# Letting z y 2 converts this to a laplace transform

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Letting z = y 2 converts this to a Laplace transform and inverting this Laplace transform gives the fundamental solution (1.9). The cal- culations are quite elementary. But what if we take the solution u 1 ? We now seek a fundamental solution p ( x, y, t ) such that Z 0 e - λy 2 u 1 ( y ) p ( x, y, t ) dy = u 1 ( x )(1 + 4 λt ) - 2+ 1+4 A 2 exp - λx 2 1 + 4 λt . Let us fix A = 3 4 . We let z = y 2 in the generalized Laplace transform, perform the inversion with the aid of Proposition 2.4 and obtain a second fundamental solution of (2.6) for this choice of A given by p ( x, y, t ) = 2 e - ( x 2 + y 2 ) 4 t y r y x xI 1 ( x y 2 t ) 4 t y + δ ( y 2 ) . (2.7) Now recall that in Section 1.1 we considered the Itˆo diffusion satis- fying the SDE dX t = 2 aX t 2 + aX t dt + p 2 X t dW t , X 0 = x > 0 , a > 0 . (2.8) The corresponding Kolmogorov forward equation u t = xu xx + 2 ax 2 + ax u x , (2.9) can be reduced to the canonical form u t = u xx - 3 4 x 2 u x (2.10) by a change of variables. This PDE has a known fundamental solution given by (1.9). From this we earlier found a fundamental solution of (2.9), but this fundamental solution was not a transition density. However we now have a second fundamental solution of equation (2.10). If we use the fundamental solution (2.7) we deduce that p ( x, y, t ) = e - ( x + y ) t (2 + ax ) t •r x y (2 + ay ) I 1 2 xy t + ( y ) , (2.11) is also a fundamental solution of (2.9) and this, as noted above, is the transition density for the diffusion. Thus we failed to produce a transition density when we first consid- ered this example, because we did not use the appropriate fundamental solution of (2.10). The point is that in order to obtain a transition density in general via the method of reduction to canonical form, it is necessary to consider all fundamental solutions of the canonical form and choose the one which leads to the desired density. This however

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FUNDAMENTAL SOLUTIONS 9 can be quite a laborious exercise. Our results will give the correct result much more efficiently. If the drift in (2.2) satisfies either of the remaining two Riccati equa- tions, Craddock and Lennox proved in [7] that fundamental solutions can be obtained by inverting a Whittaker transform. Unfortunately Whittaker transforms are difficult to invert, so we would prefer a re- sult which gives fundamental solutions in terms of generalized Laplace transforms. Fortunately this is possible. This discovery is one of the main contributions of this paper. The first result is the following. Theorem 2.5. Consider the PDE u t = σx γ u xx + f ( x ) u x - g ( x ) u, γ 6 = 2 , x 0 , (2.12) and suppose that g and h ( x ) = x 1 - γ f ( x ) satisfy σxh 0 - σh + 1 2 h 2 + 2 σx 2 - γ g ( x ) = A 2(2 - γ ) 2 x 4 - 2 γ + B 2 - γ x 2 - γ + C, where A > 0 , B and C are arbitrary constants. Let u 0 be a stationary, analytic solution of (2.12). Then (2.12) has a solution of the form U ² ( x, t ) = (1 + 2 ² 2 (cosh( At ) - 1) + 2 ² sinh( At )) - c × fl fl fl fl fl cosh( At 2 ) + (1 + 2 ² ) sinh( At 2 ) cosh( At 2 ) - (1 - 2 ² ) sinh( At 2 ) fl fl fl fl fl B 2 σ A (2 - γ ) e - 1 2 σ F ( x ) - Bt σ (2 - γ ) × exp ( - A²x 2 - γ (cosh( At ) + ² sinh( At )) σ (2 - γ ) 2 (1 + 2 ² 2 (cosh( At ) - 1) + 2 ² sinh( At ) ) × exp ( 1 2 σ F ˆ x (1 + 2 ² 2 (cosh( At ) - 1) + 2 ² sinh( At )) 1 2 - γ !) ×
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