The invariant-measure conjecture for geometric stochastic heat equations

Let $(\mathcal{M},g)$ be a Riemannian manifold, let $A$ be a smooth vector field, and let $\mu$ be the loop measure determined by the diffusion with generator

$$L f = \frac{1}{2}\Delta f + df(A).$$

Assume that the diffusion is nonexplosive in finite time and that $x \mapsto p_1(x,x)$ is integrable on $\mathcal{M}$. Let $\sigma_i$ be any collection of smooth vector fields such that $g=\sigma_i\otimes\sigma_i$, and let $\Gamma^\alpha_{\beta\gamma}$ be the Christoffel symbols of the Levi-Civita connection on $\mathcal{M}$. Invariant-measure conjecture. There exists a universal constant $c$ such that the unique process given by the canonical solution of Theorem~ for the stochastic PDE

$$\partial_t u = \nabla_{\partial_xu}\partial_xu + \bigl((dA^\flat)(\partial_xu)\bigr)^\sharp - \nabla_AA - \frac{1}{2}\nabla\operatorname{div}A + c\nabla R(u) + \sqrt{2}\sigma_i(u)\,\xi_i$$

where $R$ denotes the scalar curvature of $\mathcal{M}$, exists for all times and has $\mu$ as its unique invariant measure. In the corresponding general equation, the coefficients are $h=-\nabla_AA-\frac{1}{2}\nabla\operatorname{div}A+c\nabla R$ and $K$ is the tensor field satisfying $K\partial_xu=\bigl((dA^\flat)(\partial_xu)\bigr)^\sharp$. This conjectures the geometric analogue of the Euclidean path-integral invariant-measure result; its validity and the universal value of $c$ are not established in the supplied text.