[main] [research] [bernoulli]mg|pages

Otto - A continuity argument in regularity structures

The lectures of Felix Otto at the second workshop of the Bernoulli Center Program “New developments and challenges in Stochastic Partial Differential Equations”

Otto | A cont. arg. in regularity structures | Lecture 1 | Tuesday July 22, 9:00–10:30

Plan: 1) Setup (informal ~ formal), definition of solutions; 2) A priori estimates on solutions; 3) Continuity argument to construct solutions.

1. Definition of solutions

1.1 Parametrization of solution manifold (informal)

To fix ideas consider an equation in the form

$\displaystyle L \varphi = \xi + a (\varphi)$

on $\mathbb{R}^{1 + d} \in (y_0, y_1, \ldots, y_d)$ where $L$ is a constant coefficients operator, maybe the heat operator $L = \tfrac{\partial}{\partial y_0} - \tfrac{\partial^2}{\partial y_1^2} - \cdots - \tfrac{\partial^2}{\partial y_1^2}$ but it is not crucial and $\xi$ is some distributional/analytic, and $a (\varphi)$ is an analytic polynomial. We decide that we will solve the equation up to an analytic function $q$:

$\displaystyle L \varphi = \xi + a (\varphi) + q$

this is not crazy since $\xi$ and $q$ are pretty much disjoint contributions. For $a = 0$ the solution manifold

$\displaystyle \{ \varphi | L \varphi = \xi + \text{analytic} | \}$

is an affine space over linear space of $\{ p \shortmid L p \text{analytic} \} = \{ p \text{analytic} \}$. Pick a $\Pi_0$ s.t. $L \Pi_0 = \xi = \Pi_0^-$ then the map $p \mapsto \Pi_0 + p$ is a parametrization of the solution manifold at $a = 0$.

Goal: Extend this parametrization to neighborhood of $a = 0$ in $a$-space in form of two maps $\Pi [a, p], \Pi^- [a, p]$ such that

$\displaystyle L \Pi = \Pi^- + \text{analytic}$

and

$\displaystyle \Pi^- = a (\Pi) + \Pi_0$

and s.t.

$\displaystyle \Pi [a = 0, p] = \Pi_0 + p$

and

$\displaystyle \Pi^- [a = 0, p] = \Pi_0^- \qquad (\operatorname{polynomial}\operatorname{sector})$

The condition $\Pi^- = a (\Pi) + \Pi_0$ is not robust, and we will eventually replace it (in the last lecture).

Try implicit function argument. Let $\delta \Pi^{(-)}$ denote the directional derivative of $\Pi^{(-)}$ in directions of $\delta a$ at $a = 0$.

$\displaystyle L \delta \Pi = L \delta \Pi^- + \text{analytic} \qquad \text{and} \qquad \delta \Pi^- = \delta a (\Pi_0 + p)$

but this PDE leaves $\delta \Pi$ underdetermined up to an analytic function.

Attempts to restrict the construction:

– translation invariance: $\varphi (z + \cdot) = \hat{\varphi}$ and $\xi (z + \cdot) = \hat{\xi}$ then $(\hat{\varphi}, \hat{\xi})$ is also a solution. Then we could try to impose covariance at the level of $\Pi^{(-)}$, $\Pi^{(-)} [a, p (z + \cdot)] = \hat{\Pi}^{(-)} [a, p] (z + \cdot)$ but this fails! Indeed look at $d = 0$, $\mathrm{d} \Pi_0 / \mathrm{d} x_0 = \Pi^-_0$ and think to $\Pi^-$ as white noise and $\Pi$ as Brownian motion and there is no way to impose this covariance.

– rescaling invariance: for $r \in (0, \infty)$, let $R y = (r^2 y_0, r y_1, \ldots, r y_d)$ is made s.t. $L \{ y \mapsto \varphi (R y) \} = r^2 (L \varphi) (R \cdot)$, so if we rescale $\varphi (R \cdot) = r^{\alpha} \hat{\varphi}$ and $\xi (R \cdot) = r^{\alpha - 2} \hat{\xi}$ and $a (r^{\alpha} \cdot) = r^{\alpha - 2} \hat{a}$ then $(\hat{a}, \hat{\varphi}, \hat{\xi})$ is again a solution. We can now try to impose that the parametrization of the solution manifold is covariant wrt. the scaling:

$\displaystyle r^{\alpha} \Pi [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Pi} [r^{\alpha - 2} a, r^{\alpha} p] (R \cdot),$ $\displaystyle r^{\alpha - 2} \Pi^- [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Pi}^- [r^{\alpha - 2} a, r^{\alpha} p] (R \cdot),$

and this works! and fixrs the degree of non-uniqueness.

Range of $\alpha$? We want to be in the singular setting so $\alpha < 0$, but we want also to be super-renormalizable, so we want $a (\varphi) = r^{\alpha - 2} \hat{a} (r^{- \alpha} \varphi) \rightarrow 0$ as $r \searrow 0$ so the l.h.s. $O (r^{\alpha - 2 - K \alpha})$ [to finish] and this will give the condition $\alpha > \cdots$.

