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Re: Solving a weakly singular integral equation - Take 2.
*To*: mathgroup at smc.vnet.net
*Subject*: [mg54625] Re: Solving a weakly singular integral equation - Take 2.
*From*: Zaeem Burq <Z.Burq at ms.unimelb.edu.au>
*Date*: Thu, 24 Feb 2005 03:21:29 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
Thanks again Paul.
> Where does this function come from? For positive a and t, c only makes
> sense for 0 < t < 1/E.
Let c[t] = (1 + a) Sqrt[2 t Log[Log[1/t]]].
1. This function comes from probability theory. It appears in the Law of
Iterated Logarithm for Brownian motion W[t] which says that the integral
\int_{0}^{t} c(t)p(c(t)/\sqrt{t}) t^{-3/2} dt
converges (diverges) if a > 0 (< 0) respectively. See, for example, Ito,
McKean, Diffusion processes and their sample paths - pg 33.
Related theorems tell us that the function
d[t_]:= (1+a) Sqrt[2t Log[1/t]]
exhibits similar behaviour near zero (i.e., the integral converges if a
> 0), but may be easier for Mathematica to handle.
2. Define T = inf {t : W[t] \geq c[t] }.
This random time T is the first time that a trajectory of Brownian motion
hits c[t]. If a > 0, then the random variable T has a density f given by
the integral equation I've been trying to solve.
3. Also, I am no expert in integral equations, but by for a Volterra type
2 eqn.
f[t] = g[t] + \int_{0}^{t} K[s,t] f[s] ds, for t \in [0,S]
to have a unique bounded solution, isn't it enough that the inhomogeneous
function g be integrable on [0,S], and the kernel K be integrable in the
triangle 0 \leq t \leq S, and 0 \leq s \leq t? (We'll let S = 1/e here.)
For example, see Jerri, Introduction to integral equations with
applications, Theorem 1 on page 137.
Integrability of the inhomogeneous function is guaranteed by a > 0 in the
Law of Iterated Logarithm above.
For the integrability of the kernel, see
http://www.ms.unimelb.edu.au/~zab/paul.pdf
4. That business about p killing the singularity rubbish rubbish. But it
seems that this is besides the point.
5. I am very new to both Mathematica and to programming. I hope you'll
forgive my fumbles.
Best, Zaeem.
________________________________
Zaeem Burq
PhD Stochastic Processes,
Dept. of Mathematics and Statistics,
Unimelb.
Room 201, Richard Berry Building
University of Melbourne,
Parkville, VIC 3052.
ph: 8344 4248.
http://www.ms.unimelb.edu.au/~zab
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