Observe that the drift is lipschitz continuous for

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Observe that the drift is Lipschitz continuous for positive a and and x is H¨older continuous, with H¨older constant 1 2 . Thus by the Yamada- Watanabe Theorem, (Theorem 5.5 of [15]), the SDE has a unique strong solution. We require a fundamental solution of u t = xu xx + 2 ax 2 + ax u x . (1.11) This can be reduced to (1.8) with A = 3 4 by making the change of variables x x and t 4 t , then eliminating the first derivative term by letting u = e ψ ( x ) ˜ u ( x, t ) for a suitable choice of ψ. We may then obtain a fundamental solution of (1.11) by applying the change of variables to Q ( x, y, t ). From which we conclude that q ( x, y, t ) = 1 t 2 + ay (2 + ax ) r x y e - ( x + y ) t I 1 2 xy t , (1.12) is a fundamental solution of (1.11). This conclusion is correct. The problem is that (1.12) is not the transition density for the diffusion. To see this, observe that l ( x, t ) = Z 0 q ( x, y, t ) dy = 1 - e - x 2 t 2 + ax 6 = 1 . Thus q is not a probability density. Notice that l ( x, t ) is a solution of the Cauchy problem for (1.11), on (0 , ) with u ( x, 0 + ) = 1 and that it has bounded second and first derivatives derivatives in x and t. However l isn’t a solution of the Cauchy problem on [0 , ), since l (0 , t ) = 1 2 6 = 1 . In [9], it was shown that the transition density is actually p ( x, y, t ) = e - ( x + y ) t (2 + ax ) t •r x y (2 + ay ) I 1 2 xy t + ( y ) . (1.13) This produces solutions of the Cauchy problem for (1.11) on [0 , ) for bounded initial data. Since (1.9) is not itself a probability density, there is no a priori reason to expect that we will obtain a density from it if we make a change of variables. In order to obtain a density, it is often necessary to include additional terms which involve generalized functions, such as the Dirac delta that occurs in (1.13). The obvious question is how do we know what these extra terms are? Exactly the same problem arises with a number of other techniques, such as the method of group invariant solutions. See [8] for a discussion of this. An advantage of our method is that the required generalized function terms appear naturally.
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6 MARK CRADDOCK 2. Generalized Laplace Transforms of Fundamental Solutions We begin with a definition. Definition 2.1. Let f : [0 , ) R be Lebesgue integrable and be of suitably slow growth. The generalized Laplace transform of f is the function F γ ( λ ) = Z 0 f ( y ) e - λy 2 - γ dy, (2.1) where λ > 0 and γ 6 = 2 . An explicit inversion theorem for this transform may be found in [14]. However the transform is usually inverted by reducing it to a Laplace transform by setting z = y 2 - γ . We need the integral resulting from (2.1) to be convergent under this change of variables, but for the transforms we consider, this is not a problem. In this paper we will be concerned with PDEs of the form u t = σx γ u xx + f ( x ) u x - g ( x ) u, x 0 , (2.2) for σ > 0 , γ 6 = 2 . Similar results can be proved in the case where γ = 2, but this is usually best handled by letting y = ln x and reducing to γ = 0. However see [8] for an explicit result when γ = 2.
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