1.1 The main theorem

We consider subcritical SPDEs of the form (1.3) such that Assumptions 2 and 3 hold. Subcriticality ensures that one can construct a problem-dependent regularity structure as in [Reference Bruned, Hairer and ZambottiBHZ19], and Assumptions 2 and 3 guarantee by [Reference Chandra and HairerCH16, Theorem 2.33] the convergence of the sequence of admissible models $\hat {Z}^{\varepsilon }$ to a random limit model $\hat {Z}$ , where $\hat {Z}^{\varepsilon }$ denotes the renormalised canonical lift of the regularised noise $\xi ^{\varepsilon }$ (see Section 2.2.2). Furthermore, we can only expect a support theorem to hold if the integration kernels associated to our equations are homogeneous on small scales, and in order to not overcomplicate the presentation, we assume that our Green’s functions are self-similar under rescaling (compare Assumption 4). For convenience, we also restrict to the case of independent (space or space-time) Gaussian white noises $\xi _i$ (but compare Remark 2.5). Our assumptions ensure that equation (1.3) can be lifted to an abstract fixed point problem as in [Reference HairerHai14, Theorem 7.8]. Finally, we need a technical assumption on the trees that appear in our regularity structure, which for ease of this introduction we will not comment on; we refer the interested reader to Assumptions 5 and 6 in Section 2.5.

In order to have a well-behaved solution map, it is convenient to be in the slightly more restrictive setting of [Reference Bruned, Chandra, Chevyrev and HairerBCCH17], which guarantees in particular that the reconstructed solution to the abstract fixed point problem for $\hat {Z}^{\varepsilon }$ satisfies the regularised and renormalised SPDE (1.4). We thus assume for the sake of the main results – Theorems 1.6 and 1.7 – that the full assumptions of [Reference Bruned, Chandra, Chevyrev and HairerBCCH17] are satisfied.

Our main result then is a support theorem for the BPHZ renormalised model $\hat {Z}$ or indeed any model differing from $\hat {Z}$ by the action of an element of the renormalisation group $\mathcal {G}_-$ associated to the class of equations under consideration. If we denote by $Z(h)$ the canonical lift of any $h \in \mathcal {C}_0^{\infty }$ , a slightly informal version of our main result reads as follows:

Theorem 1.5. There exist a subgroup $\mathcal {H} \subset \mathcal {G}_-$ and a left coset $g\mathcal {H}$ of $\mathcal {H}$ such that

$$ \begin{align*} \operatorname{\mathrm{supp}} \hat{Z} = \overline{\left\{ \mathcal{R}^h Z(f): f \in \mathcal{C}_0^{\infty}, h \in g\mathcal{H}\right\}}. \end{align*} $$

One important remark here is that $\mathcal {H}$ is not determined solely by the regularity structure of the problem. Instead, it also incorporates information about the symmetries satisfied by the integration kernels associated to the problem. Extracting this ‘rigid’ algebraic data out of ‘soft’ analytic data is one of the main difficulties of this article.

We also have a more concrete statement at the level of solutions, which we state now. Regarding solutions, our support theorem applies for u in any space $\mathcal {X}= \bigoplus _{i} \mathcal {X}_i$ such that the solution operator (mapping the space of admissible models for the regularity structure $\mathcal {T}$ into $\mathcal {X}$ ) is continuous. For instance, one could define the space $\mathcal {X}_i$ as a version of the usual Hölder spaces allowing for finite-time blow-up as in [Reference Bruned, Chandra, Chevyrev and HairerBCCH17]. In situations where we know a priori that the solution survives until some deterministic time $T>0$ almost surely, one can take alternatively for $\mathcal {X}_i$ the usual Hölder–Besov spaces $\mathcal {C}^{-\frac {\lvert {\mathfrak {s}}\rvert }{2} + \beta _i -\kappa }_{ \mathfrak {s}_{\Lambda } }((0,T) \times \mathbf {T}^e)$ . (Here $\beta _i>0$ and the scaling ${ \mathfrak {s}_{\Lambda } } : \{0, \dotsc , e\} \to {\mathbf N}$ are determined by the linear part $\partial _t - \mathcal {L}_i$ of our equations; see Assumption 4. The statement holds for any $\kappa>0$ .) The main theorem of this article is the following description of the topological support of u:

Theorem 1.6. Under Assumption 1, let $u^{\varepsilon }$ denote the classical solutions to the regularised and renormalised equation (1.4) with noise $\xi ^{\varepsilon }$ and renormalisation constants $c^{\varepsilon }_{\tau } = h\circ g^{\varepsilon }_{\textrm{BPHZ}}(\tau )$ for some fixed $h \in \mathcal {G}_-$ , and set $u := \lim _{\varepsilon \to 0}u^{\varepsilon }$ . Then one has the identity

$$ \begin{align*} \operatorname{\mathrm{supp}} u=\bigcap_{\varepsilon>0} \overline{\bigcup_{\delta<\varepsilon} \operatorname{\mathrm{supp}} u^{\delta}} \end{align*} $$

in $\mathcal {X}$ .

In Theorem 3.14 we show that Assumptions 2–5, 7 and 8 imply an analogous support theorem for the random models associated to u and $u^{\delta }$ . More precisely, we show that

(1.9) $$ \begin{align} \operatorname{\mathrm{supp}} \hat{Z}=\bigcap_{\varepsilon>0}\overline{\bigcup_{\delta<\varepsilon} \operatorname{\mathrm{supp}} \hat{Z}^{\delta}}, \end{align} $$

where $\hat {Z}^{\delta }$ denotes the BPHZ model associated to the noise $\xi ^{\delta }$ and $\hat {Z}$ denotes its limit, namely the BPHZ model associated to the limiting white noise $\xi $ .

Once we know equation (1.9), Theorem 1.6 is a direct consequence of the continuity of the solution operator given in [Reference Bruned, Chandra, Chevyrev and HairerBCCH17, Theorem 2.21], combined with the fact that given a measure $\mu $ and a continuous map F, $\operatorname {\mathrm {supp}} F^*\mu $ is given by the closure of $F(\operatorname {\mathrm {supp}} \mu )$ . It will become clear from our proof that for a ‘tweaked’ choice of renormalisation constants $\tilde {c}^{\varepsilon }_{\tau } = k^{\varepsilon } \circ h \circ g^{\varepsilon }_{\textrm{BPHZ}}(\tau )$ with $k^{\varepsilon } \to \mathbf {1}^*$ as $\varepsilon \to 0$ , one can show that denoting by $\tilde {u}^{\varepsilon }$ the classical solution to the system(1.3) with renormalisation constants $\tilde {c}^{\varepsilon }$ , one still has $\tilde {u}^{\varepsilon } \to u$ in probability in $\mathcal {X}$ , but one has the stronger statement

$$ \begin{align*} \operatorname{\mathrm{supp}} \tilde{u}^{\varepsilon}\subseteq \operatorname{\mathrm{supp}} u \end{align*} $$

for any $\varepsilon>0$ .

