the Hamilton-Jacobi equation in the sense of viscosity solutions. In Section 3, reasoning similar to the previous section is demonstrated for the discretized Hamilton-Jacobi equations with the most elementary finite difference method, where a stochastic feature appears due to numerical viscosity. The convergence of approximation is proven in terms of stochastic and variational techniques, yielding a viscosity solution, its derivative and characteristic curves all at once. This approach is substantially applied to weak Kolmogorov-Arnold-Moser (KAM) theory.