Stochastic-integral representation conjecture for the renormalised Amperean area

Let $W$ be the Brownian motion and let $G$ be the Green kernel appearing in the definition of the renormalised Amperean area. Write $I_W$ for the recentred self-intersection local time, and interpret ${\rm d}\overleftarrow{W}_t$ as a backward stochastic integral. Stochastic-integral representation conjecture. The renormalised Amperean area $\tilde{\mathcal{A}}_W$ can be expressed as

$$\mathcal{A}_W=-2\int_0^{\bar{T}}\int_0^sG(W_s-W_t)\,{\rm d}\overleftarrow{W}_t\,{\rm d}W_s-I_W/2$$

and equivalently as

$$\mathcal{A}_W=-2\int_0^{\bar{T}}\int_0^sG(W_s-W_t)\,{\rm d}W_t\,{\rm d}W_s+I_W/2.$$

The counterterm needed to define $\mathcal{A}_W$ is absorbed into the counterterm defining $I_W$. The corresponding Stratonovich integral is not known to be well-defined, although it may be defined as the average of the backward and forward integrals.