We also have a characterisation of the support in the spirit of Stroock and Varadhan’s support theorem for SDEs [Reference Stroock and VaradhanSV72]. The ‘correct’ way to resolve the issue of divergent renormalisation constants in such a description turns out to be the following:

Theorem 1.7. Under Assumption 1, let $h \in \mathcal {G}_-$ and u be as in Theorem 1.6. There exist a Lie subgroup $\mathcal {H} \subseteq {\mathcal {G}_-}$ of the renormalisation group and a character $f \in {\mathcal {G}_-}$ independent of the choice of h in Theorem 1.6 such that the following holds: The support $\operatorname {\mathrm {supp}} u$ is given by the closure of all solutions $\varphi $  to

(1.10) $$ \begin{align} \partial_t \varphi_{i} = \mathcal{L}_i \varphi_i + F_i(\varphi, \nabla \varphi, \dotsc) + \sum_{j\le n} F_i^j(\varphi, \nabla \varphi, \dotsc) \psi_j + (\Upsilon_i k)(\varphi, \nabla \varphi, \dotsc), \end{align} $$

for any character $k \in h \circ f \circ \mathcal {H}$ , initial conditionFootnote ¹ $\varphi (0,\cdot ) \in \Phi _0^k$ and smooth deterministic functions $\psi _j$ , $j=1, \dotsc , n$ (depending only on space if $\xi _j$ is purely spatial white noise). Here we write $\Upsilon _i k := \left (\Upsilon M^{k}\Omega _F \right )_i$ for simplicity.

This theorem follows from Proposition 3.8, the properties of the shift operator, Theorem 2.4 and the continuity of the solution operator. The Lie subgroup $\mathcal {H}$ is given as the annihilator of a finite number of linear ‘constraints’ between the renormalisation constants. We refer the reader to Definition 3.3 for a precise definition. The tweaking by f is necessary, since the BPHZ characters respect these constraints only up to order $1$ (a by-product of the fact that we use truncated integration kernels for its definition).

Remark 1.11. The classical Stroock–Varadhan support theorem can be viewed as the case $d=0$ of our result with $\mathcal {L}_i = 0$ . In this case, one has $\mathcal {G}_- \simeq (\mathbf {R},+)$ Footnote ² and, in the notation of Theorem 1.7,

$$ \begin{align*} (\Upsilon_i c)(u) = c F_k^j(u) \partial_k F_i^j(u),\quad c \in \mathbf{R} \simeq \mathcal{G}_-, \end{align*} $$

with summation over j and k implied. Furthermore, using the BPHZ model (and therefore setting $h = 0$ ) leads to solutions in the Itô sense. Since there is only one renormalisation constant in this case and the ‘heat kernel’ is given by the Heaviside function, which is nontrivial, Definition 3.3 readily leads us to the conclusion that $\mathcal {J}$ is the unit ideal, so that its annihilator is given by $\mathcal {H} = \{0\}$ .

Theorem 1.7 then states that there exists some constant c such that the support of the Itô solutions to

$$ \begin{align*} du_i = F_i(u)dt + \sum_{j \le n} F_i^j(u)dW_j \end{align*} $$

is given by the closure of all solutions to

(1.11) $$ \begin{align} \dot u_i = F_i(u) + \sum_{j \le n} F_i^j(u)\psi_j + c \sum_{k \le m} \sum_{j \le n} F_k^j(u) \partial_k F_i^j(u), \end{align} $$

with smooth controls $\psi _j$ . Note that the correct value of c (corresponding to the character f in the statement) is not specified by the theorem. On the other hand, one can explicitly compute the ‘BPHZ character’ in this case and show that (again identifying $\mathcal {G}_-$ with $\mathbf {R}$ ) it converges as $\varepsilon \to 0$ to $-\textstyle {\frac {1}{2}}$ , and we conclude from Remark 1.10 that $c = -\textstyle {\frac {1}{2}}$ in equation (1.11), thus recovering the Stroock–Varadhan support theorem.

Before we proceed, let us briefly discuss how these results compare to the existing literature. There are of course many support theorems for stochastic PDEs that do not require renormalisation – see, for example, [Reference Bally, Millet and Sanz-SoléBMS95, Reference Cardon-Weber and MilletCWM01, Reference Chueshov and MilletCM11, Reference Delgado-Vences and Sanz-SoléDVSS14]. In all of these cases, the statement is the one that one would expect, namely that the support is given by the closure of all solutions obtained by replacing the noises by suitable controls. In the case of singular SPDEs, information on the support follows in some special cases. For example, Jona-Lasinio and Mitter [Reference Jona-Lasinio and MitterJLM85] construct solutions to a type of Langevin equation for the $\Phi ^4_2$ measure by using Girsanov’s theorem, which yields full support as an immediate by-product. One of the earliest results on the support in cases that cannot be dealt with in this way is the work by Chouk and Friz [Reference Chouk and FrizCF18] in which they consider a generalised parabolic Anderson model of the form $\partial _t u = \Delta u + g(u) \xi $ in dimension $2$ and show that a suitably renormalised version of it has support given by the closure of all solutions to control problems of the type $\partial _t u = \Delta u + g(u) \phi + c (gg')(u)$ with $\phi $ a smooth function (constant in time) and c an arbitrary constant. This can be viewed as a special case of our result in a situation where $\mathcal {H} = \mathcal {G} \approx (\mathbf {R},+)$ . The way we deal with the presence of renormalisation, while inspired by [Reference Chouk and FrizCF18], substantially differs from the construction given there. See the discussion at the start of Section 5 for more details.

Using similar techniques, Tsatsoulis and Weber [Reference Tsatsoulis and WeberTW18] showed that the $\Phi ^4_2$ dynamic has full support. Finally, proofs of support theorems for stochastic ordinary differential equations based on rough-path techniques are by now very classical. It was already mentioned in [Reference LyonsLyo98] that the continuity properties of the solution map can be used for a straightforward proof of a support theorem, provided one has a support theorem for the enhanced Brownian motion. The latter was shown in a series of results – see for instance [Reference Ledoux, Qian and ZhangLQZ02] (for a support theorem for rough paths in the p-variation topology), [Reference Friz, Waymire and DuanFri05] (in Hölder topology), [Reference Friz and VictoirFV06] (for enhanced fractional Brownian motions) and [Reference Friz and VictoirFV10a] (for an implementation using deterministic shifts). For an introduction to the topic and more details, see [Reference Friz and HairerFH14, Section 9.3] or [Reference Friz and VictoirFV10b, Chapter 19].

1.2 Applications

1.2.1 The $\Phi ^p_d$ equation

The $\Phi ^p_d$ equation formally is given by

(1.12) $$ \begin{align} \partial_t u = \Delta u + \sum_{1 \le k \le p-1} a_k u^k + \xi \end{align} $$

with space-time white noise $\xi $ on ${\mathbf D} = \mathbf {R} \times \mathbf {T}^d$ . This equation is subcritical in the sense of [Reference HairerHai14, Reference Bruned, Hairer and ZambottiBHZ19], provided that $p < 2d/(d-2)$ . As already pointed out, in a formal sense, one can also consider equation (1.12) in dimension $d-\varepsilon $ , either by replacing $\Delta $ by $-(-\Delta )^{1+\varepsilon }$ or by convolving $\xi $ with a slightly regularising Riesz kernel. We will restrict ourselves here to the cases $d=2$ and p even, $d =3$ and $p=4$ , and $d = 4-\varepsilon $ and $p=4$ . We denote by ‘the’ solution to equation (1.12) the BPHZ solution in the sense of [Reference Bruned, Hairer and ZambottiBHZ19, Reference Chandra and HairerCH16] for any fixed truncation K of the heat kernel. All statements that follow are independent of the choice of cutoff.

