Time-dependent tempered generalized functions and Itô’s formula

Abstract

The paper introduces a novel Itô’s formula for time- dependent tempered generalized functions. As an application, we study the heat equation when initial conditions are allowed to be a generalized tempered function. A new proof of the Üstunel-Itô’s formula for tempered distributions is also provided.

Keywords:

• generalized functions
• Itô’s formula
• Hermite expansions
• Hermite functions

AMS Subject Classifications:

• 46F10
• 46F30
• 60H99