Skip to content

Writing an SPDE

The input DSL is four classes — Unknown, Noise, an Operator, and SPDE — plus plain SymPy for the right-hand side.

The pieces

Unknown

u = Unknown("u", dim=1)

A solution component in dim space dimensions. It exposes the SymPy objects you build the right-hand side from:

  • u.field — the function \(u(t, x_1, \dots)\);
  • u.t, u.x — the time symbol and the list of space symbols;
  • write \(\partial_{x_1} u\) as Derivative(u.field, u.x[0]).

All components of a system must share the same dim.

Noise

from counterterms import kappa
xi = Noise("xi", regularity=Rational(-3, 2) - kappa)

regularity is the noise’s Hölder regularity \(\beta_0\) directly (see Conventions §1). It may be a rational, or a rational minus a multiple of kappa (the positive infinitesimal); white noises always carry a - kappa. Use xi.symbol in the right-hand side.

Operator

Parabolic(dim=1)                          # ∂_t − Δ            (the default choice)
Parabolic(dim=1, mass=1)                  # ∂_t − Δ + 1
Parabolic(dim=2, order=4)                 # order-4 parabolic — warns, see below
FractionalHeat(dim=1, sigma=Rational(3,4))  # ∂_t + (−Δ)^σ     — warns, see below

The engine reads exactly two numbers from the operator: the metric scaling \(\mathfrak{s}\) and the Schauder order \(m\) of \(L^{-1}\) (it maps \(\mathcal{C}^\alpha \to \mathcal{C}^{\alpha+m}\)). Everything downstream — homogeneities, \(S(\tau)\), the Υ-map — is computed from that pair; the display strings are cosmetic. You can subclass Operator to model any linear operator whose inverse has a Schauder estimate in a scaled Hölder scale.

Order ≠ 2 runs with a warning

The regularity-structure theory behind the output is proven for 2nd-order parabolic operators. For other orders the engine computes the combinatorics anyway (the tree enumeration is complete for any order) and warns that the analytic theory is unverified — see Scope & limitations. Treat order ≠ 2 output as exploratory.

SPDE

Scalar form:

spde = SPDE(operator=op, unknown=u, noises=[xi], rhs=...)

System form — a list of (unknown, operator, rhs) triples:

spde = SPDE(noises=[xi], equations=[
    (u, op, a(v.field) * xi.symbol + g(u.field) * Derivative(u.field, u.x[0])**2),
    (v, op, b(u.field) * xi.symbol),
])

Then:

eq = spde.renormalize()      # -> RenormalizedEquation

The right-hand side

Plain SymPy, with three kinds of atoms: noise symbols (xi.symbol), the fields (u.field, v.field), and their first spatial derivatives (Derivative(u.field, u.x[i])). Coefficient functions are undeclared SymPy Functions of the fields — f(u.field), g(u.field, v.field), etc.

What the parser accepts (and the theory covers):

  • affine in the noise: f(u)·ξ + (noise-free part) — a ξ² or f(ξ) is rejected;
  • at most quadratic in \(\partial u\) (Assumption D2), counting total gradient degree — g(u)·(∂ₓu)² is fine, (∂ₓu)³ is rejected;
  • polynomial or symbolic-function dependence on the fields themselves is unrestricted.

Systems and multiple noises

xi  = Noise("xi",  regularity=Rational(-1) - kappa)
eta = Noise("eta", regularity=Rational(-1) - kappa)
res = SPDE(operator=op, unknown=u, noises=[xi, eta],
           rhs=f(u.field) * xi.symbol + h(u.field) * eta.symbol).renormalize()
  • Noises are treated as independent; with several noises, subcriticality binds through the worst regularity, and Wick pairing (in the canonical constants) is within-type only.
  • In systems, one tree can produce counterterms in several equations; the free constant is shared — res.per_component[a] lists each equation’s counterterms, and the same k_i symbol may appear in several components (that is the theorem, not a bug).
  • res.n_components, res.per_component[comp], res.counterterm_rhs(comp) give per-equation access.

What gets rejected, and why

The engine enforces its scope with explicit errors rather than wrong output:

Input Response
Supercritical \(\beta_0 \le -\text{order}\), polynomial additive noise (\(\Phi^4_2, \Phi^4_3\)) ValueError pointing to daprato_lift
Supercritical, not liftable (sine-Gordon — needs Wick exponentials) rejected
Nonlinearity not affine in the noise rejected
\(g\) more than quadratic in \(\partial u\) rejected (Assumption D2)
Derivative factors with ( p
Quasilinear / non-parabolic \(L\) not expressible in the DSL