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-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.
Probabilistic representation of fundamental solutions to partial derivative u/partial derivative t = K-m partial derivative(m)u/partial derivative x(m) / Orsingher, Enzo; D'Ovidio, Mirko. - In: ELECTRONIC COMMUNICATIONS IN PROBABILITY. - ISSN 1083-589X. - ELETTRONICO. - 17:0(2012), pp. 1-12. [10.1214/ecp.v17-1885]