A Hamiltonian Path Integral for a Degenerate Parabolic Pseudo-Differential Operator
Naoto Kumano‐go · Institutional Repositories DataBase (IRDB) · 1996
In this $\mathrm{p}\mathrm{a}\mathrm{p}e\dot{\mathrm{r}}$ , using a Hamiltonian path integral, we give an expression of the symbol of the fundamental solution for a degenerate parabolic pseudo-differential operator.This Hamiltonian path integral converges in the topology of the symbol class $s_{x_{\beta}^{m}}^{2},,\delta$ and in the weak topology of the symbol class $S_{\lambda,\rho,\delta}^{0}$ . $0$ . IntroductionIn this paper, we construct the fundamental solution for a degenerate parabolic pseudo-differential operator in a different way from that in C.Tsutsumi [10].In [10], she constructed the fundamental solution by Levi-Mizohata method.On the other hand, in this paper, we construct the fundamental solution by a Hamiltonian path integral.If we use a Hamiltonian path integral, we can actually write the symbol of the fundamental solution.Furthermore, this Hamiltonian path integral converges in the topology of the synbol class $S_{\lambda,\rho,\delta}^{2m}$ and in the weak topology of the symbol class $S^{0}\lambda,\rho,\delta$ .In Section 1, we introduce some basic properties of pseudo-differential operators, which we use in Section 2. For the details, see Chapter 7 \S 1 and \S 2 in H. .In Section 2, we construct the fundamental solution for a degenerate parabolic pseudo-differential operator by a Hamiltonian path integral.Theorem 2.1 is the main theorem in this paper.