Return to Answer

It is hard to say more as an MO answer, but you can see these preprints for further explanations:

1. "A second-quantized Kolmogorov-Chentsov theorem"
2. "Towards three-dimensional conformal probability"
3. "QFT, RG, and all that, for mathematicians, in eleven pages"

It is hard to say more as an MO answer, but you can see these papers for further explanations:

1. "A second-quantized Kolmogorov-Chentsov theorem via the operator product expansion", published here.
2. "Towards three-dimensional conformal probability", published here.
3. "QFT, RG, and all that, for mathematicians, in eleven pages"

Update: The article by Mourrat and Weber mentioned by Martin in his comment below has now appeared in CMP, see here. It provides a new construction of the scalar $\phi^4$ model in three dimensions, in finite volume.

Update (Jan 27, 2018): The article by Mourrat and Weber mentioned by Martin in his comment below has now appeared in CMP, see here. It provides a new construction of the scalar $\phi^4$ model in three dimensions, in finite volume.

Update (Jan 21, 2019): Progress in the area has been quite fast and, since my last update, the limitation to finite volume has been overcome as mentioned in Martin's comment below. For the infinite volume treatment of $\phi_3^4$ via the stochastic quantization method see this article by Gubinelli and Hofmanová (with the paracontrolled approach) and the one by Moinat and Weber mentioned below (with the regularity structures approach).

Update: The article by Mourrat and Weber mentioned by Martin in his comment below has now appear in CMP, see here. It provides a new construction of the scalar $\phi^4$ model in three dimensions, in finite volume.

Update: The article by Mourrat and Weber mentioned by Martin in his comment below has now appeared in CMP, see here. It provides a new construction of the scalar $\phi^4$ model in three dimensions, in finite volume.