1.2. Centered model $(\Pi, \Gamma)$ (informal)

No natural origin for space–time. $\Pi^{(-)}_x$ for every origin = “base-point” $x$. We recover covariance wrt. translations:

$\displaystyle \Pi_x^{(-)} [a, p (z + \cdot)] = \Pi_{x + z}^{(-)} [a, p] (z + \cdot) .$

By construction, there exists

$\displaystyle (a, p) \mapsto \Gamma [a, p] \in (\tilde{a}, \tilde{p}) \text{-space}$

such that

$\displaystyle \Pi_x \circ \Gamma_{x, x'} = \Pi_{x'}$

and

$\displaystyle \Gamma [a, p] = (a, \underbrace{\Gamma' [a, p]}_{\text{\scriptsize{$\begin{array}{c} \operatorname{space}-\operatorname{time}\\ \operatorname{analytic}\\ \operatorname{functions} \end{array}$}}})$

Covariance properties:

$\displaystyle \Gamma'_{x, x'} [a, p (z + \cdot)] = \hat{\Gamma}'_{z + x, z + x'} [a, p], \qquad$ $\displaystyle r^{\alpha} \Gamma_{x, 0}' [a (r^{\alpha} \cdot), p (R \cdot)] = \Gamma'_{R x, 0} [r^{\alpha - 2} a, r^{\alpha} p] .$

Remark.

— The big difference with RS is that we are more parcimonious on the parametrization of the objects, we reduce to the solution manifold and have the right amount of counterterms, the price to pay is that we are not able to formulate a linear fixpoint problem and we will use a continuity argument to handle the non-linear setting.

— There is no single canonical parametrization of the solution manifold, we have charts (caused by the choice of base-point) and transition maps given by the $\Gamma$s.

— We parametrize the rough manifold of solutions with analytic function (exploiting the property that they are indeed solutions).

1.3 Algebraization (informal to formal)

Recall informal:

$\displaystyle \text{analytic } \ni (a, p) \mapsto \Pi^{(-)} [a, p] \in \text{rough}$

Power-series Ansatz:

$\displaystyle \Pi_x^{(-)} [a, p] = \sum_{\beta} \Pi_{x \beta}^{(-)} \prod_{k \in \mathbb{N}_0} \left( \frac{1}{k!} \frac{\mathrm{d}^k a}{\mathrm{d} \varphi^k} (0) \right)^{\beta (k)} \prod_{h \in \mathbb{N}_0^{1 + d}} \left( \underbrace{\frac{1}{n!} \frac{\mathrm{d}^n p}{\mathrm{d} y^n} (x)}_{z_n [p (x + \cdot)]} \right)^{\beta (n)}$ $\displaystyle z_n [p] := \frac{1}{n!} \frac{\mathrm{d}^n p}{\mathrm{d} y^n} (0), \qquad n = (n_0, n_1, \ldots, n_d) \in \mathbb{N}_0^{1 + d}$

and $\beta$ is a multi-index on index set $\mathbb{N}_0 \sqcup \mathbb{N}_0^{1 + d}$ and $\left( z^{\beta} := \prod_k z_k^{\beta (k)} \prod_n z_n^{\beta (n)} \right) [a, p]$.

we may construct these coefficients $\Pi_{x \hspace{1em} \beta} \in \mathcal{S}' (\mathbb{R}^{1 + d})$. We put them in a single object as a formal power-series:

$\displaystyle \Pi_x \in \mathcal{S}' (\mathbb{R}^{1 + d}) [[(z_k), (z_n)]] .$

Polynomial sector: there are three classes of multi-indeces: 1) the polynomial ones,

$\displaystyle T^{\operatorname{pol}} := \{ \delta_n \shortmid n \in \mathbb{N}_0^{1 + d} \}, \qquad T^- := \{ 0 \} \cup \{ \beta \shortmid \beta (k) \neq 0 \text{for some $k \in \mathbb{N}_0$} \}, \qquad T := T^{\operatorname{pol}} \cup T^-$

and by construction in 1.1, (i.e. $\Pi^- [a = 0, p] = \Pi^-_0$ and $\Pi [a = 0, p] = \Pi_0 + p$) we find

$\displaystyle \begin{array}{lll} \Pi^-_{x \beta} = 0, & & \text{unless $\beta \in T^-$},\\ \Pi_{x \beta} = 0, & & \text{unless $\beta \in T$},\\ \Pi_{x \delta_n} = (\cdot - x)^n . & & \end{array}$

Recall 1.2 informally:

$\displaystyle \text{analytic } \ni (a, p) \mapsto \Gamma_{x, x'} [a, p] \in \text{analytic}$

We make a power-series Ansatz

$\displaystyle z^{\gamma} [\Gamma_{x, x'} [a, p]] = \sum_{\beta} (\Gamma_{x, x'}^{\star})^{\gamma}_{\beta} z^{\beta} [a, p]$

i.e. $\Gamma_{x, x'}^{\star}$ is an algebra automorphism in the formal power series in the variables $((z_k), (z_n))$ with coefficient in $\mathbb{R}$:

$\displaystyle \Gamma_{x, x'}^{\star} \in \widetilde{\operatorname{Alg}} (\mathbb{R} [[z_k, z_n]])$ $\displaystyle \Gamma_{x, x'}^{\star} (\pi \pi') = \Gamma_{x, x'}^{\star} (\pi) \Gamma_{x, x'}^{\star} (\pi')$

By definition in 1.2. we get the axioms of regularity structures”

$\displaystyle \Pi_x^{(-)} = \Gamma_{x, x'} \Pi_{x'}^{(-)}$

and this makes sense since $\Pi_{x'}^{(-)} \in \mathcal{S}' [[z_k, z_n]]$ and $\Gamma_{x, x'} \in \operatorname{End} (\mathbb{R} [[z_k, z_n]])$. The property $\Gamma_{x, x'} [a, p] = (a, \ast)$ translates to the fact that $\Gamma_{x, x'}^{\star} =\operatorname{id}$ on $\mathbb{R} [[z_k]]$ and the scaling covariance translate to

