Variations of the solution to a stochastic heat equation
Abstract
We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, $F(t)$, has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical Itô calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of $t$, converge weakly to Brownian motion.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2005
 arXiv:
 arXiv:math/0601007
 Bibcode:
 2006math......1007S
 Keywords:

 Mathematics  Probability;
 60F17 (Primary) 60G15;
 60G18;
 60H05;
 60H15 (Secondary)
 EPrint:
 Published in at http://dx.doi.org/10.1214/009117907000000196 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)