# U 1 y q x y t dy u 1 x thus we have two fundamental

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0 u 1 ( y ) q ( x, y, t ) dy = u 1 ( x ) . / Thus we have two fundamental solutions which integrate to one, but they cannot both be the transition density, as the transition density for squared Bessel processes is known to be unique. (See [18] for a detailed exposition of the theory of squared Bessel processes). The existence of these additional fundamental solutions does not seem to have been previously observed. Example 2.2. Take n = 3. Then the transition density for the squared Bessel process of dimension 6 is p ( x, y, t ) = 1 2 t y x e - x + y 2 t I 2 xy t . (2.21) From the stationary solution u 0 ( x ) = x - 2 we obtain a second funda- mental solution p 2 ( x, y, t ) = p ( x, y, t ) + e - x + y 2 t y 2 2 tx δ ( y ) + 2 t y x · 2 δ 0 ( y ) , (2.22) and R 0 p 2 ( t, x, y ) dy = 1 . It acts on test functions which have finite derivative at zero. Notice however that the solution of the Cauchy problem u t = 2 xu xx + 6 u x , u ( x, 0) = φ ( x ) given by this fundamental solution will not be continuous as x 0 . So how can we tell that a fundamental solution is in fact the desired density? A reasonably straightforward one in the γ = 1 case is the following. Similar results can be proved for any γ. Proposition 2.11. Let X = { X t : t 0 } be an Itˆo diffusion which is the unique strong solution of X t = X 0 + Z t 0 f ( X s ) ds + Z t 0 p 2 σX t dW t , (2.23)

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FUNDAMENTAL SOLUTIONS 15 where W = { W t : t 0 } is a standard Wiener process. Suppose further that f is measurable and there exist constants K > 0 , a > 0 such that | f ( x ) | ≤ Ke ax for all x. Then there exists a T > 0 such that u ( x ; t, λ ) = E x £ e - λX t / is the unique solution of the first order PDE ∂u ∂t + λ 2 σ ∂u ∂λ + λ E x £ f ( X t ) e - λX t / = 0 , (2.24) subject to u ( x ; 0 , λ ) = e - λx , for 0 t < T, λ > a . Proof. The Itˆo formula gives e - λX t = e - λx + Z t 0 e - λX s ( λ 2 σX s - λf ( X s )) ds - λM t , (2.25) where M t = R t 0 e - λX s 2 σX s dW s is a local martingale. Obviously we have E x •Z t 0 ( p X s e - λX s ) 2 ds = E x •Z t 0 X s e - 2 λX s ds E x •Z t 0 ds 2 < , so M t is a martingale, from which E x [ M t ] = M 0 = 0. Taking expecta- tions in (2.25) therefore gives E x £ e - λX t / - λ 2 σ Z t 0 E x £ X s e - λX s / ds = e - λx - λ Z t 0 E x £ f ( X s ) e - λX s / ds. (2.26) Further, E x £ X s e - λX s / = - ∂λ E x £ e - λX s / . Differentiation of (2.26) with respect to t gives (2.24). Now f ( x ) e - λx is bounded and measurable for λ > a. So E x £ f ( X t ) e - λX t / is continuous in t . (For example, Proposition 15.49 of Breiman [2]). For each fixed t it is analytic in λ. The coefficients of the PDE are analytic as is the initial data. The uniqueness of the solution follows from the Cauchy-Kovalevskaya Theorem for first order systems. (See Tr` eves’ book [19] for a proof of the Cauchy-Kovalevskaya Theorem). / Example 2.3. Let X = { X t : t 0 } be the squared Bessel process sat- isfying the SDE, dX t = ndt + 2 X t dW t . Then equation (2.24) implies that u ( x ; t, λ ) = E x £ e - λX t / is the unique solution of the first order PDE u t +2 λ 2 u λ + λnu = 0 , u ( x ; 0 , λ ) = e - λx . This PDE is easily solved by the method of characteristics, giving E x £ e - λX t / = 1 (1+2 λt ) n 2 exp ( - λx 1+2 λt ) .
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• Fall '16
• Dr Salim Zahir
• Fourier Series, Dirac delta function, fundamental solution

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