$\displaystyle r^{| \beta |} \Pi^-_{\beta} = \hat{\Pi}_{\beta} (R \cdot), \qquad r^{| \beta | - 2} \Pi^-_{\beta} = \hat{\Pi}_{\beta}^- (R \cdot), \qquad r^{| \beta | - | \gamma |} (\Gamma_{x, 0}^{\star})^{\gamma}_{\beta} = (\Gamma_{R x, 0}^{\star})^{\gamma}_{\beta} .$

with homogeneity

$\displaystyle | \beta | - \alpha = - \underbrace{(\alpha - 2)}_{\text{scaling of $a$}} \sum_k \beta (k) + \underbrace{(\alpha)}_{\text{scaling of $\varphi$}} \sum_k k \beta (k) - \underbrace{(\alpha)}_{\text{scaling of $p$}} \sum_n \beta (n) + \underbrace{(1)}_{\text{scaling of $y$}} \sum_n | n | \beta (n)$

with $| n | = 2 n_0 + n_1 + \cdots + n_d$.

Recall what $\hat{\Pi}$ is. We rescale the solution and the parametrization of the solution manifold, and its coordinates

$\displaystyle \varphi (R \cdot) = r^{\alpha} \hat{\varphi}, \quad \xi (R \cdot) = r^{\alpha - 2} \hat{\xi}, \quad a (r^{\alpha} \cdot) = r^{\alpha - 2} \hat{a},$ $\displaystyle r^{\alpha} \Pi [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Pi} [r^{\alpha - 2} a, r^{\alpha} p] (R \cdot)$ $\displaystyle r^{| \beta |} \Pi_{\beta} = \hat{\Pi}_{\beta} (R \cdot)$

LOTT: We may construct $\Pi_x \in \mathcal{S}' [[z_k, z_n]]$, $\Gamma_{x, x'}^{\star} \in \operatorname{Aut} (\mathbb{R} [[z_n, z_k]])$ such that

$\displaystyle \Pi_{x, \beta, \gamma} (x) = O (r^{| \beta |}), \qquad \Pi_{x, \beta, \gamma}^- (x) = O (r^{| \beta | - 2}), \qquad (\Gamma_{x, x'}^{\star})^{\gamma}_{\beta} = O (| x' - x |^{| \beta | - | \gamma |}) .$

1.4. Notion of solution (informal to formal)

$\displaystyle \varphi = \text{solution for some fixed nonlinearity $a$}$ $\displaystyle = \Pi [a, p] \qquad \text{for some $p$ with $L \Pi [a, p] = \Pi^- [a, p]$},$

point in $(a, p)$-space which can be identified wth Dirac measures, which is nothing more than a multiplicative linear functionals on functions $\pi$ on $(a, p)$ space. So it is convenient to think about this as an algebra homomorphism (character) of the algebra $\mathbb{R} [[z_k, z_n]]$, i.e. $\operatorname{Alg} (\mathbb{R} [[z_k, z_n]], \mathbb{R})$ and this is for us a modelled distribution.

[end of the first lecture]

Otto | Solution theory for singular SPDE | Lecture 2 | Tuesday July 22, 14:15–15:45

Recall our setting:

$\displaystyle \{ \varphi \shortmid L \varphi = a (\varphi) + \xi + \text{analytic} \}$ $\displaystyle \varphi = \Pi_x [a, p], \qquad a (\varphi) + \xi = \Pi_x^- [a, p], \qquad \Gamma_{x, x'}$ $\displaystyle \Pi_x \in \mathcal{S}' [[z_k, z_n]], \qquad \Gamma^{\star}_{x, x'} \in \operatorname{Alg} (\mathbb{R} [[z_k, z_n]])$

plus additional covariance properties of these objects.

1.4 Notion of solution (informal to formal)

Fix a non-linearity $a$.

Then $\varphi = \Pi [a, p]$ for some $p$ with $L \Pi [a, p] = \Pi^- [a, p]$ $\sim$ this is a point in $(a, p)$-space $=$ Dirac measure on $(a, p)$-space $=$ multiplicative functional on the space of functions on $(a, p)$-space $=$ an element $f$ of the set of algebra homomorphism from $\mathbb{R} [z_k, z_n]$ to $\mathbb{R}$, i.e. $(f. \pi \pi') = (f. \pi) (f. \pi')$, $f. 1 = 1$.

Recall that $\mathbb{R} [z_k, z_n]$ is the space of polynomials, which in the dual picture give more freedom in $\operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})$.

Ansatz: let's fix a base point $x$ and let $f_x \in \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})$

$\displaystyle \frac{1}{k!} \frac{\mathrm{d}^k a}{d \varphi^k} (0) = f_x .z_k, \qquad \varphi = \underbrace{f_x}_{\in (\mathbb{R} [z_k, z_n])^{\star}} . \underbrace{\Pi_x}_{\in \mathcal{S}' [[z_k, z_n]]}$

and

$\displaystyle \varphi^- = f_x . \Pi_x^-, \qquad \text{satisfies} \qquad L \varphi = \varphi^-$

Could have taken a different base point $x'$ and

$\displaystyle \varphi = f_{x'} . \Pi_{x'} = f_x . \Gamma_{x, x'} \Pi_{x'} .$

We should think of $f = (f_x)_x$ satisfying the covariance property

$\displaystyle f_{x'} = f_x . \Gamma_{x, x'} .$

This is the space of “modelled distributions”. But there is a mistmatch also here between $\mathbb{R} [z_k, z_n] $ and $\mathbb{R} [[z_k, z_n]]$. We need a truncation to make these two structures compatible. Fix $\kappa > 2$ and consider

$\displaystyle Q : \mathbb{R} [[z_k, z_n]] \rightarrow \mathbb{R} [z_k, z_n]$

defined as

$\displaystyle Q z^{\beta} = \left\{ \begin{array}{lll} z^{\beta}, & \qquad & \text{if $| \beta | < \kappa$},\\ 0, & & \text{otherwise} . \end{array} \right.$

