Generalized Feynman path integrals and applications to higher-order heat-type equations
From the introduction: One of the aims of the present paper is precisely to give a detailed account of the theory and applications of infinite dimensional Fresnel integrals to the construction of a functional integral representation of the solution of PDEs of the form \[ \frac{\partial}{\partial t}u(x,t) = (-i)^p\alpha \frac{\partial^p}{\partial x^p}u(x,t) +V(x)u(x,t), \] where \(t\in [0,+\infty), x\in \mathbb{R}\), \(p\in \mathbb{N}\), \(\alpha\in \mathbb{C}\) and \(V:\mathbb{R} \to \mathbb{C}\) is a continuous bounded function. In Section 2 we review the theory of infinite dimensional Fresnel integrals as well as their applications to the mathematical definition of Feynman path integrals. Section 3 presents a recent generalization of infinite dimensional Fresnel integrals which covers the case of polynomial phase functions and the application to the representation of the solution of higher-order equations of the above form. In Section 4 we prove a Girsanov type formula for the higher-order equation above form. Eventually Section 5 presents a new phase space path integral solution formula for Eq. above form.
scientific article; zbMATH DE number 7000847
scientific article