Some parabolic equations for measures and Gaussian semigroups

TL;DR

This paper introduces explicit solutions and properties of infinite-dimensional parabolic equations related to quantum physics, including generalizations of the Mehler formula and Gaussian semigroups, with applications to measures and quantum theory.

Contribution

It provides new explicit formulas for solutions of infinite-dimensional parabolic equations and explores their properties, extending classical formulas to a broader context.

Findings

Explicit solutions for infinite-dimensional parabolic equations

Generalization of the Mehler formula

Construction of an analogue of the Ornstein-Uhlenbeck measure

Abstract

This short communication (preprint) is devoted to mathematical study of evolution equations that are important for mathematical physics and quantum theory; we present new explicit formulas for solutions of these equations and discuss their properties. The results are given without proofs but the proofs will appear in the longer text which is now under preparation. In this paper, infinite-dimensional generalizations of the Euclidean analogue of the Schr\"odinger equation for anharmonic oscillator are considered in the class of measures. The Cauchy problem for these equations is solved. In particular cases, explicit formulas for fundamental solutions are obtained, which are a generalization of the Mehler formula, and the uniqueness of the solution with certain properties is proved. An analogue of the Ornstein-Uhlenbeck measure is constructed. The definition of Gaussian semigroups is given and their connection with the considered parabolic equations is described.