Definition of a solution. A solution is a family $f$ satisfying three conditions:

1.	$f = \{ f_x \in \operatorname{Alg} (R [z_k, z_n], \mathbb{R}) \}_x$

and

2.	$f_x .z_k = \frac{1}{k!} \frac{\mathrm{d}^k a}{d \varphi^k} (0), \qquad (f_{x'} - f_x . \Gamma_{x, x'})^{\gamma} = O (\| x - x' \|^{\kappa - \| \gamma \|})$

As a consequence of this condition and of the estimates of $\Pi^{(-)}_x$ we have

$\displaystyle (f_{x'} .Q \Pi_x - f_x .Q \Pi_x)_r (x') = O (\max_{| \gamma | < \kappa} (| x' - x |^{\kappa - | \gamma |} + r^{| \gamma |}))$ $\displaystyle (f_{x'} .Q \Pi_x^- - f_x .Q \Pi_x^-)_r (x') = O (\max_{| \gamma | < \kappa} (| x' - x |^{\kappa - | \gamma |} + r^{| \gamma | - 2}))$

and by reconstruction there exists unique (!) functions $\varphi, \varphi^- \in \mathcal{S}'$ such that

$\displaystyle (\varphi - f_x .Q \Pi_x)_r (x) = O (r^{\kappa}), \qquad (\varphi^- - f_x .Q \Pi^-_x)_r (x) = O (r^{\kappa - 2}),$

now, for some $\psi \in \mathcal{S}$:

$\displaystyle \xi_r (x) = (\xi, r^{- D} \psi (R^{- 1} (x + \cdot))), \qquad R^{- 1} y = (r^{- 2} y_0, r^{- 1} y_1, \ldots, r^{- 1} y_d),$

and $D = 2 + d$ is the effective dimension. And moreover we impose:

3.	$L \varphi = \varphi^-$

2. Apriori estimates of $f$

2.1 Overview

Fix $\kappa > 2$ and $\kappa \not{\in} \mathbb{N}$. Introduce norms

$\displaystyle M := \max_{| \gamma | < \kappa} \sup_{x, x'} \frac{| (f_{x'} - f_x Q \Gamma_{x, x'})^{\gamma} |}{| x' - x |^{\kappa - | \gamma |}}, \qquad M^- := \max_{| \gamma | < \kappa, \marked{\gamma \in T^-}} \sup_{x, x'} \frac{| (f_{x'} - f_x Q \Gamma_{x, x'})^{\gamma} |}{| x' - x |^{\kappa - | \gamma |}},$ $\displaystyle M^{\operatorname{pol}} := \max_{| \gamma | < \kappa, \marked{\gamma \in T^{\operatorname{pol}}}} \sup_{x, x'} \frac{| (f_{x'} - f_x Q \Gamma_{x, x'})^{\gamma} |}{| x' - x |^{\kappa - | \gamma |}},$

where recall that

and by construction in 1.1, (i.e. $\Pi^- [a = 0, p] = \Pi^-_0$ and $\Pi [a = 0, p] = \Pi_0 + p$) we find

$\displaystyle \begin{array}{lll} \Pi^-_{x \beta} = 0, & & \text{unless $\beta \in T^-$},\\ \Pi_{x \beta} = 0, & & \text{unless $\beta \in T$},\\ \Pi_{x \delta_n} = (\cdot - x)^n . & & \end{array}$ $\displaystyle R := \sup_{\psi \in \mathcal{B}} \sup_{\gamma, x} \frac{| (\varphi - f_x .Q \Pi_x)_r (x) |}{r^{\kappa}}, \qquad R^- := \sup_{\psi \in \mathcal{B}} \sup_{\gamma, x} \frac{| (\varphi^- - f_x .Q \Pi^-_x)_r (x) |}{r^{\kappa - 2}}$

with $\mathcal{B}$ a bounded set of Schwarz functions.

Auxiliary norms:

$\displaystyle \| f \| = \max_{| n | < \kappa} \sup_x | f_x .z_n |, \qquad \lambda = \max_{k \leqslant K} \sup_x | f_x .z_k | = \max_{k \leqslant K} \left| \frac{1}{k!} \frac{\mathrm{d}^k a}{\mathrm{d} \varphi^k} (0) \right| \ll 1.$

We can think to $M$ as a $C^{\kappa}$-norm while to $\| f \|$ as a $C^{\lfloor k \rfloor}$-norm, so slightly weaker. But these numbers refer to non-linear objects.

Four tasks:

1) Algebraic continuity argument: we can estimate $M$ by $M^{\operatorname{pol}}$ by a straighforward argument to get: $\displaystyle M \lesssim_{\Gamma} C (\\| f \\|) (M^{\operatorname{pol}} + 1),$ $\displaystyle M^- \lesssim_{\Gamma} \lambda (M + C (\\| f \\|))$	2) Reconstruction: $\displaystyle R^- \lesssim_{\Pi} M^-$
3) Integration: $\displaystyle R \lesssim R^- \qquad \left( \kappa \not{\in} \mathbb{N} \right)$	4) Three point argument $\displaystyle M^{\operatorname{pol}} \lesssim_{\Gamma} M^- + R$

All together these give

$\displaystyle M \lesssim_{\Pi, \Gamma} C (\| f \|) (\lambda M + 1)$

which looks quite good and useful, by taking $\lambda$ small. To beat $(\| f \|)$ we need an interpolation argument. Estimate $\| f \|$ as follows

$\displaystyle \| f \| \leqslant \varepsilon M + C_{\varepsilon} (\sup_x | f_x .z_0 |)$

but the real problem comes from the term $C (\| f \|) \ast 1$ in the estimate for $M$, this cannot be beaten easily.

