Good question! Before going further in your investigations on rigorous nonperturbative implementations of the renormalization group (RG) philosophy used for the construction of QFTs in the continuum, you should at least read my previous answer

https://physics.stackexchange.com/questions/372306/what-is-the-wilsonian-definition-of-renormalizability/375571#375571

If you have time, also look at

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

There is nothing sacrosanct about Polchinski's approach. It is just one among many ways of implementing the RG method. It is characterized by the use of a continuous flow, or ODE on a very infinite-dimensional space of effective actions/potentials. It looks nonperturbative but it isn't really because no one was able to find norms on space of functionals (functions of functions) where one can prove a local in time well posedness result for this ODE. As far as I know, there is only one rigorous result with a continuous flow RG: the article "Continuous Constructive Fermionic Renormalization" by Disertori and Rivasseau in Annales Henri Poincaré 2000. They didn't use Polchinski's equation but the older Calan-Symanzik equation, and this only works for Fermions which are easier than Bosons because perturbation series converge in the case of Fermion (with cutoffs).

As to what has been done rigorously with Polchinski's equations, it concerns rigorous proofs of perturbative renormalizability, i.e., in the sense of formal power series. A good introduction to this is the book "Renormalization: An Introduction" by Manfred Salmhofer. For recent results with this approach see the works of Christoph Kopper and Stefan Hollands.

For nonperturbative rigorous results for Bosons, at least for what is known so far, one has to abandon the idea of continuous flow as in Polchinski's equation and work with a discrete RG transformation. Again there are several approaches. If you want something that is closest to the Polchinski philosophy, then you might want to look at the approach by David Brydges and collaborators developed over many years and culminating with a series of five articles in J. Stat. Phys., with Gordon Slade and Roland Bauerschmidt. An pedagogical introduction is now available with the book "Introduction to a Renormalisation Group Method" by these three authors.

Good question! Before going further in your investigations on rigorous nonperturbative implementations of the renormalization group (RG) philosophy used for the construction of QFTs in the continuum, you should at least read my previous answer

https://physics.stackexchange.com/questions/372306/what-is-the-wilsonian-definition-of-renormalizability/375571#375571

If you have time, also look at

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

There is nothing sacrosanct about Polchinski's approach. It is just one among many ways of implementing the RG method. It is characterized by the use of a continuous flow, or ODE on a very infinite-dimensional space of effective actions/potentials. It looks nonperturbative but it isn't really because no one was able to find norms on space of functionals (functions of functions) where one can prove a local in time well posedness result for this ODE. As far as I know, there is only one rigorous nonperturbative result with a continuous flow RG: the article "Continuous Constructive Fermionic Renormalization" by Disertori and Rivasseau in Annales Henri Poincaré 2000. They didn't use Polchinski's equation but the older Callan-Symanzik equation, and this only works for Fermions which are easier than Bosons because perturbation series converge in the case of Fermion (with cutoffs).

As to what has been done rigorously with Polchinski's equations, it concerns rigorous proofs of perturbative renormalizability, i.e., in the sense of formal power series. A good introduction to this is the book "Renormalization: An Introduction" by Manfred Salmhofer. For recent results with this approach see the works of Christoph Kopper and Stefan Hollands.

For nonperturbative rigorous results for Bosons, at least for what is known so far, one has to abandon the idea of continuous flow as in Polchinski's equation and work with a discrete RG transformation. Again there are several approaches. If you want something that is closest to the Polchinski philosophy, then you might want to look at the approach by David Brydges and collaborators developed over many years and culminating with a series of five articles in J. Stat. Phys., with Gordon Slade and Roland Bauerschmidt. A pedagogical introduction is now available with the book "Introduction to a Renormalisation Group Method" by these three authors.

Good question! Before going further in your investigations on rigorous nonperturbative implementations of the renormalization group (RG) philosophy used for the construction of QFTs in the continuum, you should at least read my previous answer

https://physics.stackexchange.com/questions/372306/what-is-the-wilsonian-definition-of-renormalizability/375571#375571

If you have time, also look at

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

There is nothing sacrosanct about Polchinski's approach. It is just one among many ways of implementing the RG method. It is characterized by the use of a continuous flow, or ODE on a very infinite-dimensional space of effective actions/potentials. It looks nonperturbative but it isn't really because no one was able to find norms on space of functionals (functions of functions) where one can prove a local in time well posedness result for this ODE. As far as I know, there is only one rigorous result with a continuous flow RG: the article "Continuous Constructive Fermionic Renormalization" by Disertori and Rivasseau in Annales Henri Poincaré 2000. They didn't use Polchinski's but the older Calan-Symanzik equation, and this only works for Fermions which are easier than Bosons because perturbation series converge in the case of Fermion (with cutoffs).

As to what has been done rigorously with Polchinski's equations, it concerns rigorous proofs of perturbative renormalizability, i.e., in the sense of formal power series. A good introduction to this is the book "Renormalization: An Introduction" by Manfred Salmhofer. For recent results with this approach see the works of Chritoph Kopper and Stefan Hollands.

For nonperturbative rigorous results for Boson, at least for what is known so far, one has to abandon the idea of continuous flow as in Polchinski's equation and work with a discrete RG transformation. Again there are several approaches. If you want something that is closest to the Polchinski philosophy, then you might want to look at the approach by David Brydges and collaborators developed over many years and culminating with a series of five articles in J. Stat. Phys. with Gordon Slade and Roland Bauerschmidt. An pedagogical introduction is now available in the book "Introduction to a Renormalisation Group Method" by these three authors.

Good question! Before going further in your investigations on rigorous nonperturbative implementations of the renormalization group (RG) philosophy used for the construction of QFTs in the continuum, you should at least read my previous answer

https://physics.stackexchange.com/questions/372306/what-is-the-wilsonian-definition-of-renormalizability/375571#375571

If you have time, also look at

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

There is nothing sacrosanct about Polchinski's approach. It is just one among many ways of implementing the RG method. It is characterized by the use of a continuous flow, or ODE on a very infinite-dimensional space of effective actions/potentials. It looks nonperturbative but it isn't really because no one was able to find norms on space of functionals (functions of functions) where one can prove a local in time well posedness result for this ODE. As far as I know, there is only one rigorous result with a continuous flow RG: the article "Continuous Constructive Fermionic Renormalization" by Disertori and Rivasseau in Annales Henri Poincaré 2000. They didn't use Polchinski's equation but the older Calan-Symanzik equation, and this only works for Fermions which are easier than Bosons because perturbation series converge in the case of Fermion (with cutoffs).

As to what has been done rigorously with Polchinski's equations, it concerns rigorous proofs of perturbative renormalizability, i.e., in the sense of formal power series. A good introduction to this is the book "Renormalization: An Introduction" by Manfred Salmhofer. For recent results with this approach see the works of Christoph Kopper and Stefan Hollands.

For nonperturbative rigorous results for Bosons, at least for what is known so far, one has to abandon the idea of continuous flow as in Polchinski's equation and work with a discrete RG transformation. Again there are several approaches. If you want something that is closest to the Polchinski philosophy, then you might want to look at the approach by David Brydges and collaborators developed over many years and culminating with a series of five articles in J. Stat. Phys., with Gordon Slade and Roland Bauerschmidt. An pedagogical introduction is now available with the book "Introduction to a Renormalisation Group Method" by these three authors.