Probabilistic representation of fundamental solutions to $$\frac{\partial u}{\partial t} = \kappa _m \frac{\partial^m u}{\partial x^m}$$.(English)Zbl 1272.35005

Summary: 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 $$X_m$$ related to the higher-order heat-type equation with positively skewed stable r.v.’s $$T^j_{1/3}, j=1,2, \dots, n$$ we obtain genuine r.v.’s whose explicit distribution is given for $$n=3$$ in terms of Cauchy asymmetric laws. We also prove that $$X_3(T^1_{1/3}(\dots(T^n_{(1/3)}(t))\dots))$$ has a stable asymmetric law.

MSC:

 35A08 Fundamental solutions to PDEs 60G52 Stable stochastic processes 35C05 Solutions to PDEs in closed form 35K30 Initial value problems for higher-order parabolic equations
Full Text: