Hairer Regularity Structures and the Renormalization of Singular SPDEs¶
Job: develop the regularity structure framework (model / modelled distributions / reconstruction), the renormalization / BPHZ (counterterms / universality), the subcriticality / scaling / power-counting, and the local well-posedness / fixed point — with the \(\Phi^4_3\) via regularity structures / BPHZ as the worked example.
Core¶
The regularity structure framework (Hairer)¶
The regularity structure (the Hairer 2014 framework: the abstract algebraic structure — the model, the regularity structure, the reconstruction).
| Regularity structure | Description |
|---|---|
| Model \(\Pi=(\Pi_x)\) | the local distributions / the jets of the renormalized multilinear functionals of the noise \(\xi\) (the canonical model from the noise) |
| Modelled distributions \(\mathcal{D}^\gamma\) | the functions valued in the regularity structure (the local expansions / the jets) |
| Reconstruction theorem | \(\mathcal{R}:\mathcal{D}^\gamma\to\mathcal{D}'\) (the reconstruction operator, the map from the modelled distributions to the space-time distributions, the continuous / the consistent) |
The renormalization and the BPHZ¶
| Renormalization / BPHZ | Description |
|---|---|
| Renormalization | the change of the model: subtract the counterterm, the renormalized model \(\hat\Pi\) |
| BPHZ | the Bogoliubov–Parasiuk–Hepp–Zimmermann / the Hairer (the renormalization of the regularity structures) |
| Counterterms | the \(C_\varepsilon\to\infty\) the divergent constants, the renormalized equation the original + the counterterm (\(\Phi^4_3\) the \(C_\varepsilon\Phi\) the mass renormalization) |
| Universality | the renormalized model independent of the noise regularization (the continuous dependence on the model, the convergence) |
The subcriticality and the scaling / the power-counting¶
| Subcriticality / scaling | Description |
|---|---|
| Subcriticality | the scaling analysis: the nonlinearity less singular than the linear part at small scales (the power-counting, the local well-posedness after renormalization) |
| Critical / supercritical | the borderline / the no-perturbative-room (the open / the different methods) |
| Regularity structure vs scaling | the regularity structure the framework for the subcritical; the critical / supercritical the open |
The local well-posedness and the fixed point¶
| Fixed point / well-posedness | Description |
|---|---|
| Fixed point | the mild equation \(X=K*\mathrm{noise}+K*F(X)\) in the regularity structure (the Picard / the fixed point in \(\mathcal{D}^\gamma\)) |
| Local well-posedness | the subcritical \(\Rightarrow\) the local solution in \(\mathcal{D}^\gamma\) (the reconstruction to the distribution) |
| Continuous dependence / stability | the solution depends continuously on the model / the noise (the universality) |
See it¶
\(\Phi^4_3\) via the regularity structures and the BPHZ renormalization. The \(\Phi^4_3\) equation \(\partial_t\Phi=\Delta\Phi-\Phi^3+\xi\) (the dynamical \(\Phi^4\) in 3D, the Parisi–Wu / the stochastic quantization). The subcriticality: the \(\Phi^3\) the subcritical in 3D (the scaling analysis: the \(\Phi^3\) less singular than the linear at small scales, the power-counting the subcritical). The regularity structure: the model \(\Pi\) from the noise \(\xi\) (the stochastic convolution \(Y=K*\xi\), the Wick powers \(:\!Y^n\!:\), the canonical model), the modelled distributions \(\mathcal{D}^\gamma\) (the \(\Phi\) as the modelled distribution, the local expansion in the jets). The renormalization: the BPHZ renormalization (the \(:\!Y^3\!:\) the Wick cube, the counterterm \(C_\varepsilon\Phi\) the mass renormalization, the \(C_\varepsilon\to\infty\)), the renormalized model \(\hat\Pi\) (the BPHZ, the universal). The fixed point: the \(\Phi=K*\xi+K*(-\Phi^3+C_\varepsilon\Phi)\) in \(\mathcal{D}^\gamma\) (the renormalized, the local well-posedness). The reconstruction: the \(\mathcal{R}(\Phi)\) the space-time distribution (the solution). The key point: the \(\Phi^4_3\) the regularity structure / the BPHZ renormalization / the local well-posedness — the singular SPDE made rigorous, the constructive QFT the dynamical \(\Phi^4_3\).
Check¶
Show that the \(\Phi^4_3\) is subcritical (the power-counting), and explain the renormalization.
