On the distribution and q-variation of the solution to the heat equation with fractional Laplacian

Abstract

We study the probability distribution of the solution to the linear stochastic heat equation with fractional Laplacian and white noise in time and white or correlated noise in space. As an application, we deduce the behavior of the q-variations of the solution in time and in space.

Estimation of the drift parameter for the fractional stochastic heat equation via power variation

“…for every T > 0 and for every measurable square integrable function H. For θ = 1, the solution to the heat equation (3) has been studied in [13]. This solution exists only if the spatial dimension is d = 1, and it is connected to the bifractional Brownian motion.…”

“…In particular, for K = 1, B H, := B H,1 is the fractional Brownian motion (fBm in the sequel) with the Hurst parameter H ∈ (0, 1). Let us recall some of the results in [13] which will be needed in the sequel.…”

“…Since the process (5) is connected to the perturbed fBm (i.e., the sum of a fBm and a smooth Gaussian process), let us recall some facts concerning the asymptotic behavior of the variation of the perturbed fBm. Some of the below results are directly taken from [13] while those concerning the rate of convergence under the Wasserstein distance are deduced from [19].…”

“…Let us recall the following result (see [13]) concerning the exact variation of the perturbed fractional Brownian motion, i.e., the sum of a fBm and a smooth Gaussian process. In the rest of this section, we will fix an interval [A 1 , A 2 ] with A 1 < A 2 and a partition t j = A 1 + j n (A 2 − A 1 ), n ≥ 1, j = 0, .…”

“…Our approach to construct and analyze the estimators for the drift parameter is based on the asymptotic behavior of the q-variations of the mild solution to (1). It is well known (see, e.g., [7,13,23]) that there exists a strong link between the law of this mild solution with θ = 1 and the fractional Brownian motion and related processes. We will use this connection in order to deduce the behavior of the q-variations (of suitable order q) of the solutions to (1) and to prove the consistency, asymptotic normality and Berry-Esséen bounds under the Wasserstein distance for the associated estimators.…”