We need a second round of estimates. Fix

$\displaystyle \tilde{\kappa} = \inf \{ | \beta | \shortmid | \beta | > 0 \} \in (0, 1)$

define $\tilde{M}, \tilde{M}^-, \tilde{M}^{\operatorname{pol}}, \tilde{R}, \tilde{R}^-$ with similar expressions as before.

Then $\tilde{M}^-$ related to $\gamma \in T^-$ with $| \gamma | < \tilde{\kappa}$ are of the form $\gamma (n) = 0$ for all $n$ because $\Gamma^{\star}_{x, x'}, f_x$ are “trivial” on $z_k$'s and $\tilde{M}^- = 0$.

$\displaystyle \tilde{M} = \tilde{M}^{\operatorname{pol}} = \sup_{x, x'} \frac{| f_{x'} .z_0 - f_x .z_0 |}{| x' - x |^{\tilde{\kappa}}} =: [f.z_0]_{\tilde{\kappa}}$

and the three steps above are:

Algebraic

$\displaystyle \tilde{R}^- \lesssim_{\Gamma} \tilde{R} + C (\| f \|) \lambda + \sup_{x, r \geqslant 1} \frac{| \varphi^-_r (x) |}{r^{\tilde{\kappa} - 2}}$

Integration

$\displaystyle \tilde{R} \leqslant \tilde{R}^-$

three point argument

$\displaystyle \tilde{M}^{\operatorname{pol}} \lesssim_{\Pi} \tilde{R}$

and all combined produce

$\displaystyle [f.z_0]_{\tilde{\kappa}} = \tilde{M} \lesssim C (\| f \|) \marked{\lambda} (M + 1) + \sup_{x, r \geqslant 1} \frac{| \varphi^-_r (x) |}{r^{\tilde{\kappa} - 2}}$

and by periodic boundary conditions we have

$\displaystyle \| f.z_0 \| \lesssim [f.z_0]_{\tilde{\kappa}}$

and we can close the loop with a small constant $\lambda$ in front of everything.

Let's go to some analytic estimates, e.g. for the algebraic argument

Lemma. a) For some $\tilde{\kappa} < \kappa$

$\displaystyle M^- \lesssim \lambda \tilde{M}$

Proof. Consider $\gamma \in T^- = \{ 0 \} \cup \{ \gamma \shortmid \exists k, \gamma (k) \neq 0 \}$. If $\gamma = 0$

$\displaystyle (f_{x'} - f_x .Q \Gamma^{\star}_{x, x'})^0 = f_{x'} .1 - f_x .Q \Gamma^{\star}_{x, x'} 1 = 0,$

Now if $\gamma (k) \neq 0$ for some $k$ we write $\gamma = \delta_k + \tilde{\gamma}$ and define a new $\tilde{\kappa}$ by $\kappa = (| \delta_k | - \alpha) + \tilde{\kappa}$ so that $| \tilde{\gamma} | < \tilde{\kappa}$ and now with $z^{\gamma} = z_k z^{\tilde{\gamma}}$ and use the properties of $\Gamma^{\star}_{x, x'}$ to have

$\displaystyle (f_{x'} - f_x .Q \Gamma^{\star}_{x, x'})^{\gamma} = f_{x'} .z^{\gamma} - f_x .Q \Gamma^{\star}_{x, x'} z^{\gamma} = f_{x'} .z^{\gamma} - f_x .Q (\Gamma^{\star}_{x, x'} z_k) (\Gamma^{\star}_{x, x'} z^{\tilde{\gamma}})$ $\displaystyle = f_{x'} . (z_k z^{\tilde{\gamma}}) - f_x . \underbrace{(\Gamma^{\star}_{x, x'} z_k)}_{z_k} \tilde{Q} (\Gamma^{\star}_{x, x'} z^{\tilde{\gamma}}) = f_{x'} .z_k f_{x'} .z^{\tilde{\gamma}} - f_x . z_k f_x . \tilde{Q} (\Gamma^{\star}_{x, x'} z^{\tilde{\gamma}})$ $\displaystyle = (f_{x'} .z_k) (f_{x'} .z^{\tilde{\gamma}} - f_x . \tilde{Q} (\Gamma^{\star}_{x, x'} z^{\tilde{\gamma}})) = \underbrace{\frac{\mathrm{d}^k a}{\mathrm{d} \varphi^k} (0)}_{\lesssim \lambda} (f_{x'} .z^{\tilde{\gamma}} - f_x . \tilde{Q} (\Gamma^{\star}_{x, x'} z^{\tilde{\gamma}}))$ $\displaystyle \lesssim \lambda \tilde{M} | x' - x |^{\tilde{\kappa} - | \tilde{\gamma} |} \lesssim \lambda \tilde{M} | x' - x |^{\kappa - | \gamma |} .$

$\Box$

[end of second lecture]

Otto | Solution theory for singular SPDEs | Lecture 3 | Wednesday July 24, 11:00–12:30

