Stochastic Navier–Stokes, Stochastic Quantization, and the Invariant Measures¶
Job: develop the stochastic Navier–Stokes (invariant measure / turbulence / ergodicity, Flandoli–Romito / Hairer–Mattingly), the Malliavin / hypoellipticity / Harris, the stochastic quantization (Parisi–Wu / Gibbs / \(\Phi^4_3\) constructive), and the invariant measures / ergodicity / rate / spectral gap — with the \(\Phi^4_3\) stochastic quantization / invariant measure / constructive QFT as the worked example.
Core¶
The stochastic Navier–Stokes¶
The stochastic Navier–Stokes \(du=(-\Delta u+u\cdot\nabla u-\nabla p)\,dt+\sigma\,dW\) (the Navier–Stokes with the additive / multiplicative noise, the stochastic forcing).
| Stochastic NS / invariant / ergodicity | Description |
|---|---|
| Stochastic NS | \(du=(-\Delta u+u\cdot\nabla u-\nabla p)\,dt+\sigma\,dW\) (the stochastic forcing) |
| Invariant measure / turbulence | the invariant measure the stationary distribution / the turbulence (the Flandoli–Romito, the Hairer–Mattingly) |
| Ergodicity | the unique ergodicity / the Markov semigroup the ergodic (the Hairer–Mattingly the 2D stochastic NS the ergodic, the Flandoli) |
The Malliavin calculus and the hypoellipticity¶
| Malliavin / hypoellipticity / Harris | Description |
|---|---|
| Malliavin calculus | the integration by parts on the Wiener space (the Malliavin / the Sobolev on the path space, the smoothness of the law) |
| Hypoellipticity (Hörmander) | the noise the hypoelliptic (the smoothness of the transition density, the Malliavin the proof) |
| Gradient / Harris | the Harris recurrence / the ergodicity (the gradient / the Malliavin, the Hairer–Mattingly) |
The stochastic quantization and the constructive QFT¶
| Stochastic quantization / Gibbs / constructive | Description |
|---|---|
| Stochastic quantization (Parisi–Wu) | the Euclidean QFT the Langevin dynamics \(\partial_t\Phi=\Delta\Phi-V'(\Phi)+\xi\) (the invariant measure the Euclidean QFT measure) |
| Invariant measure / Gibbs | the \(\Phi^4\) invariant measure \(d\mu\propto e^{-\int V(\Phi)}d\mathcal{D}\Phi\) (the Gibbs / the Euclidean QFT, the stochastic quantization the construction) |
| Ergodicity / constructive | the SPDE well-posedness + the ergodicity \(\Rightarrow\) the constructive QFT control (the Hairer \(\Phi^4_3\) the modern analytic apex) |
The Gibbs measure and the \(\Phi^4\) / the constructive QFT¶
| Gibbs / quantization / constructive / AQFT | Description |
|---|---|
| Gibbs measure | \(d\mu\propto e^{-\int(\Phi^4+m^2\Phi^2)}d\mathcal{D}\Phi\) (the Euclidean \(\Phi^4\) / the constructive QFT measure, the AQFT / constructive connection) |
| Stochastic quantization | the Langevin / the SPDE the dynamics (the invariant measure the Gibbs) |
| Constructive | the SPDE + ergodicity \(\Rightarrow\) the measure exists / the constructive (the Hairer the \(\Phi^4_3\)) |
| AQFT connection | the constructive QFT the rigorous construction (the SPDE the modern approach, the AQFT / constructive) |
The invariant measures and the ergodicity / the long-time¶
| Invariant / ergodicity / rate / spectral gap | Description |
|---|---|
| Invariant measures | the stationary / the invariant (the SPDE the long-time, the convergence to the invariant) |
| Ergodicity | the unique ergodic / the convergence to the invariant (the Hairer–Mattingly, the Flandoli) |
| Rate / spectral gap | the spectral gap / the Harris (the exponential convergence, the ergodic theory / Ornstein the connection) |
See it¶
Stochastic quantization of \(\Phi^4_3\) and the constructive QFT / the invariant measure. The stochastic quantization: the \(\Phi^4_3\) SPDE \(\partial_t\Phi=\Delta\Phi-\Phi^3+\xi\) (the Parisi–Wu, the Langevin). The invariant measure: the \(\Phi^4\) Gibbs measure \(d\mu\propto e^{-\int(\Phi^4+m^2\Phi^2+C_\varepsilon\Phi^2)}d\mathcal{D}\Phi\) (the Euclidean \(\Phi^4_3\), the renormalized the \(C_\varepsilon\) the mass, the constructive QFT measure). The ergodicity: the \(\Phi^4_3\) SPDE the ergodic (the Hairer \(\Phi^4_3\) the well-posedness + the ergodicity \(\Rightarrow\) the invariant measure the unique, the convergence to the Gibbs). The constructive: the SPDE + the ergodicity \(\Rightarrow\) the \(\Phi^4_3\) measure the constructive QFT (the rigorous construction, the AQFT / constructive the modern approach, the Hairer the analytic apex). The key point: the \(\Phi^4_3\) the stochastic quantization / the invariant measure / the ergodicity \(\Rightarrow\) the constructive QFT — the SPDE the modern approach to the constructive QFT, the Hairer \(\Phi^4_3\) the apex.
