The Cauchy Problem for the Heat Operator on \({\mathbf {R}}^{n}\)

In this chapter we consider a second order, strictly elliptic differential operator \(P(x,D)\) with smooth coefficients, called the drift Laplacian, and construct the fundamental solution operator \(u(t;x,D)\) of the Cauchy problem for the heat operator on Euclidean space \({\mathbf {R}}^{n}\) . Our approach here due to C. Tsutsumi (Proc. Jpn. Acad. 50:11–15, 1974), C.T. Iwasaki (Osaka J. Math. 14:569–592, 1977) and C. Iwasaki–N. Iwasaki (Publ. Res. Inst. Math. Sci. 17:577–655, 1981) may be considered as a modern version of the pioneering work of Levi (Ann. Mat. 14:187–264, 1908) in the framework of the symbolic calculus of pseudo-differential operators (see Theorems 14.19 and 14.23). For the classical parametrix method of Levi (Ann. Mat. 14:187–264, 1908), the reader might refer to Èǐdel’man (Am. Math. Soc. Trans. (2) 41:1–48, 1964; Am. Math. Soc. Trans. (2) 41, 49–120, 1964; Trudy Moscov. Mat. Obšč. 23:179–234, 1970), Ladyženskaja et al. (Linear and Quasilinear Equations of Parabolic Type (Russian). Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence, 1968) and Friedman (Partial Differential Equations of Parabolic Type. Dover Publications Inc., Mineola, 1964/1992/2008). strictly elliptic