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For the distribution-valued field called the GFF (Gaussian free field) over a domain $\Omega\subset \mathbb{R}^{2}$: $$ h_{\Omega}=\sum a_{k}f_{k,\Omega}(z),$$

where $a_{k}\sim N(0,1)$ and $f_{k},\Omega$ are an orthonormal basis for $H^{1}(\Omega)$ with the Dirichlet inner product $\langle g,f\rangle=\int_{\Omega} \nabla g \nabla f dz$, we have a generalization of the Markov property: for compact $D\subset \Omega$, there is a random harmonic function $g_{D,\Omega}$ s.t. we can decompose the field as $$h_{\Omega}=h_{D}+g_{D,\Omega},$$ where $h_{D}:=\sum b_{k}f_{k,D}(z)$ for $b_{k}\sim N(0,1)$ and it is independent of $g_{D,\Omega}$. In words, the field $h_{D}$ depends only to the filtration information up to the boundary $\partial D$ and fields supported outside D are independent of it.

The additive stochastic heat equation (SHE) is $$ v(t,z)=\Delta v(t,z)+\xi(t,z), $$ where $\xi(t,z)$ is a space time white noise in domain $\Omega$ and we will let $v_{0}\equiv 0$ and zero boundary data. The stationary solution for SHE is the GFF.

Q1: can we obtain a similar property for SHE ? Namely $$v=v_{D}+g,$$ where $v_{D}$ solves the SHE in $D\subset \Omega$ and it is independent of g.

In The Stochastic Heat Equation Driven by a Gaussian Noise: germ Markov Property,they prove a markov property for SHE where $\xi(t,z)$ is white in time and with a Bessel-kernel correlation in space. Using the reproducing kernel Hilbert space method (RKHS), they manage to write: $$v=v_{1}+v_{2},$$ where $v_{1}\in H_{\bar{D}},v_{2}\in H_{\bar{D^{c}}}\setminus H_{\partial D}$ and $H_{D}$ is the Hilbert space for SHE generated from correlations of v evaluated in $D\subset \Omega$ via RKHS.

Q2: can this result be extended to space-time white noise?

Motivation

This property was essential in building a measure based on GFF, which was then used to study GFF's "zero level set" and to identify it as SLE(4), a random planar curve that enjoys conformal invariance.

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