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 It\^{o} 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.Comment: 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

We consider the solution u(x, t) to a stochastic heat equation. For fixed x, the process F (t) = u(x, t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions g(x, t), a stochastic integral g(F (t), t) dF (t) exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of F . . This reprint differs from the original in pagination and typographic detail. 1 2 K. BURDZY AND J. SWANSON 0, whereẆ is a space-time white noise on R × [0, ∞). That is, u(x, t) = R×[0,t]

Let B be a fractional Brownian motion with Hurst parameter H = 1/6. It is known that the symmetric Stratonovich-style Riemann sums for g(B(s)) dB(s) do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of B.as the mesh of the partition {t j } goes to zero. Typically, we regard (1.1) as a process in t, and require that it converges uniformly on compacts in probability (ucp). * Supported in part by the (french) ANR grant 'Exploration des Chemins Rugueux'. † Supported by DFG research center Matheon project E2. ‡ Supported in part by NSA grant H98230-09-1-0079.This is closely related to the so-called symmetric integral, denoted by t 0 X(s) d • Y (s), which is the ucp limit, if it exists, of2) as ε → 0. The symmetric integral is an example of the regularization procedure, introduced by Russo and Vallois, and on which there is a wide body of literature. For further details on stochastic calculus via regularization, see the excellent survey article [13] and the many references therein. A special case of interest that has received considerable attention in the literature is when Y = B H , a fractional Brownian motion with Hurst parameter H. It has been shown independently in [2] and [5] that when Y = B H and X = g(B H ) for a sufficiently differentiable function g(x), the symmetric integral exists for all H > 1/6. Moreover, in this case, the symmetric integral satisfies the classical Stratonovich change of variable formula,g(B H (t)) = g(B H (0)) + t 0 h n (x) = (−1) n e x 2 /2 d n dx n (e −x 2 /2 ). (2.1) Note that the first few Hermite polynomials are h 0 (x) = 1, h 1 (x) = x, h 2 (x) = x 2 − 1, and h 3 (x) = x 3 − 3x. The following orthogonality property is well-known: if U and V are jointly normal with E(U) = E(V ) = 0 and E(U 2 ) = E(V 2 ) = 1, then E[h p (U)h q (V )] = q!(E[UV ]) q if p = q, 0 otherwise. (2.2)

We consider the median of n independent Brownian motions, denoted by M n (t), and show that √ n M n converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the increments of the median process. An explicit formula is given for the covariance function of the limit process. The limit process is also shown to be Hölder continuous with exponent γ for all γ < 1/4.

The transmission control protocol (TCP) is a transport protocol used in the Internet. In Ott (2005), a more general class of candidate transport protocols called ‘protocols in the TCP paradigm’ was introduced. The long-term objective of studying this class is to find protocols with promising performance characteristics. In this paper we study Markov chain models derived from protocols in the TCP paradigm. Protocols in the TCP paradigm, as TCP, protect the network from congestion by decreasing the ‘congestion window’ (i.e. the amount of data allowed to be sent but not yet acknowledged) when there is packet loss or packet marking, and increasing it when there is no loss. When loss of different packets are assumed to be independent events and the probabilitypof loss is assumed to be constant, the protocol gives rise to a Markov chain {Wn}, whereWnis the size of the congestion window after the transmission of thenth packet. For a wide class of such Markov chains, we prove weak convergence results, after appropriate rescaling of time and space, asp→ 0. The limiting processes are defined by stochastic differential equations. Depending on certain parameter values, the stochastic differential equation can define an Ornstein-Uhlenbeck process or can be driven by a Poisson process.