## The Annals of Probability

### Variations of the solution to a stochastic heat equation

Jason Swanson

#### 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.

#### Article information

Source
Ann. Probab., Volume 35, Number 6 (2007), 2122-2159.

Dates
First available in Project Euclid: 8 October 2007

https://projecteuclid.org/euclid.aop/1191860418

Digital Object Identifier
doi:10.1214/009117907000000196

Mathematical Reviews number (MathSciNet)
MR2353385

Zentralblatt MATH identifier
1135.60041

#### Citation

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

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