Open Access
November 2007 Variations of the solution to a stochastic heat equation
Jason Swanson
Ann. Probab. 35(6): 2122-2159 (November 2007). DOI: 10.1214/009117907000000196


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.


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Jason Swanson. "Variations of the solution to a stochastic heat equation." Ann. Probab. 35 (6) 2122 - 2159, November 2007.


Published: November 2007
First available in Project Euclid: 8 October 2007

zbMATH: 1135.60041
MathSciNet: MR2353385
Digital Object Identifier: 10.1214/009117907000000196

Primary: 60F17
Secondary: 60G15 , 60G18 , 60H05 , 60H15

Keywords: fractional Brownian motion , iterated Brownian motion , long-range dependence , Quadratic Variation , Quartic variation , Self-similar processes , stochastic integration , Stochastic partial differential equations

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 6 • November 2007
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