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What kind of approaches can be used to study the following quasilinear parabolic pde for a scalar function $u=u(x,t)$ ? $$ u_t= u \ u_{x x} $$ The physical problem where this pde comes from dictates …

It turns out that the above generating function pde is related to a Lax pair of Painlevé III/Sine-Gordon, at least for the parameter value g=2. As soon as I have written this up, I'll post details

This differential-difference three-term recursion occurs in conjunction with the study of an imaginary exponential functional of Brownian motion (another question I asked on MO). For those interested …

For those interested, we have released today an ArXiv preprint with what we were able to find: http://arxiv.org/abs/1101.1173

Thanks to the work by M. Yor and colleagues, much is known about the following exponential of Brownian motion: $X= \int_0^{\infty}{\rm d}t \ e^{-t + g \ B(t)}$ where $g$ is a real scale parameter. …

Thanks but yes! Actually the problem as stated comes from the Fourier coefficients of the solution of the heat-like pde (with initial conditions $G(x,0)=1 \ \forall x$): $t \partial_t G(x,t) - \part …

Trying to solve a PDE coming from the computation of some functional of Brownian motion, I have came across the following Bessel-looking like functional recurrence: $n^2 g_n(t) + t g_n'(t) = (t/2)(g …