Note that in dimension $d=2$ , Assumption 6 is violated, but as is pointed out in Remark 2.23, Assumption 6 can be replaced by Assumptions 7 and 8, which are trivially true in this case (one has $\mathcal {J}:=\{ 0 \}$ and $\mathcal {H} = \mathcal {G}_-$ ). In dimension $d=3$ , all assumptions are satisfied. However, the ‘black-box’ theorem of [Reference Bruned, Chandra, Chevyrev and HairerBCCH17] only allows us to start the approximate equation at a perturbation of $\mathcal {S}^-_{\varepsilon }(\xi )(0,\cdot )$ (compare Remark 1.2; in this case $\mathcal {S}^-_{\varepsilon }(\xi )(0,\cdot )$ is in law a smooth approximation to the Gaussian free field). As was already noticed in [Reference HairerHai14, Section 9.4], this issue can be circumvented, but this requires working with a model topology which is slightly stronger than the usual one. We show in Appendix C.1 that the support theorem still holds for this topology. If we emulate dimension $d=4-\varepsilon $ by slightly regularising the noises, then our assumptions on the noises are violated (since they are no longer white), but it is again possible to resolve this issue (see Appendix C.2). We will be interested in showing the ergodicity of equation (1.12), so that we will always assume that $a_{p-1} <0$ . Under this condition, we have the following consequence of Theorem 1.7:

Theorem 1.12. Set $u_0 \in \mathcal {C}^{\eta }(\mathbf {T}^3)$ , where $\eta> -\frac {2}{3}$ if $d=2,3$ and $\eta>-(\varepsilon \wedge \frac 23 )$ if $d=4-\varepsilon $ . Let u denote the solution to the $\Phi ^p_d$ equation with the combinations of p and d already mentioned, with initial condition $u_0 + \mathcal {S}^-(\xi )(0,\cdot )$ (in the sense of Remark 1.2). Then for any $T>0$ , u has full support in $\mathcal {C}^{\alpha }_{\mathfrak {s}}\left ((0,T) \times \mathbf {T}^d\right )$ for $\alpha = \frac {2-d}{2}-\kappa $ for any $\kappa>0$ .

For $d=2,3$ , let $\alpha $ be as in the foregoing and consider the solution u with fixed initial condition $u_0 \in \mathcal {C}^{\eta }\left (\mathbf {T}^3\right )$ for some $\eta \in (-\frac 23,\alpha ]$ . Then u has support in $\mathcal {C}\left ([0,T],\mathcal {C}^{\eta }\left (\mathbf {T}^d\right )\right )$ given by all functions with value $u_0$ at time $0$ .

Proof . Global existence for these equation was shown in [Reference Tsatsoulis and WeberTW18] in $d=2$ and in [Reference Mourrat and WeberMW17, Reference Moinat and WeberMW20] in $d=3$ . For $d=4-\varepsilon $ it will be a consequence of a forthcoming paper [Reference Chandra, Moinat and WeberCMW19]. The first statement then follows directly from Theorem 1.7, which shows that any trajectory can be realised, since the equation is driven by additive noise.

The second statement does not follow immediately, because the topology of our model space is too weak for the solution map to be continuous as a map with values in $\mathcal {C}\left ([0,T],\mathcal {C}^{\eta }\left (\mathbf {T}^d\right )\right )$ . We show in Appendix C.1 that one can endow it with a slightly stronger topology in such a way that the solution map becomes continuous and our support theorem still holds.

A particular application of our support theorem in dimension $d \le 3$ is to the uniqueness of the invariant measure and exponential convergence to this measure:

Corollary 1.13. Assume that p, $d\le 3$ and $a_{p-1}$ are as before. Then the $\Phi ^p_d$ equation admits a unique invariant measure $\mu $ on $\mathcal {C}^{\alpha }\left (\mathbf {T}^d\right )$ .

Moreover, if $p \ge 4$ , then we have uniform exponential convergence of the dynamical model to the invariant measure in the following sense: Let u be the solution starting from $u_0$ as in Theorem 1.12. Then

(1.13) $$ \begin{align} \lVert (u_t)_*\mathbf{P} - \mu \rVert_{\mathrm{TV}} \le 1 \wedge C\exp (- \lambda t), \end{align} $$

for some $C,\lambda>0$ , uniformly over $t \ge 0$ and $u_0 \in \mathcal {C}^{\alpha }\left (\mathbf {T}^d\right )$ . (Here, $f_*\mathbf {P}$ denotes the push-forward of the measure $\mathbf {P}$ under the random variable f.)

Proof . This follows from Doeblin’s theorem (see, for instance, [Reference HairerHai16a, Theorem 3.6] with $V=0$ ) that it suffices to show that for some $t>0$ , one hasFootnote ³

(1.14) $$ \begin{align} \left\lVert \left(u_t^{v}\right)_*\mathbf{P} - \left(u_t^{w}\right)_*\mathbf{P} \right\rVert_{\mathrm{TV}} \le 1-\delta \end{align} $$

for some $\delta>0$ and all $v,w \in \mathcal {C}^{\alpha }\left (\mathbf {T}^d\right )$ . Here $u^{v}$ denotes the solution to equation (1.12) with initial condition v.

As a consequence of the ‘coming down from infinity’ property (see [Reference Tsatsoulis and WeberTW18, Equation 3.24] for $d=2$ , [Reference Mourrat and WeberMW17, Equation 1.27] for $d=3$ ; see also [Reference Moinat and WeberMW20]), there exists a compact set $K \subseteq \mathcal {C}^{\alpha }\left (\mathbf {T}^d\right )$ such that

$$ \begin{align*} \inf_{v \in \mathcal{C}^{\alpha}\left(\mathbf{T}^d\right)} \mathbf{P}\left[u_1^{v} \in K\right] \ge {\frac{1}{2}}. \end{align*} $$

By the strong Feller property for $\Phi ^p_d$ shown in [Reference Hairer and MattinglyHM18] (see also [Reference Tsatsoulis and WeberTW18] for $d=2$ ), the transition probabilities are continuous in the total variation norm, so that for some $\varepsilon>0$ one has

$$ \begin{align*} \left\lVert \left(u_1^{v}\right)_*\mathbf{P} - \left(u_1^{w}\right)_*\mathbf{P} \right\rVert_{\mathrm{TV}} \le {\frac{1}{2}} \end{align*} $$

for any v, w in the centred $\varepsilon $ -ball $B_{\varepsilon }$ in $\mathcal {C}^{\alpha }\left (\mathbf {T}^d\right )$ . Again by the continuity of the transition probabilities and the compactness of K, the infimum

$$ \begin{align*} \rho:=\inf_{v \in K} \mathbf{P}\left[u^{v}_1 \in B_{\varepsilon}\right] \end{align*} $$

is attained for some $\bar {v} \in K$ , and by Theorem 1.12, one has $\rho> 0$ . It follows that formula (1.14) holds for $t=3$ with $\delta = \frac {1}{4} \rho $ .

Remark 1.14. We have to restrict to $d \le 3$ in Corollary 1.13 because it is not known whether the solution to $\Phi ^4_{4-\varepsilon }$ is a Markov process (although it is expected). Actually, at the current state it is even unclear whether one can start the equation at a fixed deterministic initial condition (compare Remark 1.2 for a discussion of this issue) or evaluate the solution at a fixed positive time.

