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Proof of Feller's Lemma 9 in "Two singular diffusion problems"
Dear all,I've been trying to replicate the proof of Feller's Lemma 9 in his paper Two singular diffusion problems. I don't see how he moves from (6.5) to (6.6) to get to . Can anybody help?Thanks
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Last edited by chertok on March 27th, 2011, 10:00 pm, edited 1 time in total.
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
I went through this paper many years ago. There are many typos. If it helps, here aresome notes on my marked up copy:1. The term 4 b^2 in (6.2) should be replaced by 12. The lhs s in (6.4) should be z3. The denominator (1-c/a) in (6.5) should be Gamma(1-c/a).Also, looking at it now, clearly Feller means "it is well known that e^(A v/s)/s is the Laplace transform ...".So, to confirm the thing, take the Laplace transform of (6.6). You are taking theLT of e^(-C x) BesselI(...) wrt x, so this just means replace s by s + C, and manipulate.
Last edited by Alan on March 27th, 2011, 10:00 pm, edited 1 time in total.
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
Alan, your paper does this for c<0, yes?
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
Sorry, which paper?
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
QuoteOriginally posted by: AlanSorry, which paper?Feller's u_t = (a x u)_xx - ((b x + c) u )_xlooks like your hitting time problem in "Application of eigenfunction expansions..."Sorry I should expand. I am a big admirer of yours and I appreciate your work very much.
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
Thank you -- very kind. You're right that there are some connections with that paper, as an alternative approach to Feller's is to take the Laplace transform with respect to t.Solving the problem that way is very instructive, also.
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
Alan, thanks for the comments. I was able to figure out 2) and 3), and Brecher and Lindsay mentioned 1) in their paper (I haven't got to that point myself). I tried twisting (6.6) this way and that yesterday, but I still can't figure out how to turn into . It looks like everything inside the parentheses for I_0 , except 2, in the first equation should be under the square root. Could you (or anyone on the forum) please provide one or two intermediate steps in the derivation?Thank you so much in advance.
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
The transform is F(s) = int e^(-s x) e^(-C x) I{2 (B v x)^(1/2)} dx, where C= b/[a (e^(bt)-1)] and B = b^2 xi e^(-b t)/[(a^2 (1-e^(-bt))^2](The expression you posted is wrong; it omits the e^(-C x).Now Feller, noted thatint e^(-s x) I{2 (B v x)^(1/2)} dx = e^(B v/s)/sSo, F(s) = e^{B v/(s+C)}/(s+C)It is simple algebra to verify that B v/(s+C) = A v/z, where (A,z) are defined in Feller.So F(s) = e^{A v/z}/(s+C)Remaining left to you ...
Last edited by Alan on March 28th, 2011, 10:00 pm, edited 1 time in total.
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
Thanks, Alan, it worked. The key is to remember that, even though RHS of (6.5) is in terms of , the inverse Laplace transform has to be taken in terms of and recall that , where . Once this is done, (6.6) follows from integrating the series term by term and remembering that and . As Alan mentioned, in (6.6) should not be there, it needs to be replaced by 1.Your help was much appreciated.
Proof of Feller's Lemma 9 in "Two singular diffusion problems"
oops: edit I misunderstood the sde for the FP eqn
Last edited by Orbit on March 30th, 2011, 10:00 pm, edited 1 time in total.