# Additive stochastic heat equation and Markov property

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.