Probabilistic representation of fundamental solutions to $\frac{\partial u}{\partial t} = \kappa_m \frac{\partial^m u}{\partial x^m}$

TL;DR

This paper introduces a stochastic representation for fundamental solutions of higher-order heat equations using damped oscillations and gamma-distributed parameters, providing explicit distributions for specific cases.

Contribution

It presents a novel probabilistic framework for fundamental solutions of higher-order heat equations, including explicit distributions for certain cases and compositions with skewed stable variables.

Findings

Explicit distribution for $n=3$ in terms of Cauchy laws.

Representation of solutions via damped oscillations and gamma distributions.

Stable asymmetric law for composed stochastic processes.

Abstract

For the fundamental solutions of heat-type equations of order $n$ we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process $X_n$ related to the higher-order heat-type equation with positively skewed stable r.v.'s $T^j_{1/3}$, $j=1,2, ..., n$ we obtain genuine r.v.'s whose explicit distribution is given for $n=3$ in terms of Cauchy asymmetric laws. We also prove that $X_3(T^1_{1/3}(...(T^n_{(1/3)}(t))...))$ has a stable asymmetric law.