Singular solution of the Feller diffusion equation via a spectral decomposition

Feller studied a branching process and found that the distribution for this process approximately obeys a diffusion equation [W. Feller, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley and Los Angeles, 1951), pp. 227–246]. This diffusion equation and its generalizations play an important role in many scientific problems, including, physics, biology, finance, and probability theory. We work under the assumption that the fundamental solution represents a probability density and should account for all of the probability in the problem. Thus, under the circumstances where the random process can be irreversibly absorbed at the boundary, this should lead to the presence of a Dirac delta function in the fundamental solution at the boundary. However, such a feature is not present in the standard approach (Laplace transformation). Here we require that the total integrated probability is conserved. This yields a fundamental solution which, when appropriate, contains a term proportional to a Dirac delta function at the boundary. We determine the fundamental solution directly from the diffusion equation via spectral decomposition. We obtain exact expressions for the eigenfunctions, and when the fundamental solution contains a Dirac delta function at the boundary, every eigenfunction of the forward diffusion operator contains a delta function. We show how these combine to produce a weight of the delta function at the boundary which ensures the total integrated probability is conserved. The solution we present covers cases where parameters are time dependent, thereby greatly extending its applicability.

Figure 1

This figure shows how the normalization of the fundamental solution found by Feller [see Eq. (3)] behaves as a function of time, $t$, for different values of the parameter $\alpha$. For this figure, we adopted a value of the parameter $\beta$ of $\beta =2$, and at an initial time of $u=0$, the random process $X_t$ was taken to have the initial value of 0.4. For all values of $\alpha$, the norm of Feller's solution decreases with time.

Figure 2

This figure shows how the probability density of the random process in Eq. (2) changes over time. For this figure, we adopted parameter values of $\alpha =0$ and $\beta =1$. At an initial time of $u=0$, the random process $X_t$ was taken to have the initial value of $0.4$. The panel for $t=0.001$ shows a schematic representation a Dirac delta function, which is the forward eigenfunction associated with zero eigenvalue (in red). The nearly opposite curve (in blue) is the contribution from eigenfunctions associated with positive eigenvalues. At time $t=0$ the red and blue curves exactly cancel one another, but as time increases, the contribution from the positive eigenvalues decreases, thereby exposing the spike at $x=0$. The other panels show the distribution at later times, including the development of the spike at $x=0$; the gray bars are the results of simulations while the black curves correspond to the theoretical result of Eq. (16). The delta functions have been broadened for the purpose of visualization and their heights represent their integrated weights.