Paracontrolled Calculus, the Da Prato–Debussche Trick, and the Renormalization¶
Job: develop the Da Prato–Debussche trick (Wick powers / \(\Phi^4_2\)), the Bony paraproduct / paracontrolled (Gubinelli–Imkeller–Perkowski), the paracontrolled solution / fixed point / renormalization, and the Besov / Hölder / Schauder estimates — with the \(\Phi^4_2\) via Da Prato–Debussche / Wick / paracontrolled as the worked example.
Core¶
The Da Prato–Debussche trick and the Wick powers¶
| Da Prato–Debussche / Wick | Description |
|---|---|
| Da Prato–Debussche trick | the \(\Phi=Y+v\) decomposition: \(Y=K*\xi\) the stochastic convolution / the Gaussian, \(v\) the remainder the PDE with random coefficients |
| Wick renormalization | the \(:\!Y^n\!:\) the Wick product / the Hermite (the renormalized products, the \(\Phi^4_2\) the Da Prato–Debussche) |
| \(d=2\) vs \(d=3\) | the \(d=2\) the Da Prato–Debussche sufficient (the \(v\) function-valued), the \(d=3\) the residual roughness the regularity structures / paracontrolled |
| \(\Phi^4_2\) | the \(\partial_t\Phi=\Delta\Phi-\Phi^3+\xi\) in 2D (the dynamical \(\Phi^4\), the Da Prato–Debussche) |
The paracontrolled calculus (Gubinelli–Imkeller–Perkowski)¶
| Bony paraproduct / paracontrolled | Description |
|---|---|
| Bony paraproduct | \(fg=T_f g+T_g f+R(f,g)\) (the paraproduct decomposition, the \(T_f\) the paraproduct, the \(R\) the resonant / the remainder) |
| Paracontrolled | the \(f\) is "paracontrolled" by \(g\) if \(f=T_g u+u'\) for some \(u'\) (the \(f\) the paracontrolled / the expansion) |
| Resonant term \(R(f,g)\) | the resonant / the product (the control via the paracontrolled assumption) |
| Commutator estimates | the commutator / the paracontrolled (the estimates, the Besov / the Hölder) |
The paracontrolled solution of the singular SPDE¶
| Paracontrolled solution / fixed point | Description |
|---|---|
| Paracontrolled solution | \(\Phi=Y+T_\Phi v+v'\) the paracontrolled expansion (the \(v\) the paracontrolled by the stochastic convolution, the \(v'\) the smoother remainder) |
| Fixed point | the mild equation in the paracontrolled / the Besov (the Picard) |
| Renormalization | the \(:\!\Phi^3\!:\) the Wick, the counterterm, the paracontrolled the renormalized |
| Comparison to regularity structures | the paracontrolled lighter / the parabolic-specific, the regularity structures the general / the abstract |
The Besov / the Hölder regularity and the estimates¶
| Besov / Hölder / Schauder | Description |
|---|---|
| Besov \(B^s_{p,q}\) | the regularity of the SPDE solutions (the Besov / the paraproduct, the Fourier-Besov) |
| Hölder \(C^\alpha\) | the regularity of the stochastic convolution (\(Y=K*\xi\) the \(C^\alpha\) for \(\alpha<1-d/2\)) |
| Schauder / fixed-point | \(K*f\in C^{\alpha+2}\) for \(f\in C^\alpha\) (the Schauder, the fixed point in the Besov / the Hölder) |
See it¶
\(\Phi^4_2\) via the Da Prato–Debussche / the Wick renormalization and the paracontrolled. The \(\Phi^4_2\) \(\partial_t\Phi=\Delta\Phi-\Phi^3+\xi\) in 2D (the dynamical \(\Phi^4\)). The Da Prato–Debussche: \(\Phi=Y+v\) (\(Y=K*\xi\) the stochastic convolution, the distribution / the Gaussian, \(v\) the remainder). The Wick: \(:\!\Phi^3\!:=Y^3-3\|Y\|^2Y+v\) (the Wick cube, the renormalization \(-3\|Y\|^2\) the counterterm), the \(v\) solves \(\partial_t v=\Delta v-:\!\Phi^3\!:\) (the renormalized, the \(v\) the function-valued in 2D). The paracontrolled: the \(\Phi=Y+T_\Phi v+v'\) the paracontrolled (the \(\Phi\) paracontrolled by \(Y\), the \(v'\) smoother), the resonant \(R(\Phi^2,\Phi)\) controlled by the paracontrolled, the fixed point in the Besov. The comparison: the \(d=2\) the Da Prato–Debussche / the paracontrolled sufficient (the \(v\) function-valued, the Wick / the counterterm), the \(d=3\) the regularity structures (the residual roughness). The key point: the \(\Phi^4_2\) the Da Prato–Debussche / the Wick / the paracontrolled — the singular SPDE the random PDE after the renormalization, the \(d=2\) the Da Prato–Debussche, the \(d=3\) the regularity structures.
Check¶
Show that the Da Prato–Debussche decomposition \(\Phi=Y+v\) makes the \(v\) function-valued in \(d=2\), and explain the Wick renormalization.
