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