11 fundamental solutions as transition densities a

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This simple result lies at the heart of the results we obtain here. 1.1. Fundamental Solutions as Transition Densities. A funda- mental solution for the Cauchy problem u t = a ( x, t ) u xx + b ( x, t ) u x + c ( x, t ) u, x Ω R , (1.4) u ( x, 0) = φ ( x ) , is a kernel p ( x, y, t ) with the property that: (i) for each fixed y , p ( x, y, t ) is a solution of (1.4) on Ω × (0 , T ] for some T > 0; and (ii) u ( x, t ) = R Ω φ ( y ) p ( x, y, t ) dy, is a solution of the given Cauchy problem for ap- propriate initial data φ . In general a fundamental solution will be a distribution in the sense of Schwartz. For an introduction to the theory of fundamental solutions for parabolic problems, see chapter one of the book by Friedman [11]. Fundamental solutions play an important role in probability theory. Consider an Itˆo diffusion X = { X t : t 0 } which satisfies the stochas- tic differential equation (SDE) dX t = b ( X t , t ) dt + σ ( X t , t ) dW t , X 0 = x, (1.5) in which W = { W t : t 0 } is a standard Wiener process. The existence and uniqueness of solutions of (1.5) depends on the coefficient functions b, σ . See [13] for conditions guaranteeing a unique strong solution to
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4 MARK CRADDOCK (1.5). Assume that b and σ are such that (1.5) has a unique strong solution. Then the expectations u ( x, t ) = E x [ φ ( X t )] = def E φ ( X t ) fl fl fl fl X 0 = x , (1.6) are solutions of the Cauchy problem u t = 1 2 σ 2 ( x, t ) u xx + b ( x, t ) u x (1.7) u ( x, 0) = φ ( x ) . The PDE (1.7) is known as the Kolmogorov forward equation. See [13] for background on stochastic calculus. Thus if p ( x, y, t ) is the appropriate fundamental solution of (1.7) then we may compute the given expectations according to E x [ φ ( X t )] = R Ω φ ( y ) p ( x, y, t ) dy. In this context, the fundamental solution is known as the probabil- ity transition density for the process. Obviously we also require that R Ω p ( x, y, t ) dy = 1 . Recall that fundamental solutions are not unique. The PDE (1.7) may have many fundamental solutions, only one of which will be the transition density. One of the strengths of the methods of this paper, is that they will always produce a fundamental solution which is a probability density. There are many other methods for which this is not the case. To illustrate, recall Lie’s result that any PDE of the form (1.4) which has a four dimensional Lie algebra of symmetries can be reduced to an equation of the form u t = u xx - A x 2 u, (1.8) where A is a constant. (See the paper [12] for a detailed discussion of this reduction method). This PDE has a well known fundamental solution Q ( x, y, t ) = xy 2 t exp - x 2 + y 2 4 t I 1 2 1+4 A xy 2 t · . (1.9) Thus a common solution strategy is to reduce an equation of the form (1.4) with a four dimensional Lie algebra of symmetries to the form (1.8) and construct from Q a fundamental solution of the original PDE. This is often referred to as the method of reduction to canonical form . This works if all we seek is a fundamental solution. But if we require the fundamental solution to also be a density, then this strategy will in general fail. As an example, suppose that we wish to find the transition density for the diffusion X = { X t : t 0 } satisfying the SDE dX t = 2 aX t 2 + aX t dt + p 2 X t dW t , X 0 = x > 0 , a > 0 . (1.10)
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FUNDAMENTAL SOLUTIONS 5
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