All we have already said in a picture:

solution manifold $\{ \varphi \}$	param. via $(a, p)$-space	model	modelled distr	apriori estimates
$\{ \varphi \| L \varphi = a (\varphi) + \xi + \text{analytic} \}$	$\displaystyle \varphi \in \Pi [a, p]$ $\displaystyle a (\varphi) + \xi = \Pi^- [a, p]$ (non robust) $\displaystyle L \Pi [a, p] = \Pi^- [a, p]$ $\displaystyle + \text{analytic}$	$\Pi, \Pi^- \in \mathcal{S}' [[z_k, z_n]]$ $\Pi [a, p] = \sum_{\beta} \Pi_{\beta} z^{\beta} [a, p]$ $z^{\beta} = \Pi_{k \in \mathbb{N}_0} z_k^{\beta (k)} \Pi_{n \in \mathbb{N}_0^{1 + d}} z_n^{\beta (n)}$ $\displaystyle z_k [a] = \frac{1}{k!} \frac{\mathrm{d}^k a}{d \varphi^k} (0)$ $\displaystyle z_n [p] = \frac{1}{n!} \frac{\mathrm{d}^n p}{\operatorname{dy}^n} (0)$	$f \in \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})$ $\displaystyle \varphi \approx f. \Pi \rightsquigarrow R$ $\displaystyle \varphi^- \approx f. \Pi^- \rightsquigarrow R^-$ and $\displaystyle L \varphi = \varphi^-$	Reconstruction $\displaystyle R^- \lesssim_{\Pi} M^-$ Integration $\displaystyle R \lesssim R^-$ 3pt argument $\displaystyle M^{\operatorname{pol}} \lesssim_{\Pi} M^- + R$ Algebraic argument $\displaystyle M \lesssim M^{\operatorname{pol}} + 1$ $\displaystyle M^- \lesssim \lambda (M + 1)$
$\displaystyle $	$\displaystyle \Pi [a = 0, p] = \Pi_0 + p$ $\displaystyle \Pi^- [a, p] = \Pi^-_0$	$\Pi_{\beta} = 0$ unless $\beta \in T$ $\Pi_{\delta_n (y)} = y^n$ for $\delta_n = T^{\operatorname{pol}}$ $\Pi_{\beta}^- = 0$
$\varphi (R \cdot) = r^{\alpha} \hat{\varphi}$ $\xi (R \cdot) = r^{\alpha - 2} \hat{\xi}$ $a (r^{\alpha} \cdot) = r^{\alpha - 2} \hat{a}$ singular: $\varphi \rightarrow \infty$ at fixed $\hat{\varphi}$ super-rinorm $\Leftrightarrow$ $\displaystyle \alpha \kappa - (\alpha - 2) > 0$	$\displaystyle r^{\alpha} \Pi [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Pi} [r^{\alpha - 2} a, r^{\alpha} p] (R \cdot)$ $\displaystyle r^{\alpha - 2} \Pi^- [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Pi}^- [r^{\alpha - 2} a, r^{\alpha} p] (R \cdot)$	$r^{\| \beta \|} \Pi_{\beta} = \hat{\Pi}_{\beta} (R \cdot)$ $\| \beta \| - \alpha = \sum_{k \leqslant K} (\alpha k - (\alpha - 2)) \beta (k) + \sum_n (\| n \| - \alpha) \beta (n)$ as motivation for $\displaystyle \Pi_{\beta r} (x) = O (r^{\| \beta \|})$ $\displaystyle \Pi_{\beta r}^- (x) = O (r^{\| \beta \| - 2})$
	$\Gamma_{x, x'}$ in $(a, p)$-space to itself $\displaystyle \Pi_x = \Pi_{x'} \circ \Gamma_{x, x'}$ $\displaystyle \Gamma_{x, x'} [a, p] = (a, \Gamma_{x, x'} [a, p])$ $r^{\alpha} {}^{\backprime} \Gamma_{x 0} [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Gamma}_{R x, 0}' [r^{\alpha - 2} a, r^{\alpha} p]$	$\Gamma^{\alpha}_{x, x'} \in \operatorname{Alg} (\mathbb{R} [[z_k, z_n]])$ $\displaystyle \Pi^{(-)}_x = \Gamma_{x, x'} \Pi^{(-)}_x$ $\Gamma^{\star}_{x x'} =\operatorname{id}$ on $\mathbb{R} [[z_k]]$ $r^{\| \beta \| - \| \gamma \|} (\Gamma^{\star}_{x 0})_{\beta}^{\gamma} = (\Gamma^{\star}_{R x 0})_{\beta}^{\gamma}$	$f_x \in \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})$ $\displaystyle f_x \approx f_{x'} \Gamma^{\star}_{x x'} \rightsquigarrow M = \left\{ \begin{array}{l} M^{\operatorname{pol}}\\ M^- \end{array} \right.$ $\displaystyle (\Gamma^{\star}_{x 0})^{\gamma}_{\beta} = O (\| x' - x \|^{\| \beta \| - \| \gamma \|})$ $f_x .z_k = \frac{1}{k!} \frac{\mathrm{d}^k a}{d \varphi^k} (0)$

3. Existence via a continuity argument

Specify to $a (\varphi) = \varphi^3$. Naively gives $L \varphi = \varphi^3 + \xi$, but this has to be renormalized.

$\displaystyle L \varphi = \lambda \varphi^3 + h^{\rho} [\lambda] \varphi + \xi_{\rho}$

On the level of the model

$\displaystyle \Pi \in \mathcal{S}' [[z_k, z_n]] \rightsquigarrow \Pi^{(\rho)} \in C^{\kappa} [[z_k, z_n]]$

which is an algebra.

Scaling property is destroyed by the regularisation, we replace it by saying

$\displaystyle (\Pi^{(\rho)})^- = z_3 (\Pi^{(\rho)})^3 + C^{(\rho)} \Pi^{(\rho)} + \xi_{\rho} \mathbb{1}$ $\displaystyle h^{\rho} [\lambda] = \sum_k \lambda^k C_k^{(\rho)}$

We drop the superscript $\rho$.