1.2.2 The generalised KPZ equation

A natural analogue to the class of SDEs (1.1) is given by the class of SPDEs recently studied in [Reference HairerHai16b, Reference Bruned, Gabriel, Hairer and ZambottiBGHZ19], which can formally be written as

(1.15) $$ \begin{align} \partial_t u = \partial_x^2 u + \Gamma(u)(\partial_xu, \partial_x u) + h(u) + \sum_{i=1}^m \sigma_i(u)\xi_i, \end{align} $$

where $u \colon \mathbf {R}_+ \times S^1 \to \mathbf {R}^n$ , the $\xi _i$ denote independent space-time white noises, $h:\mathbf {R}^n \to \mathbf {R}^n$ and $\sigma _i :\mathbf {R}^n \to \mathbf {R}^n$ are smooth functions and $\Gamma $ is a smooth map from $\mathbf {R}^n$ into the space of symmetric bilinear maps $\mathbf {R}^n \times \mathbf {R}^n \to \mathbf {R}^n$ . This should be viewed as a connection on $\mathbf {R}^n$ , which is why we use the customary symbol $\Gamma $ for it, and it gives rise to a notion of covariant differentiation:

(1.16) $$ \begin{align} (\nabla_X Y)^i(u) = X^j(u)\partial_j Y^i(u) + \Gamma^{i}_{j,k}(u)X^j(u)Y^k(u), \end{align} $$

for any two smooth vector fields $X,Y \colon \mathbf {R}^n \to \mathbf {R}^n$ .

One problem in trying to even guess the form of a support theorem for an equation like equation (1.15) is that there is typically no canonical notion of a solution associated to it. Instead, one has a whole family of solution theories that can be parametrised by a renormalisation group $\mathcal {G}_-$ . This already happens for SDEs where one has a natural one-parameter family of solution theories which include solutions in the sense of Itô, Stratonovich, backwards Itô, and so on, so that $\mathcal {G}_- = (\mathbf {R},+)$ in this case. While $\mathcal {G}_-$ is always a finite-dimensional Lie group, it can be quite large in general: Even after taking into account the $x \leftrightarrow -x$ symmetry and the fact that the noises $\xi _i$ are Gaussian and independent and identically distributed, one has $\mathcal {G}_- = \left (\mathbf {R}^{54},+\right )$ in the case of equation (1.15) (at least for n large enough; see [Reference Bruned, Gabriel, Hairer and ZambottiBGHZ19, Proposition 6.8]). Furthermore, there is typically no naïve analogue of the Wong–Zakai theorem: If one simply replaces $\xi $ by a mollified version $\xi ^{(\varepsilon )}$ , the resulting sequence of solutions $u^{(\varepsilon )}$ typically fails to converge to any limit whatsoever. Instead, one needs to modify the right-hand side of the equation in an $\varepsilon $ -dependent way in order to obtain a well-defined limit.

In some cases, imposing additional desirable properties on the solution theory results in a reduction of the number of degrees of freedom but still leads to mollifier-dependent counterterms. For example, it is shown in [Reference Bruned, Gabriel, Hairer and ZambottiBGHZ19] that equation (1.15) admits a natural one-parameter family of solution theories, all of which satisfy all of the following properties simultaneously:

• • The usual chain rule holds in the sense that if u solves equation (1.15) and $v = \phi (u)$ for some diffeomorphism $\phi \colon \mathbf {R}^n \to \mathbf {R}^n$ , then v solves the equation obtained from equation (1.15) by formally performing the corresponding change of variables as if the $\xi _i$ were smooth. (This is analogous to the property of Stratonovich solutions to SDEs.)

• • If $\{\tilde {\sigma }_j \}_{j=1}^{\tilde {m}}$ is a collection of smooth vector fields on $\mathbf {R}^n$ such that

  $$ \begin{align*} \sum_{i=1}^m \sigma_i(u) \otimes \sigma_i(u) = \sum_{j=1}^{\tilde{m}} \tilde{\sigma}_j(u) \otimes \tilde{\sigma}_j(u), \end{align*} $$
  then the solution to equation (1.15) is identical in law to the solution with the $\sigma _i$ replaced by the $\tilde {\sigma }_i$ . (This is analogous to the property of Itô solutions to SDEs.)

• • Given equation (1.15), there exists a collection of $12$ vector fieldsFootnote ⁴ $W_i$ on $\mathbf {R}^n$ such that for any mollifier $\rho $ , there exist constants $c_i^{(\varepsilon )}$ such that, setting $\xi _i^{(\varepsilon )} = \rho _{\varepsilon } \star \xi _i$ , solutions to equation (1.15) are given by $u = \lim _{\varepsilon \to 0}u_{\varepsilon }$ with

  (1.17) $$ \begin{align} \partial_t u_{\varepsilon} = \partial_x^2 u_{\varepsilon} + \Gamma(u_{\varepsilon})(\partial_xu_{\varepsilon}, \partial_x u_{\varepsilon}) + h(u_{\varepsilon}) + \sum_{i=1}^m \sigma_i(u_{\varepsilon})\xi_i^{(\varepsilon)} - \sum_{j=1}^{12}c_j^{(\varepsilon)} W_j(u_{\varepsilon}). \end{align} $$
  Furthermore, the $W_j$ are such that for every $u_{\star } \in \mathbf {R}^n$ such that $\Gamma (u_{\star }) = 0$ and $D \sigma _i(u_{\star }) = 0$ (for $i> 0$ ), one has $W_j(u_{\star }) = 0$ for every j.

Given equation (1.15), we then define a number of auxiliary vector fields. First, for $\mu ,\nu =1,\dotsc ,m$ , we set

$$ \begin{align*} X_{\mu\nu}(u) = (\nabla_{\sigma_{\mu}}\sigma_{\nu})(u), \end{align*} $$

and we also write $V_{\star }$ for the vector field $H_{\Gamma ,\sigma }$ defined in [Reference Bruned, Gabriel, Hairer and ZambottiBGHZ19, Equation 1.9]. We then use the $X_{\mu \nu }$ to define two additional vector fields

$$ \begin{align*} V = X_{\mu\mu},\qquad \hat{V} = \nabla_{X_{\mu\nu}} X_{\mu\nu}, \end{align*} $$

with implied summation over repeated indices.

As already mentioned, this class of equations admits a one-parameter canonical family of solution theories that combine the formal properties of both Stratonovich and Itô solutions. We fix once and for all one of these solution theories and call it henceforth ‘the’ solution to equation (1.15). Again, our statement is independent of the precise choice of solution theory as long as it belongs to the canonical family. (Actually, this can be further weakened; see Appendix C.3.) Under the assumption that $\Gamma $ , h and $\sigma $ are smooth functions, we have the following result, the proof of which is postponed to Appendix C.3:

Theorem 1.15. Let u be the solution to equation (1.15) with deterministic initial condition $u(0) = u_0 \in \mathcal {C}^{\alpha }(\mathbf {T})$ for some $\alpha \in (0, {\frac {1}{2}} )$ . Then there exists a constant $\hat {c}$ such that the support of the law of u in $\mathcal {C}^{\alpha }(\mathbf {R}_+\times \mathbf {T})$ is given by the closure of all solutions to

(1.18) $$ \begin{align} \partial_t u^i &= \partial_x^2 u^i + \Gamma^{i}_{j,k}(u) \partial_x u^j \partial_x u^k + h^i ( u ) + \hat{c} \hat{V}^i(u) + K_{\star} V_{\star}^i(u) + K V^i(u) + \sigma_{\mu}^i ( u ) \eta^{\mu} \end{align} $$

for arbitrary smooth controls $\eta ^{\mu }$ and arbitrary constants $K, K_{\star }$ .

