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Lie group symmetries as integral transforms of fundamental solutions. (English) Zbl 1147.35009

There are several interesting aspects to this paper and the choice of the most interesting could be very much a matter of the ‘eye of the beholder’. The main part of this paper is the computation of fundamental solutions for a class of $1+1$ linear evolution partial differential equations possessing a sufficiently large symmetry algebra, namely $\left\{\text{sl}\left(2,ℝ\right)\oplus {A}_{1}\right\}{⨁}_{s}\infty {A}_{1}$, where the infinite-dimensional subalgebra is composed of the solution symmetries. The class of equations arises in financial mathematics in the modelling of bond pricing with a general drift function and a specific form for the volatility. The coefficient functions in the $\text{sl}\left(2,ℝ\right)$ subalgebra are found from the solution of one of three possible Riccati equations in which the drift function and volatility play major roles. The authors show that, if the action of the exponentiated vector field on a solution $u\left(t,x\right)$ gives $\sigma \left(t,x;\epsilon \right)u\left({a}_{1}\left(t,x;\epsilon \right),{a}_{2}\left(t,x;\epsilon \right)\right)$, where $\epsilon$ is the parameter of the transformation, then the so-called multiplier, $\sigma$, (maybe slightly modified) is always a classical integral transform of the fundamental solution of the partial differential equation. For example in the case of the heat equation the transform is the Laplace transform. This is certainly a very curious result and one hopes that the authors pursue their suggested investigation of the connection.

The authors claim that their approach has many benefits. For example it obviates the need for changes of variables or the solution of an ordinary differential equation. On the other hand it does require obtaining the inverse transform and this does not seem to be a completely transparent task.

The attraction of the paper is marred by a number of misprints. It is unfortunate that the Journal of Differential Equations does not have a competent copy editor. One becomes more than a little irked at the umpteenth misspelling of Riccati!

##### MSC:
 35A08 Fundamental solutions of PDE 35A22 Transform methods (PDE) 44A10 Laplace transform 91B70 Stochastic models in economics 58J70 Invariance and symmetry properties 35A30 Geometric theory for PDE, characteristics, transformations