# Approximation by convolutions with probability densities and applications to PDEs

## TL;DR

This paper introduces new convolution operators based on probability densities, deriving associated PDEs through Fourier analysis, with applications to solving initial and final value problems in differential equations.

## Contribution

It presents novel convolution operators generated by probability densities and establishes their connection to PDEs via Fourier transform techniques.

## Key findings

- New convolution operators derived from probability densities
- PDEs associated with these convolutions identified
- Methods applicable to initial and final value problems

## Abstract

The purpose of this paper is to introduce several new convolution operators, generated by some known probability densities. By using the inverse Fourier transform and taking inverse steps (in the analogues of the classical procedures used for, e.g., the heat or Laplace equations), we deduce the initial and final value problems satisfied by the new convolution integrals.