Remark 1.16. The appearance of the additional constants K and $\hat {c}$ in equation (1.18) may seem strange at first, although we have of course already seen in the discussion preceding equation (1.8) that one cannot expect to obtain the support of u by simply replacing noises by smooth controls in equation (1.18).

Remark 1.17. At this stage, we do not know whether one actually has $\hat {c} = 0$ (which would be natural) or whether the description already given even depends on the value of $\hat {c}$ . We do, however, know that both terms $V_{\star }$ and V are required for the result to hold, as follows from the example

$$ \begin{align*} \partial_t u_1 = \partial_x^2 u_1 + \xi,\qquad \partial_t u_2 = \partial_x^2 u_2 + (\partial_x u_1)^2,\qquad \partial_t u_3 = \partial_x^2 u_3 + (\partial_x u_2)^2, \end{align*} $$

with $u(0) = 0$ , say. In this case, $V \propto (0,1,0)$ and $V_{\star } \propto (0,0,1)$ , so that Theorem 1.15 (when combined with Lemma C.4) shows that the law of u has full support, whereas we would have $u_2(t) \ge C_2 t$ , $u_3(t) \ge C_3 t$ if we placed some constraints on the possible values of $K_{\star }$ and K.

1.3 Outline

All equations in our setting can be lifted to abstract fixed point problems [Reference HairerHai14, Theorem 7.8] in a problem-dependent regularity structure $\mathcal {T}$ . Exploiting the continuity of the solution map – mapping the space of admissible models $\mathcal {M}_0$ (see Section 2.2.1) continuously into some solution space $\mathcal {X}$ – we can redirect our focus toward showing Theorem 3.14, which gives a characterisation of the topological support of random models in complete analogy with Theorem 1.6. We are interested in random models $\hat {Z}$ obtained as the limit of a sequence of smooth random models $\hat {Z} = \lim _{\varepsilon \to 0} \hat {Z}^{\varepsilon }$ . The upper bound for the support of $\hat {Z}$ then follows from elementary probabilitytheory arguments. The basic idea to show the lower bound is to fix a deterministic model Z, for which we want to show $Z \in \operatorname {\mathrm {supp}} \hat {Z}$ , and to construct a sequence of ‘shifts’ $\xi +\zeta _{\delta }$ of the underlying Gaussian noise $\xi $ by a smooth random function $\zeta _{\delta } = \zeta _{\delta }(\xi )$ such that the ‘shifted model’ $\hat {Z}_{\delta }$ , formally given by $\hat {Z}_{\delta }(\xi ) = \hat {Z}(\xi + \zeta _{\delta })$ , converges to Z almost surely as $\delta \to 0$ . Since $\operatorname {\mathrm {supp}} \hat {Z}_{\delta } \subseteq \operatorname {\mathrm {supp}} \hat {Z}$ for any $\delta>0$ (this is not completely obvious, since $\zeta _{\delta }$ is not adapted in general, so Girsanov’s theorem need not apply; but see Lemma 3.12 for a proof) and $\operatorname {\mathrm {supp}} \hat {Z}$ is closed, this shows that $Z \in \operatorname {\mathrm {supp}} \hat {Z}$ . While this is the broad strategy already used in [Reference Bally, Millet and Sanz-SoléBMS95, Reference Chouk and FrizCF18, Reference Tsatsoulis and WeberTW18], the identification of a suitable shift $\zeta _{\delta }$ is significantly more involved in this case.

We want to consider random shifts for reasons outlined in detail later (most crucially, our shifted noises are still of the type considered in [Reference Chandra and HairerCH16]). It is then not even clear a priori what we mean by a ‘shifted model’, since the law of $\xi + \zeta _{\delta }(\xi )$ is not necessarily absolutely continuous with respect to the law of $\xi $ , so that simply evaluating the random limit model $\hat {Z}$ at $\xi + \zeta _{\delta }(\xi )$ is in general not well defined. Instead we rely on a purely analytic shift operator $T_f$ (Theorem 2.4; see also [Reference Hairer and MattinglyHM18, Theorem 3.1]), acting continuously on the space of admissible models and satisfying $\hat {Z}(\xi + f) = T_f \hat {Z}(\xi )$ for deterministic, smooth, compactly supported functions f (in which case $\hat {Z}(\xi + f)$ is well defined by the Cameron–Martin theorem), and we call $\hat {Z}_{\delta }(\xi ):=T_{\zeta _{\delta }(\xi )} \hat {Z}(\xi )$ the shifted model. From the deterministic continuity of the shift operator, we infer in particular that any shift maps the support of $\hat {Z}$ into itself (this also works for random shifts; see Lemma 3.12), so that we are left to find the set of models Z for which a shift as discussed can be constructed.

For the type of statement we are looking for, it suffices to consider models Z of the form $Z = \mathcal {R}^h Z_{\mathrm {c}}(f)$ for some tuple of smooth functions $f = (f_i)_{i \le m}$ , where $f_i \in \mathcal {C}_c^{\infty }(\mathbf {R} \times \mathbf {T}^e)$ for any $i \le m$ , and some character h in the renormalisation group ${\mathcal {G}_-}$ . (See Section 2.2 for the notation used here; $f \mapsto Z_{\mathrm {c}}(f)$ denotes the canonical lift, $\mathcal {R}:{\mathcal {G}_-} \times {\mathcal {M}_0} \to {\mathcal {M}_0}$ denotes the action of the renormalisation group onto the set of admissible models.) In fact, since the shift operator commutes with the action of the renormalisation group (Theorem 2.4), it suffices to consider $f=0$ in the sense that we aim to find a set $H \subseteq {\mathcal {G}_-}$ which is as large as possible such that for any $h \in H$ one can find a sequence of smooth random shifts $\zeta _{\delta }$ such that

(1.19) $$ \begin{align} \lim_{\delta \to 0} T_{\zeta_{\delta}} \hat{Z}(\xi) = \mathcal{R}^h Z_{\mathrm{c}}(0), \end{align} $$

where the limit is taken in the sense of convergence in probability in the space of models. Actually, since our proof draws on the results of [Reference Chandra and HairerCH16], we will automatically have convergence in $L^p$ for any $p\ge 1$ .

Since the limit we aim for as $\delta \to 0$ is deterministic, we are left to choose $\zeta _{\delta }$ in such a way that the variance of the models goes to zero, while the expected value has the correct behaviour in the limit. The first point is ensured if $\xi +\zeta _{\delta } \to 0$ in a strong enough sense, which will be formalised in Definition 2.13. Note that the space of noises introduced there is a subset of the one used in [Reference Chandra and HairerCH16], and our distance (formula (2.21)) is stronger (see Lemma 2.18). Our noises always live in a fixed inhomogeneous Wiener chaos with respect to some fixed Gaussian noise, which in particular allows us to work with a linear space of noises, and our distance is an actual norm on this space. The main issue is then to obtain equation (3.6), namely to ‘control’ the expected value $\hat {\Upsilon }^{\delta }\tau := {\mathbf E} T_{\zeta _{\delta }} \hat {\boldsymbol {\Pi }}^{\xi }\tau (0)$ of the finite number of trees $\mathscr {T}_-$ of negative homogeneity, so that in the limit $\delta \to 0$ they equal $h(\tau )$ . Here $\hat {\boldsymbol {\Pi }}^{\xi }$ denotes the renormalised canonical lift of $\xi $ and $T_{\zeta _{\delta }}$ is, as before, the shift operator acting on the space of admissible models.

