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

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 $T_{1/3}^j$, $j=1,2,...,n$ we obtain genuine r.v.'s whose explicit distribution is given for $n=3$ in terms of Cauchy asymmetric laws. We also prove that $X_3(T_{1/3}^1(...(T_{(1/3)}^n(t))...))$ has a stable asymmetric law.