Lemma. (consistency) If $\varphi \in C^{\kappa}$ satisfies

$\displaystyle L \varphi = \lambda \varphi^3 + h \varphi + \xi_{\rho} =: \varphi^-$

then there exists a unique $f = \{ f_x \in \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R}) \}_x$ such that

$\displaystyle M = \max_{| \gamma | < \kappa} \sup_{x, x'} \frac{| (f_{x'} - f_x Q \Gamma^{\star}_{x, x'})^{\gamma} |}{| x' - x |^{\kappa - | \gamma |}} < \infty$

and

$\displaystyle R = \sup_{\psi \in \mathcal{B}} \sup_{x, r} \frac{| (\varphi - f_x .Q \Pi_x)_r (x) |}{r^{\kappa}} < \infty, \qquad R^- = \sup_{\psi \in \mathcal{B}} \sup_{x, r} \frac{| (\varphi^- - f_x .Q \Pi_x^-)_r (x) |}{r^{\kappa - 2}} < \infty$

Classical result. Exists $\varphi \in C^{\kappa}$ of $L \varphi = \lambda \varphi^3 + h \varphi + \xi_{\rho}$ plus boundary conditions, and it is unique if it is not too large for $| \lambda | \ll_{\rho} 1$. The bounds are not uniform in $\rho$.

Strategy: monitor $\dot{\varphi} = \frac{\mathrm{d} \varphi}{\mathrm{d} \lambda}$ and it satisfies

$\displaystyle L \dot{\varphi} = (3 \lambda \varphi^2 + h) \dot{\varphi} + \varphi^3$

and establish solvability of

$\displaystyle L_{\varphi} \dot{\varphi} = [L - (3 \lambda \varphi^2 + h)] \dot{\varphi} = \zeta \in C^{\kappa - 2}$

uniformly in $| \lambda | \ll_{\Pi, \Gamma} 1$ then we are done, by a continuity argument.

Need a robust formulation of $L_{\varphi} \dot{\varphi} = \zeta$.

$\displaystyle f \in \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})$ $\displaystyle \dot{f} \in T_f \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})$

means that $\dot{f}$ is a derivation:

$\displaystyle \dot{f} . (\pi \pi') = (\dot{f} . \pi) (f. \pi') + (f. \pi) (\dot{f} . \pi'), \qquad \dot{f} .1 = 0.$

We then construct associated norms $\dot{M}, \dot{R}, \ldots$

Consider $\dot{f} \in T_f \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})$ with

$\displaystyle \dot{M} = \max_{| \gamma | < \kappa} \sup_{x, x'} \frac{| (\dot{f}_{x'} - \dot{f}_x Q \Gamma^{\star}_{x, x'})^{\gamma} |}{| x' - x |^{\kappa - | \gamma |}} < \infty,$

and by reconstruction

$\displaystyle \dot{R} = \sup_{\psi \in \mathcal{B}} \sup_{x, r} \frac{| (\dot{\varphi} - \dot{f}_x .Q \Pi_x)_r (x) |}{r^{\kappa}} \lesssim_{\Pi} \dot{M},$ $\displaystyle \dot{R}^- = \sup_{\psi \in \mathcal{B}} \sup_{x, r} \frac{| (\dot{\varphi}^- - \dot{f}_x .Q \Pi_x^-)_r (x) |}{r^{\kappa - 2}} \lesssim_{\Pi} \dot{M}$

we encode

$\displaystyle L_{\varphi} \dot{\varphi} = \zeta$

$\displaystyle L \dot{\varphi} = \dot{\varphi}^- + \zeta .$

Task: estimate $M$ (norm of $f$) in terms of $[\zeta]_{\kappa - 2}$. This will require again the four arguments, but nothing more.

As in 2 we obtain apriori estimates in the linearized problem by running through the four tasks:

1) Algebraic argument (via the algebraic structure of the model distribution as the derivative of a coherent model distribution)

$\displaystyle \dot{M} \lesssim C (M, \| f \|) (\dot{M}^{\operatorname{pol}} + \| \dot{f} \|)$

2) 3pt argument:

$\displaystyle \dot{M}^{\operatorname{pol}} \lesssim_{\Pi } \dot{M}^- + \dot{R}$

3) Integration

$\displaystyle \dot{R} \lesssim R + [\zeta]_{\kappa - 2}$

4) Reconstruction

$\displaystyle \dot{R}^- \lesssim_{\Pi} \dot{M}^-$

5) Algebraic argument

$\displaystyle \dot{M}^- \lesssim_{\Gamma} C (M, \| f \|) \lambda (\dot{M} + \| \dot{f} \|)$

and putting all together:

$\displaystyle \dot{M} \lesssim_{\Pi, \Gamma} C (M, \| f \|) (\lambda \dot{M} + \| \dot{f} \| + [\zeta]_{\kappa - 2})$

and $\lambda \dot{M}$ can be absorbed, while $\| \dot{f} \|$ can be interpolated using the argument of the previous lecture, to obtain for $\lambda$ small depending only on $\| \Pi, \Gamma \|$,

$\displaystyle \dot{M} \lesssim [\zeta]_{\kappa - 2}$

which allows to close the continuity argument only with the condition $| \lambda | \ll 1 / \| \Pi, \Gamma \|$ which is stable for the passage to the limit.

