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

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.


Introduction
The problem of studying the form of fundamental solutions of higher-order heat equations of the form where κ m = (−1) m/2+1 if m is even ±1 if m is odd has been tackled in some particular cases by mathematicians of the caliber of Bernstein [3]; Lévy [7]; Pòlya [11] and Burwell [4]. By applying the steepest descent method some recent papers by Li and Wong [8], Accetta and Orsingher [1], Lachal [6] have explored the form of the fundamental solutions of equation (1.1). The aim of this note is to give an explicit representation of the solutions to (1.1) for the case where the order of the equation is odd, alternative to the inverse Fourier transform and capable of representing the sign-varying behavior of the fundamental solutions to (1.1). Our result is that the fundamental solutions to (1.1) have the probabilistic representation (1.3) u 2n+1 (x, t) = 1 xπ E e −bnxG 2n+1 (1/t) sin a n xG 2n+1 (1/t) , x ∈ R, t > 0 in the odd-order case, and (1.4) u 2n (x, t) = 1 πx E sin xG 2n (1/t) , x ∈ R, t > 0 for the even-order case. In (1.3) and (1.4) by G γ (t) we denote the generalized gamma r.v. with density The parameters a n , b n appearing in (1.3) and (1.4) are a n = cos π 2(2n + 1) , b n = sin π 2(2n + 1) .
Results (1.3) and (1.4) show that the fundamental solutions have an oscillating behavior which has been explored in several papers by many researchers. In our view our result represents a concluding picture of the solutions to higher-order heat equations. For all values of the degree n of the equation (1.1) we have solutions which have the behavior of damped oscillations where the probabilistic ingredients (the generalized gamma or Weibull-type distributions) depend only on n ∈ N. An alternative universal representation of the fundamental solution in the odd-order case reads Functions u 2n+1 display ascillations which fade off as the degree 2n + 1 of the equation increases. A special attention has been devoted to third-order equations where we have that In the fourth-order case (biquadratic heat-equation) in Orsingher and D'Ovidio [10] we have shown that In a recent paper we have shown that the composition of an odd-order pseudoprocess X 2n+1 with a positivly skewed stable r.v. T 1 2n+1 of order 1 2n+1 yields a genuine r.v. with asymmetric Cauchy distribution, that is π (x + t sin π 2(2n+1) ) 2 + t 2 cos 2 π 2(2n+1) dx.
For n = 1 from (1.9) we can extract a very interesting relationship for the Airy function which reads We show here that the m-times iterated pseudo-process (with T j 1 2n+1 We have also explored the connection between solutions of fractional equations with the solutions of higher-order heat-type equations (1.1) for α = 1 m , m ∈ N.

Pseudo-processes
Some basic facts about the fundamental solutions of higher-order heat equations had been established many years ago essentially by applying the steepest descent method. In particular, Li and Wong [8] have shown that the number of zeros is infinite for solutions to even-order equations. The steepest descent method was applied by Accetta and Orsingher [1] for the analysis of the third-order equation. The oscillating behavior of the solutions of higher-order heat-type equations is confirmed by our analysis. Furthermore, for the odd-order case our results show that the asymmetry of solutions decreases as the order 2n + 1 increases. The result of Theorem 2.1 below shows that solutions of all odd-order heat equations can be constructed by means of damped oscillating functions with gamma distributed parameters.
We pass now to our principal result.
Theorem 2.1. The solution to is given by and a n = cos π 2(2n + 1) , b n = sin π 2(2n + 1) Proof. We start by evaluating the Fourier transform of (2.2) where in the last step we used the integral representation of the Heaviside function In the above steps we used the fact that (a n + ib n ) 2n+1 = i, and (a n − ib n ) 2n+1 = −i.
The integral (2.4) can be performed in two different ways. First we can take the Laplace transform dt.
This shows that We can arrive at the some result by means of the following trick We have thus shown that the Fourier transform of (2.2) coincides with the Fourier transform of the solution to the Cauchy problem (2.1).
For the special case of the third-order heat equation we have the following result.
Theorem 2.2. The solution of the Cauchy problem can be written as Proof. It is convenient to work with the following series expansion of the Airy function (see Orsingher and Beghin [9, formula (4.10)]) If we expand the function we establish a relationship which is useful in transforming (2.9) as This proves, in a different way, that Theorem 2.3. We can write the fundamental solution u 2n+1 in the following alternative form e −xy sin π 2(2n+1) sin xy cos π 2(2n + 1) y 2n e −ty 2n+1 dy = (2n + 1)t πx ∞ 0 e −xy cos nπ (2n+1) sin xy sin nπ (2n + 1) y 2n e −ty 2n+1 dy which is in accord with Lachal [6, formula 11].
Theorem 2.6. The solution to can be written as Clearly, in force of the symmetry and by considering formula (2.13), we obtain that Remark 2.8. It is well-known that the solution to the fractional diffusion equation is given by The folded solution to the equation (2.17) reads x t α r Γ(α(r + 1)) r! sin (πα(r + 1)) .

(2.18)
This represents a probability density of a r.v. X(t) on the half-line (0, ∞) which can be expressed in terms of positively skewed stable densities.
where in the last step the expression of the stable density The calculations above show that the r.v. X(t) with distribution (2.18) can be expressed as where Y (t) is a positively skewed stable-distributed r.v. of order α ∈ (0, 1). In other words the stable law of Y (t) is related to the folded solution of the fractional diffusion equation X(t) in the sense that

This is because
We give also the Laplace transforms with respect to time t and space where in the last step, formula has been applied (see formula (5.1) of Beghin and Orsingher [2]). Furthermore, Formulas above help to check that q α satisfies the fractional equation Remark 2.10. For α = 1/2, the formula (2.19) yields Indeed, we have that and therefore ∂q ∂t By observing that and the fact that y (z) − zy(z) = 0 that is, y satisfies the Airy equation, we get that By recursive arguments it can be shown that is also a solution to (2.27) for m > 1.
Remark 2.12. We have shown in a previous paper that the r.v.
Remark 2.14. We note that by adjusting the derivation of (2.34) we can obtain result (1.10) of the introduction.