Statistical inference for rough Burgers-type stochastic partial differential equations

Source abstract

We consider the Burgers-type stochastic PDE $$du = θ\,\partial_x^2 u\,dt + f(u)\,dt + g(u)\,\partial_x u\,dt + σ\,dW(t),$$ driven by space--time white noise on the one-dimensional torus, where $θ>0$ is the diffusivity parameter and $σ>0$ the noise intensity. The transport term $g(u)\,\partial_x u$ admits no classical interpretation, and we work with the mild solutions defined via rough paths following~\cite{Hairer2011}, for which the solution decomposes as $u=ψ+R$ with $ψ$ a stationary centred Gaussian process solving a damped stochastic heat equation and $R$ a remainder. We first show that, away from the initial time, $R$ is Lipschitz in space and Hölder in time of order strictly greater than $1/2$, hence strictly smoother than $ψ$ in both variables. We then compute exactly the spatial quadratic variation and the temporal quartic variation of $ψ$, almost surely and not merely in $L^{2}(Ω)$, and prove that the corresponding variations of $u$ are unaffected by the remainder. Combined with an exact time-scaling identity, this separation of regularity scales yields strongly consistent pathwise estimators of $θ$. To our knowledge this is the first inference procedure for an SPDE whose solution is defined only pathwise, as a measurable function of a rough path, rather than as an Itô process.