The \(\Phi^4_2\) in 2D: \(\partial_t\Phi=\Delta\Phi-\Phi^3+\xi\). The Da Prato–Debussche: \(\Phi=Y+v\), \(Y=K*\xi\) (the stochastic convolution, the Gaussian, the \(Y\) the distribution / the \(C^\alpha\) for \(\alpha<0\) in \(d=2\): the \(Y\) the \(C^\alpha\) for \(\alpha<1-1=0\), the \(Y\) the distribution-valued). The \(v\): \(\partial_t v=\Delta v-(Y+v)^3\) (the \(v\) the remainder, the \(v\) the function-valued). The Wick renormalization: the \((Y+v)^3=Y^3+3Y^2 v+3Yv^2+v^3\), the \(Y^3\) and \(Y^2 v\) the ill-defined (the \(Y\) the distribution, the products the ill-defined), the Wick: \(:\!Y^3\!:=Y^3-3\|Y\|^2 Y\) (the subtract \(3\|Y\|^2\) the counterterm, the \(Y^3\to:\!Y^3\!:\) the Wick cube), the \(3Y^2 v\to3:\!Y^2\!:v\) (the \(Y^2\to:\!Y^2\!:\) the Wick square, the \(\|Y\|^2\) the counterterm). The \(v\) the function-valued in \(d=2\): the \(v\) the \(C^\beta\) for \(\beta>0\) (the \(v\) the function, the renormalized equation the \(v\) the better regularity), the Da Prato–Debussche the \(v\) function-valued (the \(d=2\) the Wick the sufficient, the \(v\) the function).
Explain the Bony paraproduct \(fg=T_f g+T_g f+R(f,g)\) and the paracontrolled, and the resonant control.
The Bony paraproduct: the Littlewood–Paley decomposition \(f=\sum \Delta_j f\), the \(fg=\sum_{j,k} \Delta_j f\cdot\Delta_k g\). The paraproduct \(T_f g=\sum_{j\le k-2}\Delta_j f\cdot\Delta_k g\) (the low-high, the \(f\) the low frequency, the \(g\) the high, the smooth in \(g\)), \(T_g f=\sum_{k\le j-2}\Delta_k g\cdot\Delta_j f\) (the high-low), \(R(f,g)=\sum_{|j-k|\le1}\Delta_j f\cdot\Delta_k g\) (the resonant / the high-high, the \(j\approx k\), the resonant). The paracontrolled: the \(f\) is paracontrolled by \(g\) if \(f=T_g u+u'\) (the \(f\) the expansion in the paraproduct \(T_g\), the \(u\) the paracontrolled coefficient, the \(u'\) the smoother remainder). The resonant control: the \(R(f,g)\) the resonant, the paracontrolled \(\Rightarrow\) the \(R(f,g)=R(T_g u,g)+R(u',g)\), the \(R(T_g u,g)\) the controlled (the commutator / the paraproduct, the \(T_g u\) the smooth in \(g\), the \(R\) the small), the \(R(u',g)\) the controlled (the \(u'\) the smoother, the \(R\) the small). So the paracontrolled \(\Rightarrow\) the resonant the controlled (the commutator estimates).
Pitfalls¶
- The Da Prato–Debussche the \(d=2\) (the \(d=3\) the regularity structures, the residual roughness) — the \(d=2\) the Wick / the \(v\) function-valued, the \(d=3\) the \(v\) the distribution, the regularity structures.
- The paracontrolled the lighter / the parabolic-specific (the regularity structures the general / the abstract) — the paracontrolled the Bony paraproduct / the parabolic, the regularity structures the general / the abstract.
- The Bony paraproduct the Littlewood–Paley (the \(T_f g\) the low-high, the \(R\) the resonant), the paracontrolled the \(f=T_g u+u'\) — the resonant the \(R(f,g)\) the control via the paracontrolled.
- The Besov / Hölder the regularity (the \(Y\) the \(C^\alpha\) for \(\alpha<1-d/2\), the \(v\) the \(C^\beta\) for \(\beta>0\) the function), the Schauder the \(K*f\in C^{\alpha+2}\) the fixed point.
- The Wick renormalization the \(:\!Y^n\!:\) (the Hermite / the Wick product, the subtract the \(\|Y\|^2\) the counterterm), the renormalized \(\ne\) the original (the counterterm the \(C_\varepsilon\)).
Refs¶
- Da Prato–Debussche, "Strong solutions to the stochastic quantization equations" (Ann. Probab. 2003)
- Gubinelli–Imkeller–Perkowski, "Paracontrolled distributions and singular PDEs" (Forum Math. Pi 2015)
- Bony, "Calcul symbolique et propagation des singularités" (1981); the paraproduct
- Bernstein–Hairer, "Almost sure regularity for the stochastic heat equation" / the \(\Phi^4_2\); the Besov / Hölder estimates
Next¶
SPDE index · Foundations · Regularity Structures · KPZ & Universality · Stochastic NS & Quantization · PDE · Functional analysis