Global in Time Solutions to Kolmogorov-Feller Pseudodifferential Equations with Small Parameter

TL;DR

This paper introduces a novel method for constructing global-in-time asymptotic solutions to pseudodifferential parabolic equations with a small parameter, utilizing Hamiltonian characteristics instead of integral representations.

Contribution

It presents a new characteristic-based approach for global solutions, contrasting with traditional schemes like Maslov's, for pseudodifferential equations with small parameters.

Findings

Successfully constructs leading-term solutions using Hamiltonian characteristics

Provides a comparison with Maslov's established scheme

Demonstrates the method's effectiveness for pseudodifferential equations

Abstract

The goal in this paper is to demonstrate a new method for constructing global-in-time approximate (asymptotic) solutions of (pseudodifferential) parabolic equations with a small parameter. We show that, in the leading term, such a solution can be constructed by using characteristics, more precisely, by using solutions of the corresponding Hamiltonian system and without using any integral representation. For completeness, we also briefly describe the well-known scheme developed by V.P.Maslov for constructing global-in-time solutions.