[end of third lecture]

1) Algebraic continuity argument: we can estimate \(M\) by \(M^{\operatorname{pol}}\) by a straighforward argument to get: \(\displaystyle M \lesssim_{\Gamma} C (\\| f \\|) (M^{\operatorname{pol}} + 1),\) \(\displaystyle M^- \lesssim_{\Gamma} \lambda (M + C (\\| f \\|))\)	2) Reconstruction: \(\displaystyle R^- \lesssim_{\Pi} M^-\)
3) Integration: \(\displaystyle R \lesssim R^- \qquad \left( \kappa \not{\in} \mathbb{N} \right)\)	4) Three point argument \(\displaystyle M^{\operatorname{pol}} \lesssim_{\Gamma} M^- + R\)

solution manifold \(\{ \varphi \}\)	param. via \((a, p)\)-space	model	modelled distr	apriori estimates
\(\{ \varphi \| L \varphi = a (\varphi) + \xi + \text{analytic} \}\)	\(\displaystyle \varphi \in \Pi [a, p]\) \(\displaystyle a (\varphi) + \xi = \Pi^- [a, p]\) (non robust) \(\displaystyle L \Pi [a, p] = \Pi^- [a, p]\) \(\displaystyle + \text{analytic}\)	\(\Pi, \Pi^- \in \mathcal{S}' [[z_k, z_n]]\) \(\Pi [a, p] = \sum_{\beta} \Pi_{\beta} z^{\beta} [a, p]\) \(z^{\beta} = \Pi_{k \in \mathbb{N}_0} z_k^{\beta (k)} \Pi_{n \in \mathbb{N}_0^{1 + d}} z_n^{\beta (n)}\) \(\displaystyle z_k [a] = \frac{1}{k!} \frac{\mathrm{d}^k a}{d \varphi^k} (0)\) \(\displaystyle z_n [p] = \frac{1}{n!} \frac{\mathrm{d}^n p}{\operatorname{dy}^n} (0)\)	\(f \in \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})\) \(\displaystyle \varphi \approx f. \Pi \rightsquigarrow R\) \(\displaystyle \varphi^- \approx f. \Pi^- \rightsquigarrow R^-\) and \(\displaystyle L \varphi = \varphi^-\)	Reconstruction \(\displaystyle R^- \lesssim_{\Pi} M^-\) Integration \(\displaystyle R \lesssim R^-\) 3pt argument \(\displaystyle M^{\operatorname{pol}} \lesssim_{\Pi} M^- + R\) Algebraic argument \(\displaystyle M \lesssim M^{\operatorname{pol}} + 1\) \(\displaystyle M^- \lesssim \lambda (M + 1)\)
\(\displaystyle \)	\(\displaystyle \Pi [a = 0, p] = \Pi_0 + p\) \(\displaystyle \Pi^- [a, p] = \Pi^-_0\)	\(\Pi_{\beta} = 0\) unless \(\beta \in T\) \(\Pi_{\delta_n (y)} = y^n\) for \(\delta_n = T^{\operatorname{pol}}\) \(\Pi_{\beta}^- = 0\)
\(\varphi (R \cdot) = r^{\alpha} \hat{\varphi}\) \(\xi (R \cdot) = r^{\alpha - 2} \hat{\xi}\) \(a (r^{\alpha} \cdot) = r^{\alpha - 2} \hat{a}\) singular: \(\varphi \rightarrow \infty\) at fixed \(\hat{\varphi}\) super-rinorm \(\Leftrightarrow\) \(\displaystyle \alpha \kappa - (\alpha - 2) > 0\)	\(\displaystyle r^{\alpha} \Pi [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Pi} [r^{\alpha - 2} a, r^{\alpha} p] (R \cdot)\) \(\displaystyle r^{\alpha - 2} \Pi^- [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Pi}^- [r^{\alpha - 2} a, r^{\alpha} p] (R \cdot)\)	\(r^{\| \beta \|} \Pi_{\beta} = \hat{\Pi}_{\beta} (R \cdot)\) \(\| \beta \| - \alpha = \sum_{k \leqslant K} (\alpha k - (\alpha - 2)) \beta (k) + \sum_n (\| n \| - \alpha) \beta (n)\) as motivation for \(\displaystyle \Pi_{\beta r} (x) = O (r^{\| \beta \|})\) \(\displaystyle \Pi_{\beta r}^- (x) = O (r^{\| \beta \| - 2})\)
	\(\Gamma_{x, x'}\) in \((a, p)\)-space to itself \(\displaystyle \Pi_x = \Pi_{x'} \circ \Gamma_{x, x'}\) \(\displaystyle \Gamma_{x, x'} [a, p] = (a, \Gamma_{x, x'} [a, p])\) \(r^{\alpha} {}^{\backprime} \Gamma_{x 0} [a (r^{\alpha} \cdot), p (R \cdot)] = \hat{\Gamma}_{R x, 0}' [r^{\alpha - 2} a, r^{\alpha} p]\)	\(\Gamma^{\alpha}_{x, x'} \in \operatorname{Alg} (\mathbb{R} [[z_k, z_n]])\) \(\displaystyle \Pi^{(-)}_x = \Gamma_{x, x'} \Pi^{(-)}_x\) \(\Gamma^{\star}_{x x'} =\operatorname{id}\) on \(\mathbb{R} [[z_k]]\) \(r^{\| \beta \| - \| \gamma \|} (\Gamma^{\star}_{x 0})_{\beta}^{\gamma} = (\Gamma^{\star}_{R x 0})_{\beta}^{\gamma}\)	\(f_x \in \operatorname{Alg} (\mathbb{R} [z_k, z_n], \mathbb{R})\) \(\displaystyle f_x \approx f_{x'} \Gamma^{\star}_{x x'} \rightsquigarrow M = \left\{ \begin{array}{l} M^{\operatorname{pol}}\\ M^- \end{array} \right.\) \(\displaystyle (\Gamma^{\star}_{x 0})^{\gamma}_{\beta} = O (\| x' - x \|^{\| \beta \| - \| \gamma \|})\) \(f_x .z_k = \frac{1}{k!} \frac{\mathrm{d}^k a}{d \varphi^k} (0)\)