These two properties are obviously necessary for the convergence (1.19) in $L^2$ in the space of models. To see this, note that if we write ${\boldsymbol {\Pi }}^g$ for the model $\mathcal {R}^g Z_{\mathrm {c}}(0)$ , then we have ${\boldsymbol {\Pi }}^g \tau (0) = g(\tau )$ for any $\tau \in \mathscr {T}_-$ . With a bit more effort (Proposition 3.21) it is possible to see that they are also sufficient. At this stage there are two main problems left to be solved, which we address respectively in Sections 4 and 5:

1. 1. What is the set H of characters h such that we can find a shift $\zeta _{\delta }$ as discussed? In particular, we have to show that this set is large enough to ‘almost’ contain the BPHZ character $g^{\varepsilon }$ (up to an $o(1)$ tweaking; see the remark after Theorem 1.6 or the second statement of Theorem 3.14).

2. 2. Given $h \in H$ , how does one construct a shift $\zeta _{\delta }$ such that $\xi +\zeta _{\delta } \to 0$ in some suitable space of admissible noises ${\mathfrak {M}_0}$ (see Definition 2.13) and such that $\lim _{\delta \to 0} {\mathbf E} T_{\zeta _{\delta }} \hat {\boldsymbol {\Pi }}^{\xi }\tau (0) = h(\tau )$ for every $\tau \in \mathscr {T}_-$ ?

Let us first discuss the second question, since our solution to this problem motivates the choice of H. It  is natural to make the ansatz $\zeta _{\delta } = -\xi ^{\delta } + k^{\delta }$ (see Section 5.1), where $\xi ^{\delta }$ is a smooth approximation of $\xi $ at scale $\delta $ and $k^{\delta }$ is a random, centred, stationary and smooth function living only on high frequencies, or equivalently on small scales (think of scales much smaller than $\delta $ ). The last property will ensure weak convergence of $k^{\delta }$ to $0$ as $\delta \to 0$ . If we simply chose $k^{\delta }=0$ , then the quantity $\hat {\Upsilon }^{\delta }\tau $ of some fixed tree $\tau \in \mathscr {T}_-$ would in general blow up, as shown in the following example:

Example 1.18. Consider the ‘cherry’ appearing in the regularity structure associated to the $\Phi ^4_3$ equation. Setting $\zeta _{\delta } = -\xi ^{\delta }$ (so $k^{\delta } = 0$ ) and using the fact that by the definition of the BPHZ character one has , one has

(1.20)

Here we use Feynman diagrams on the right-hand side to encode real constants in the same way as in, for example, [Reference HairerHai18] or [Reference HairerHai14, Section 10.5]. Straight lines represent the heat kernel, dotted lines represent the $\delta _0$ -distribution and wavy lines represent an approximation to $\delta _0$ at scale $\delta>0$ .

To see how a ‘high-frequency perturbation’ can solve this issue, consider adding a term of the form $k^{\delta } = a^{\delta } \xi ^{\lambda }$ with $\lambda =\lambda _{\delta } \ll \delta $ and $a^{\delta } \in \mathbf {R}$ . Similar to equation (1.20), one obtains

Fix now a number

. Then provided $\lambda \ll \delta $ , one can find $a^{\delta }$ such that

. To see this, observe that in the regime $\lambda \ll \delta $ and $a^{\delta } \ll 1$ , one has $\delta ^{-1} \ll \lambda ^{-1}$ and $\left (a^{\delta }\right )^2 \ll a^{\delta }$ , so that the third term dominates all other terms, and one can solve the fixed point problem

This strategy is complicated by two hurdles. First, one has to control various trees simultaneously, and it is a priori not clear that a perturbation designed to control one tree does not destroy the desired expected value of another. Indeed, it is not hard to see that with our strategy we are in general not able to control all trees $\tau \in \mathscr {T}_-$ at the same time to arbitrary values $h(\tau )$ , but we have to respect certain linear constraints between them – see Examples 4.2–4.5 for examples of such linear constrains in the context of various interesting SPDEs. (It is a crucial insight that these constraints are ‘almost’ satisfied by the BPHZ character; see the outline to follow and Assumption 8.)

The second problem comes from the fact that we also have to bound the expected values of trees with more than two leaves. If one tries to use high-frequency perturbations which are Gaussian, then in general trees with one white noise replaced by such a perturbation would not dominate the expression $\hat {\Upsilon }^{\delta }\tau $ . There are even trees for which these expressions vanish identically for any Gaussian shift ${ { \zeta } }_{\delta }$ . An example is the tree from the $\Phi ^5_2$ equation, for which we obtain (in case of a Gaussian shift ${ { \zeta } }_{\delta }$ )

(1.21)

Here, red nodes are new noise types and should be thought of as placeholders for the shift ${ { \zeta } }_{\delta }$ . Formally, the trees on the right-hand side of equation (1.21), which we call ‘shifted trees’, are elements of an enlarged regularity structure (see Section 5.1). The renormalisation group ${\mathcal {G}_-}$ acts naturally on by only considering contractions of original trees. In this way one can build for any $\varepsilon ,\delta>0$ a ‘renormalised’ model $\hat {\boldsymbol {\Pi }}^{\xi _{\varepsilon },\zeta _{\delta }}_{\mathrm {en}}$ , which converges in the limit $\varepsilon \to 0$ to a model $\hat {\boldsymbol {\Pi }}^{\xi ,\zeta _{\delta }}_{\mathrm {en}}$ , and we introduce the notation $\hat {\Upsilon }_{\mathrm {en}}^{\delta } \tau := {\mathbf E}\hat {\boldsymbol {\Pi }}^{\xi , \zeta _{\delta }}_{\mathrm {en}} \tau (0)$ . (Note that $\hat {\boldsymbol {\Pi }}^{\xi , \zeta _{\delta }}_{\mathrm {en}}$ is very different from the BPHZ renormalisation $\hat {\boldsymbol {\Pi }}^{\xi _{\varepsilon },\zeta _{\delta }}$ on the large regularity structure, in which case these quantities would vanish by the definition of the BPHZ character.) We will define just after equation (5.3) a shift operator , formally given by replacing blue nodes with red nodes in all possible ways, and we will show in Lemma 5.1 that $\hat {\Upsilon }^{\delta } = \hat {\Upsilon }_{\mathrm {en}}^{\delta } \mathscr {S}$ .

