Why One Control Can Be Enough for a Stochastic Heat Equation

Can a control acting on a small arc bring an entire noisy temperature profile exactly to zero?

Consider the stochastic heat equation on a circle:

The control acts only in the drift and can use only information already available. Our paper proves that one such control is enough.¹ The cosine is a small example with the genuine difficulty: the noise coefficient depends on space.

An unknown that will not stay in its own frequency

The adjoint equation has two unknowns,

One drift control calls for an estimate observing only \(z\) on \(G\). Familiar stochastic Carleman arguments generally retain a term involving \(Z\). Spectral arguments encounter another obstruction:

A mode outside a frequency cutoff can feed into the retained modes through the unobserved \(Z\). The projected equation no longer closes as it does for spatially constant noise.

The first exponential: move into complex space

Write \(D=-i\partial_x\). Applying \(e^{\sigma D}\) multiplies the \(k\)th Fourier mode by \(e^{\sigma k}\)—equivalently, it translates \(x\) to \(x-i\sigma\).

Work first with finite Fourier approximations, leaving their projections implicit. Set \(X=e^{\sigma D}z\) and \(Y=e^{\sigma D}Z\). The coupling becomes

For \(|\sigma|\le\rho\), the coefficient is bounded by \(\cosh\rho\). The weighted unknown \(Y\) may be huge; the important point is that it appears with a bounded multiplier.

On an interval \([t,s]\), choose \(\sigma(r)=\rho(s-r)/(s-t)\). The translation measures an analytic norm at \(t\) but vanishes at \(s\). Ordinary square-integrable terminal data are therefore enough.

The second exponential: pay for what remains

Itô's formula contributes a positive \(\|Y\|^2\). Together with the coupling, it gives

Now \(Y\) is inside a nonnegative square. The negative remainder involves only \(X\). The moving translation creates one more state-only loss, controlled by \(|\sigma'|^2\|X\|^2/2\) using the Laplacian.

Both losses are compensated by the time weight \(\Gamma(r)=e^{a(r-t)}\), where \(a=\cosh^2\rho+\rho^2/[2(s-t)^2]\). Integrating and taking expectations gives

The opposite translation handles the other Fourier direction. Passing to the limit gives quantitative spatial analyticity of \(z\), without assuming analyticity of \(Z\).

From a local arc to the whole circle

View the analytic state as taking values in \(L^2(\Omega)\). Propagation of smallness transfers a local observation, together with the analytic bound, to a global estimate. A telescoping argument in time removes the intermediate global norms. Only the observation of \(z\) remains; duality supplies the single drift control.

The full result allows higher-dimensional tori and random time-dependent coefficients with a common analytic strip and a pathwise uniform time-integrability bound. General smooth coefficients and domains with boundary require different arguments.

The two exponentials have distinct jobs: the spatial one keeps the coupling usable; the temporal one compensates for the losses left after the stochastic unknown has been put into a positive square.