arXiv subjects
Publications and source records attributed to Niclas Petersson.
In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T), covered by the process in the time interval [0,T]. The Laplace transform of A(T) follows as a consequence. The main proof involves a double Laplace transform of A(T) and is based on excursion theory and local time for Brownian motion.
We consider $u_t=u^α u_{xxx}+n(u)u_xu_{xx}+m(u)u_x^3+ r(u)u_{xx} +p(u)u_x^2 + q(u)u_x+s(u)$ with $α=0$ and $α=3$, for those functional forms of $m, n, p, q, r, s$ for which the equation is integrable in the sense of an infinite number of Lie-Bäcklund symmetries. Local $x$- and $t$-independent recursion operators that generate these infinite sets of symmetries are obtained for the equations. A combination of potential forms, hodograph transformations and $x$-generalised hodograph transformations are applied to the obtained equations.
In our article "A tree of linearisable second-order evolution equations by generalised hodograph transformations" [J. Nonlin. Math. Phys. {\bf 8} (2001), 342-362] we presented a tree of linearisable (C-integrable) second-order evolution equations in (1+1) dimensions. Expanding this result we report here the complete set of recursion operators for this tree and present several linearisable (C-integrable) hierarchies in (1+1) dimensions.