In the example earlier, $\hat {\Upsilon }_{\mathrm {en}}^{\delta }$ vanishes on any ‘shifted’ tree which does not appear on the right-hand side of equation (1.21). To clarify why, let us write $\Upsilon _{\mathrm {en}}^{\varepsilon ,\delta } \tau := {\mathbf E} {\boldsymbol {\Pi }}^{\xi _{\varepsilon },{ { \zeta } }_{\delta }} \tau (0)$ , where ${\boldsymbol {\Pi }}^{\xi _{\varepsilon },{ { \zeta } }_{\delta }}$ denotes the canonical lift of $(\xi _{\varepsilon },{ { \zeta } }_{\delta })$ (think of $\varepsilon \ll \delta $ ) to a model in the enlarged regularity structure. Using equation (5.4), one shows that

(1.22)

The second identity in this equation holds only if ${ { \zeta } }_{\delta }$ is Gaussian in general. This can be seen by using Wick’s rule of calculating the expected values of all trees involved, which shows that it identically vanishes for any fixed $\varepsilon> 0$ . Note also that the renormalisation constant of this tree vanishes identically – that is, one has for any smooth Gaussian noises $\eta $ and the BPHZ character $g^{\eta }$ , but the expectation after shifting the noise does not vanish, and with the choice $\zeta _{\delta }=-\xi ^{\delta }$ would blow up as $\delta \to 0$ .

One could now try to use shifted trees with more than one shifted noise to dominate the expression, but that would lead to two issues which seem difficult to resolve. First, in general it would now be subtrees of $\tau $ that dominate the behaviour of the shifted tree (in our example, it would be ), and one may see constraints between these trees. Contrary to the constraints we end up with, such constraints (between trees of different homogeneity with different number of leaves) are not seen at the level of the BPHZ characters. Second, while the equation we needed to solve for was a perturbation of a linear equation, we would now have to solve a polynomial equation, which introduces nonlinear constraints (for example, $\left (a^{\delta }\right )^2$ is always positive), and it is not clear if these polynomial equation can be solved (to worsen the matter, recall that we need to control various trees simultaneously, so that we end up with a system of polynomial equations).

We opt for a different way. We introduce a shift $k^{\delta }$ such that trees with one noise replaced by a shifted noise give a nonvanishing contribution. We ensure this by choosing $k^{\delta }$ such that the cumulant of $\left (k^{\delta },\xi ,\dotsc ,\xi \right )$ , with $m(\tau ):=\#L(\tau )-1$ instances of white noise $\xi $ , does not vanish. (Here $L(\tau )$ denotes the number of ‘leaves’ of $\tau $ .) The easiest way to guarantee this is to choose $k^{\delta }$ in the $m(\tau )$ th homogeneous Wiener chaos with respect to $\xi $ .

Example 1.20. Consider the tree

from the generalised KPZ equation, where we draw

and

to distinguish two different (hence independent) noise types. In this case we would choose our shift

where

is a suggestive way to write a kernel of the form

for some kernels

, and

denotes a third-order stochastic integral with respect to the joint law of

(see formula (2.9)). One then has the following graphical representation:

(1.23)

Here, a dark-red node represents an instance of $k^{\delta }$ . We would now rescale the kernels

to a scale $\lambda =\lambda _{\delta } \ll \delta $ at a homogeneity $\alpha $ , which is determined by the homogeneity

and

(see formula (5.7)).

(See Section 2.1 for the definition of the domain $\bar {\mathbf D}$ .)

The strategy we have outlined is implemented in Section 5 as follows: In Section 5.1 we construct an enlarged regularity structure, containing additional noise types (5.2), large enough to be able to represent the regularised noise $\xi ^{\delta }_{\Xi }$ (for any noise type $\Xi $ ) and the highly oscillating perturbation $k_{\left (\Xi ,\tau \right )}^{\delta }$ (for any tree $\tau $ and noise type $\Xi \in \mathfrak {t}(L(\tau ))$ appearing in $\tau $ ). We thus construct the shift operator as in equation (5.3). We determine the set of trees ${ \mathscr {S}^{\uparrow } }[\tau ]$ in the image of the shift operator which will dominate the expected value in Definition 5.2. In Section 5.2 we construct in formula (5.12) a ‘highly oscillating perturbation’ $\eta _{\left (\Xi ,\tau \right )}^{\delta }$ in the $m(\tau )$ th Wiener chaos for any tree $\tau \in \mathfrak {T}_-$ (see later for the definition of $\mathfrak {T}_- \subseteq \mathscr {T}_-$ ) and any $\Xi \in \mathfrak {t}(L(\tau ))$ . The kernel $K_{\left (\Xi ,\tau \right )}^{\delta }$ (with respect to Gaussian integration) of this perturbation is a rescaled version of a fixed kernel $\Phi _{\left (\Xi ,\tau \right )}$ (see formula (5.6)), at a homogeneity $\alpha _{\left (\Xi ,\tau \right )}$ (see formula (5.7)), to a scale $\lambda ^{\delta }_{\tau }$ (we will discuss shortly the choice of these scales). The kernels $\Phi _{(\Xi ,\tau )}$ are chosen along the lines of Example 1.20 (there is a slight subtlety here in case of log-divergencies – see Example 5.7 – which we ignore for the sake of this introduction).

A key result is Lemma 5.10, which determines the behaviour of the ‘dominating’ trees ${\tilde {\tau }} \in { \mathscr {S}^{\uparrow } }[\tau ]$ . It will be useful to introduce the function $F_{\tau }(a,\lambda ) := \hat {\Upsilon }^{\delta } \tau $ for $\tau \in \mathfrak {T}_-$ (see formula (5.20)), where $a=a^{\delta }_{\tau }$ and $\lambda =\lambda ^{\delta }_{\tau }$ , $\tau \in \mathfrak {T}_-$ . In Proposition 5.19 in Section 5.4, we then, for fixed $\lambda $ , recast the equation $F_{\tau }(a,\lambda ) = h(\tau )$ for $\tau \in \mathfrak {T}_-$ into a fixed point problem for a. This problem is a small perturbation of a solvable linear problem (linear because of the definition of $k^{\delta }$ , solvable thanks to Lemma 5.10 and a small perturbation thanks to Lemma 5.17), which is therefore straightforward to solve. The tricky issue is that in order for Lemma 5.17 to hold, one needs to choose the scales $\lambda _{\tau }^{\delta }$ carefully. In Section 5.3 we determine an order $\le $ on the set of trees $\mathfrak {T}_-$ , and we choose the scales such that $\lambda _{\tau }^{\delta } \ll \lambda _{{\tilde {\tau }}} ^{\delta }$ whenever $\tau ,{\tilde {\tau }} \in \mathfrak {T}_-$ with $\tau \le {\tilde {\tau }}$ . To formalise this idea, we introduce in Definition 5.14 the notion of an attainable statement, and we show at the end of Section 5.4 that the necessary bound of Lemma 5.17 is attainable in this sense.

We now outline how we will address the first point mentioned earlier – that is, how to define the set H, which we do in Section 4. Every tree $\tau \in \mathcal {T}$ can be mapped onto a function $\mathcal {K}\tau : \mathbf {R}^{(d+1)L(\tau )} \to \mathbf {R}$ (see formula (3.1)). One should think of $\mathcal {K}\tau $ as the function obtained by anchoring the root to the origin and integrating out all other vertices except for the leaves.

Example 1.21. In the case of the $\Phi ^4_3$ equation, one has, for instance,

where we identify the set of leaves

, with $1$ denoting the leaf directly attached to the root, and where K denotes a truncation of the heat kernel.