The \(\Phi^4_3\) equation \(\partial_t\Phi=\Delta\Phi-\Phi^3+\xi\) in 3D. The scaling: the parabolic scaling \(x\to\lambda x\), \(t\to\lambda^2 t\), the linear part the \(\Delta\) the order \(-2\) (the \(\Delta\Phi\) the \(\lambda^{-2}\Phi\)), the noise \(\xi\) the order \(-d/2-1=-5/2\) (the white noise the \(-5/2\) in 3D parabolic), the stochastic convolution \(Y=K*\xi\) the order \(-5/2+2=-1/2\) (the \(K\) the \(+2\), the \(Y\) the \(-1/2\)). The nonlinearity \(\Phi^3\): if \(\Phi\) the order \(\alpha\) (the \(Y\) the \(-1/2\)), then \(\Phi^3\) the order \(3\alpha\). The subcritical: \(3\alpha<\alpha+2\) (the nonlinearity less singular than the linear part \(\Delta\Phi\) the \(\alpha-2\)) \(\Leftrightarrow\) \(2\alpha<2\) \(\Leftrightarrow\) \(\alpha<1\). The \(Y\) the \(-1/2<1\) (the subcritical, the power-counting the room). The renormalization: the \(\Phi^3\) the product of the distributions (the \(Y^3\) the ill-defined, the \(Y\) the \(-1/2<0\)), the Wick renormalization \(:\!Y^3\!:\) (the Hermite / the Wick product, the subtract \(3\|Y\|^2 Y\) the counterterm), the BPHZ the renormalized model (the \(C_\varepsilon\Phi\) the mass renormalization, the \(C_\varepsilon\to\infty\)). The subcritical \(\Rightarrow\) the regularity structure the local well-posedness after the renormalization.
Explain the reconstruction theorem \(\mathcal{R}:\mathcal{D}^\gamma\to\mathcal{D}'\), and why it is the key output.
The modelled distributions \(\mathcal{D}^\gamma\) (the functions valued in the regularity structure, the local expansions \(f(x)=\sum_\alpha f_\alpha(x)\tau_\alpha\) the jets in the regularity structure, the \(\gamma\) the regularity). The reconstruction theorem: the \(\mathcal{R}:\mathcal{D}^\gamma\to\mathcal{D}'\) the reconstruction operator (the map from the modelled distributions to the space-time distributions), the \(\mathcal{R}(f)\) the distribution that locally looks like \(\Pi_x f(x)\) (the model applied to the jet, the local distribution). The consistency: the \(\mathcal{R}(f)-\Pi_x f(x)\) the small (the \(\gamma\)-regular, the local expansion the consistent). The continuous: the \(\mathcal{R}\) the continuous (the \(\|\mathcal{R}(f)\|\le C\|f\|_{\mathcal{D}^\gamma}\), the Lipschitz). The key output: the regularity structure the abstract (the model / the modelled distributions), the reconstruction the concrete (the distribution the solution). The fixed point in \(\mathcal{D}^\gamma\) (the abstract), the reconstruction to \(\mathcal{D}'\) (the concrete the distribution the solution). So the reconstruction the key output: the abstract framework to the concrete distribution.
Pitfalls¶
- The regularity structures the subcritical (the critical / supercritical the open) — the regularity structure the framework for the subcritical, the critical / supercritical the open / the different methods.
- The renormalization / the BPHZ the universal (the renormalized model independent of the regularization, the universality — but the counterterms the \(C_\varepsilon\to\infty\) the divergent, the renormalized equation \(\ne\) the original). The renormalized equation the original + the counterterm (the \(C_\varepsilon\Phi\)), the renormalized \(\ne\) the formal.
- The model \(\Pi\) the canonical (from the noise) vs the renormalized \(\hat\Pi\) (the BPHZ) — the two the different (the canonical the ill-defined, the renormalized the well-defined).
- The subcriticality the scaling / the power-counting (the nonlinearity less singular than the linear) — the borderline the critical, the beyond the supercritical.
- The continuous dependence / the universality the model the convergence (the renormalized model \(\hat\Pi_\varepsilon\to\hat\Pi\) the BPHZ, the independent of the \(\varepsilon\)), the solution the convergence (the \(\Phi_\varepsilon\to\Phi\) the universal).
Refs¶
- Hairer, "A theory of regularity structures" (Invent. Math. 2014); Hairer, "Regularity structures and renormalisation" (2015)
- Bruned–Chandra–Hairer–Zambotti, "Renormalisation of regularity structures" (2019); the BPHZ for the regularity structures
- Chandra–Hairer, An Introduction to Regularity Structures (2018, lecture notes)
- Hairer, "Solving the KPZ equation" (Annals 2013); the \(\Phi^4_3\) the constructive QFT
Next¶
SPDE index · Foundations · Paracontrolled & Renormalization · KPZ & Universality · Stochastic NS & Quantization · PDE · AQFT / constructive