On a new probabilistic representation for the solution of the heat equation

Abstract

We obtain a new probabilistic representation for the solution of the heat equation in terms of a product for smooth random variables which is introduced and studied in this paper. This multiplication, expressed in terms of the Hida–Malliavin derivatives of the random variables involved, exhibits many useful properties that are to some extents opposite to some peculiar features of the Wick product.

Keywords:

• heat equation
• Hida–Malliavin derivative
• second quantization operators
• smooth random variables

AMS 2000 Classification::

• 60H30
• 60H07

Acknowledgement

The authors are grateful to Aurel I. Stan for stimulating discussions.

Notes

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