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

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

2012-03-14

arxiv.org/abs/1203.3133v1

Mathematics

Probability

Scientific paper

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 $T^j_{1/3}$, $j=1,2, ..., n$ 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.

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