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

   
@article{ECP1885,
	author = {Enzo Orsingher and Mirko D'Ovidio},
	title = {Probabilistic representation of fundamental solutions to $\frac{\partial u}{\partial t} = κ_m \frac{\partial^m u}{\partial x^m}$},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {17},
	year = {2012},
	keywords = {Pseudo-process; higher-order heat equation; Airy functions; Cauchy distribution; stable laws; fractional diffusion equations},
	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 $X_m$ 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.},
	pages = {no. 34, 1-12},
	issn = {1083-589X},
	doi = {10.1214/ECP.v17-1885},    
        url = {http://ecp.ejpecp.org/article/view/1885}}