Stochastic Partial Differential Equations¶
Job: organize SPDE as stochastic evolution equations — noise regularity, mild solutions, and the singular regime (Hairer / paracontrolled).
Core¶
SPDEs couple PDE dynamics with space-time noise. The analytic difficulty is that white noise is too rough for classical products; solution theory tracks regularity (Hölder / Besov) against the scaling of the equation.
| Layer | Content |
|---|---|
| Motivations | continuum limits, stochastic quantization, fluid models |
| Noise | space-time white noise; cylindrical Wiener; colored noise |
| Mild solutions | Duhamel + stochastic convolution |
| Regular regime | globally Lipschitz / monotone nonlinearities in Hilbert spaces |
| Singular regime | regularity structures; paracontrolled distributions |
| Quantization | \(\Phi^4_d\) as SPDE (Da Prato–Debussche / Hairer) |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | motivations, white noise, mild solutions, regularity structures pointer, stochastic quantization |
| Regularity Structures | Hairer regularity structures / model / modelled distributions / reconstruction; BPHZ renormalization / counterterms / universality; subcriticality / scaling; local well-posedness / fixed point |
| Paracontrolled & Renormalization | Da Prato–Debussche / Wick / \(\Phi^4_2\); Bony paraproduct / paracontrolled (Gubinelli–Imkeller–Perkowski); paracontrolled solution / fixed point / renormalization; Besov / Hölder / Schauder |
| KPZ & Universality | KPZ / Hopf–Cole / SHE; KPZ universality / \(1/3\) / Tracy–Widom / TASEP; 1:2:1 scaling / RG / Airy; directed polymers / strong-weak disorder |
| Stochastic NS & Quantization | stochastic Navier–Stokes / invariant measure / turbulence / ergodicity (Hairer–Mattingly); Malliavin / hypoellipticity / Harris; Parisi–Wu stochastic quantization / Gibbs / \(\Phi^4_3\) constructive |
Dependency order¶
- Stochastic processes (Itô, martingales, BM)
- PDE (heat kernel, mild form, energy estimates)
- Functional analysis (semigroups, Sobolev)
- This section; singular SPDEs need distribution theory / paradifferential calculus
- Microlocal analysis (wavefront sets / singularities when present)
Where it appears¶
- KPZ equation / interface growth; continuous directed polymers
- Stochastic Navier–Stokes / turbulence models
- Euclidean QFT constructive program via dynamical \(\Phi^4\)
- Filtering of SPDEs; numerical SPDE schemes (splitting, regularity-preserving)
See it¶
Stochastic heat equation \(\partial_t u=\Delta u+\xi\) on torus: mild solution \(u=G*\xi\) (space-time convolution with heat kernel) is a Gaussian field with explicit Hölder regularity \(<1-d/2\) in space for \(d\) dimensions — already a distribution when \(d\ge 2\). Multiplicative noise \(u\cdot\xi\) fails classical definition precisely when \(u\) is distribution-valued.
Check¶
Why is space-time white noise on \(\mathbb{R}_+\times\mathbb{T}^d\) not a function for \(d\ge 1\)?
Covariance is \(\delta(t-t')\delta(x-x')\); the formal series \(\sum_k \dot\beta_k(t) e_k(x)\) has \(\mathbb{E}\| \xi(t)\|_{H^{-s}}^2=\infty\) unless \(s>d/2\). Realizations lie in negative Sobolev / Hölder–Zygmund spaces almost surely.
Mild vs weak vs strong (classical) solution — one-line distinctions.
Classical: pointwise PDE. Weak: integrated against test functions. Mild: variation of constants \(u=S(t)u_0+\int S(t-s)F(u)\,ds+\int S(t-s)\,dW\) in a Banach space — the standard SPDE notion for semilinear problems.
Pitfalls¶
- Cylindrical Wiener process is not \(H\)-valued; need Hilbert–Schmidt embeddings \(U_0\hookrightarrow H\) to define stochastic convolutions in \(H\).
- Pathwise vs Itô: rough path / regularity structure solutions are often pathwise; classical Da Prato–Zabczyk theory is Itô in Hilbert space — different toolkits.
- Renormalization: singular products require subtracting infinite constants/counterterms; solution depends on renormalization convention unless fixed by probability or physics.
Refs¶
Da Prato–Zabczyk, Stochastic Equations in Infinite Dimensions; Hairer, A theory of regularity structures; Gubinelli–Imkeller–Perkowski (paracontrolled); Hairer–Quastel surveys on KPZ.
Next¶
Foundations · Regularity Structures · Paracontrolled & Renormalization · KPZ & Universality · Stochastic NS & Quantization · Stochastic processes · PDE · Functional analysis