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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
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