Check¶
Show that the stochastic quantization of \(\Phi^4_3\) has the Gibbs invariant measure, and explain the constructive QFT.
The stochastic quantization: the \(\Phi^4_3\) SPDE \(\partial_t\Phi=\Delta\Phi-\Phi^3+\xi\) (the Parisi–Wu Langevin). The Gibbs invariant measure: the \(\partial_t\Phi=\Delta\Phi-V'(\Phi)+\xi\) with \(V(\Phi)=\Phi^4/4+m^2\Phi^2/2\) the potential, the invariant measure \(d\mu\propto e^{-\int V(\Phi)}d\mathcal{D}\Phi\) (the Gibbs / the Euclidean \(\Phi^4\), the Langevin the gradient flow + the noise, the invariant the Gibbs). The renormalized: the \(V(\Phi)=\Phi^4/4+(m^2+C_\varepsilon)\Phi^2/2\) (the \(C_\varepsilon\) the mass renormalization, the BPHZ), the \(d\mu\propto e^{-\int(\Phi^4+(m^2+C_\varepsilon)\Phi^2)/2}d\mathcal{D}\Phi\) the renormalized Gibbs. The constructive: the SPDE the well-posedness (the Hairer \(\Phi^4_3\) the regularity structures), the ergodicity (the convergence to the invariant, the unique), the \(\Phi^4_3\) measure the constructive QFT (the rigorous construction of the \(\Phi^4_3\) measure, the AQFT / constructive the modern approach). The key: the SPDE + ergodicity \(\Rightarrow\) the measure the constructive.
Explain the Malliavin / hypoellipticity (Hörmander) and the Harris, and the ergodicity of the stochastic NS.
The Malliavin calculus: the integration by parts on the Wiener space (the \(\mathbb{E}[\partial_i F]=\mathbb{E}[F\cdot L_i]\) the Malliavin / the Sobolev on the path space, the \(L_i\) the Malliavin derivative). The hypoellipticity (Hörmander): the noise the hypoelliptic (the Lie algebra the full, the Hörmander condition, the smoothness of the transition density \(p_t(x,y)\) the \(C^\infty\)). The Malliavin the proof: the Malliavin the integration by parts \(\Rightarrow\) the smoothness of the law (the \(p_t\) the smooth, the Malliavin the proof of the Hörmander). The Harris: the Harris recurrence / the ergodicity (the minorization / the Lyapunov, the Harris the theorem, the exponential convergence / the spectral gap). The stochastic NS: the Malliavin / the hypoellipticity (the noise the hypoelliptic, the transition density the smooth), the Harris (the recurrence / the Lyapunov, the ergodicity), the Hairer–Mattingly the 2D stochastic NS the ergodic (the unique invariant measure, the convergence). The 3D the open (the stochastic NS the 3D the open, the PDE / Navier–Stokes the deterministic the open, the stochastic the partial).
Pitfalls¶
- The stochastic NS the invariant measure (the Flandoli–Romito / the Hairer–Mattingly, the 2D the ergodic, the 3D the open) — the 2D the ergodic (the Hairer–Mattingly), the 3D the open (the stochastic the partial).
- The Malliavin / the hypoellipticity the Hörmander (the noise the hypoelliptic, the smoothness of the law) — the Malliavin the proof, the Hörmander the condition.
- The stochastic quantization the Parisi–Wu (the invariant measure the Gibbs / the Euclidean QFT) — the Langevin the dynamics, the Gibbs the invariant.
- The \(\Phi^4_3\) the Hairer (the well-posedness + the ergodicity \(\Rightarrow\) the constructive, the modern analytic apex) — the SPDE + ergodicity \(\Rightarrow\) the measure the constructive, the Hairer the apex.
- The constructive QFT the AQFT / constructive the modern approach (the SPDE the recent, the classical the Nelson / the Glimm–Jaffe the earlier) — the SPDE the recent, the classical the Nelson / the Glimm–Jaffe.
- The rate / the spectral gap the Harris (the exponential convergence), the ergodic theory / Ornstein the connection (the SPDE the ergodic, the Ornstein the structure).
Refs¶
- Flandoli–Romito, "Markov selections for the stochastic Navier–Stokes" (2008); Hairer–Mattingly, "Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing" (2006)
- Malliavin, Stochastic Analysis (1997); the Hörmander / the hypoellipticity; Harris, "The existence of stationary measures" (the recurrence)
- Parisi–Wu, "Perturbation theory without gauge fixing" (1981); the stochastic quantization
- Hairer, "Solving the KPZ equation" / the \(\Phi^4_3\) (the constructive QFT, the regularity structures); the AQFT / constructive the modern approach
Next¶
SPDE index · Foundations · Regularity Structures · Paracontrolled & Renormalization · KPZ & Universality · PDE / Navier–Stokes · AQFT / constructive · Ergodic theory / Ornstein