The Annals of Probability
- Ann. Probab.
- Volume 35, Number 6 (2007), 2122-2159.
Variations of the solution to a stochastic heat equation
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.
Ann. Probab. Volume 35, Number 6 (2007), 2122-2159.
First available in Project Euclid: 8 October 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G15: Gaussian processes 60G18: Self-similar processes 60H05: Stochastic integrals 60H15: Stochastic partial differential equations [See also 35R60]
Quartic variation quadratic variation stochastic partial differential equations stochastic integration long-range dependence iterated Brownian motion fractional Brownian motion self-similar processes
Swanson, Jason. Variations of the solution to a stochastic heat equation. Ann. Probab. 35 (2007), no. 6, 2122--2159. doi:10.1214/009117907000000196. https://projecteuclid.org/euclid.aop/1191860418.