Denote now by $\hat {\mathcal {K}}\tau $ the function defined in same way, but with K replaced by the actual (i.e., not truncated) heat kernel $\hat {K}$ (we will later write $\mathcal {K}_{\hat {K}}\tau $ for this). It is a priori not clear that these integrals are well defined on large scales, but we show in Theorem 4.19 that at least for trees $\tau $ of nonpositive homogeneity this is always the case. Let us furthermore write $\mathcal {K}_{\mathrm {sym}} \tau $ and $\hat {\mathcal {K}}_{\mathrm {sym}}\tau $ for the kernels obtained from $\mathcal {K} \tau $ and $\hat {\mathcal {K}} \tau $ by symmetrisation under spatial reflections $(t,x) \mapsto (t,-x)$ and permutation of the variables. (If $\tau $ contains more than one noise type, one should only symmetrise variables corresponding to the same noise type.)

From the foregoing discussion, it is clear that we cannot hope to control two trees $\tau ,{\tilde {\tau }}$ independently if $\hat {\mathcal {K}}_{\mathrm {sym}}\tau $ and $\hat {\mathcal {K}}_{\mathrm {sym}}{\tilde {\tau }}$ are linearly dependent. To make this more clear, consider the following example:

Example 1.22. Continuing Example 1.20, one has

(1.24)

where we identify the leaves of with $\{1,2,3,4\}$ from left to right. Since one should think of as being rescaled to scales $\lambda \ll \delta $ , only the small-scale behaviour of $\mathcal {K}_{\mathrm {sym}}\tau $ matters, which is (essentially) the behaviour of the self-similar kernel $\hat {\mathcal {K}}_{\mathrm {sym}}\tau $ . (The last statement is justified by Lemma 5.10, where we show that the difference between equation (1.24) with $\mathcal {K}_{\mathrm {sym}}$ and with $\hat {\mathcal {K}}_{\mathrm {sym}}$ vanishes in the limit $\delta \to 0$ .) It follows that if ${\tilde {\tau }} \in \mathscr {T}_-$ is another tree carrying the same noise types as $\tau $ and such that $\hat {\mathcal {K}}_{\mathrm {sym}}{\tilde {\tau }} = c \hat {\mathcal {K}}_{\mathrm {sym}} \tau $ for some $c \in \mathbf {R}$ , then the shifted trees which are dominating (i.e., elements of ${ \mathscr {S}^{\uparrow } }[\tau ]$ and ${ \mathscr {S}^{\uparrow } }[{\tilde {\tau }}]$ ) satisfy the same linear relation in the limit $\delta \to 0$ .

Motivated by this example, we introduce in Definition 3.3 an ideal $\mathcal {J}\subset {\mathcal {T}_-}$ generated by linear combination of trees $\sigma \in \mathcal {T}$ carrying the same noise types and such that $\hat {\mathcal {K}}_{\mathrm {sym}} \sigma =0$ . Here we introduce the notation ${\mathcal {T}_-}$ for the free, unital, commutative algebra generated by $\mathscr {T}_-$ . We recall at this point [Reference Bruned, Hairer and ZambottiBHZ19] that ${\mathcal {T}_-}$ is naturally endowed with a Hopf algebra structure with coproduct ${\Delta _-}$ (the character group of ${\mathcal {T}_-}$ is precisely the renormalisation group ${\mathcal {G}_-}$ already mentioned); see Section 2.2.1 for details and precise references.

We show in Section 4 that $\mathcal {J}$ is a Hopf ideal (see Assumption 7). The crucial implication is that its annihilator $\mathcal {H}$ is a Lie subgroup of ${\mathcal {G}_-}$ . We show further Assumption 8, which states that the BPHZ character $g^{\varepsilon }$ of the regularised noise $\xi ^{\varepsilon }$ ‘almost’ belongs to this group, in the sense that one has $g^{\varepsilon } \in f^{\varepsilon } \circ \mathcal {H}$ , for a sequence of characters $f^{\varepsilon } \in {\mathcal {G}_-}$ which converges to a finite limit $f^{\xi }$ as $\varepsilon \to 0$ . It is crucial to note that we show this also for a class of non-Gaussian approximations $\xi ^{\varepsilon }$ which is rich enough to contain the shift $\zeta _{\delta }$ . Assumption 8 finally justifies the assertion already made that $g^{\varepsilon }$ ‘almost’ satisfies the linear constraints. (In a perfect world, $g^{\varepsilon }$ would satisfy these constraints precisely. The discrepancy stems from the fact that we use truncated kernels to define $g^{\varepsilon }$ .) Moreover, we have identified that the set $H \subseteq {\mathcal {G}_-}$ for which we can construct a shift as before is equal to the coset $f^{\xi } \circ \mathcal {H}$ . It  may be useful to observe that while the character $f^{\xi }$ is not uniquely defined, the coset $f^{\xi } \circ \mathcal {H}$ is unique.

Section 4 shows that a under a technical assumption (Assumption 6), Assumptions 7 and 8 always hold. The latter two are formulated as assumptions (rather than theorems) because there are a range of interesting equations in which Assumption 6 is violated, whereas one can simply show Assumptions 7 and 8 by hand. (Examples are the $\Phi _2^p$ equations discussed in Section 1.2.1 and the 2D parabolic Anderson model.) The general proof, assuming Assumption 6 and given in Section 4, is motivated and outlined at the beginning of this section.

We are left to link the two constructions we have outlined. In Definition 3.18 we define a set $\mathfrak {T}_- \subseteq \mathscr {T}_-$ , which is a maximal set with the property that $\mathop {\mathrm {Vec}} \mathfrak {T}_-$ and $\mathcal {J}$ are linearly independent (in other words, one has $\mathop {\mathrm {Vec}} \mathfrak {T}_- \oplus \mathcal {J} = \mathcal {T}_-$ , where $\oplus $ denotes the direct sum of vector spaces). For any fixed character $h \in f^{\xi } \circ \mathcal {H}$ , we will tailor a shift of the noise in Section 5 (see earlier in the outline) such that $\hat {\Upsilon }^{\delta } \tau \to h(\tau )$ as $\delta \to 0$ for any $\tau \in \mathfrak {T}_-$ . Using the fact that $\mathop {\mathrm {Vec}} \mathfrak {T}_-$ has a complement in $\mathop {\mathrm {Vec}} \mathscr {T}_-$ which is a subset of the ideal $\mathcal {J}$ , we will show in Proposition 3.21 that the sequence of shifted models converges to $\mathcal {R}^h Z_{\mathrm {c}}(0)$ almost surely, which shows in particular that

$$ \begin{align*} \left\{ \mathcal{R}^h Z_{\mathrm{c}}(f) : h \in f^{\xi} \circ \mathcal{H} , f \in \mathcal{C}_c^{\infty} \right\} \subseteq \operatorname{\mathrm{supp}} \hat{Z}. \end{align*} $$

Philosophically, Proposition 3.21 fills in the ‘gap’ between $\mathfrak {T}_-$ and $\mathscr {T}_-$ , in the sense that we do not need any a priori information on how the shifted models behave on trees $\tau \in \mathscr {T}_- \backslash \mathfrak {T}_-$ . This step relies of course on the relation between the set $\mathfrak {T}_-$ and the ideal $\mathcal {J}$ , and the fact that we choose $h \in f^{\xi } \circ \mathcal {H}$ , where $\mathcal {H}$ is the annihilator of $\mathcal {J}$ . What is less obvious, it also uses crucially the fact that $\mathcal {H}$ is indeed a subgroup of ${\mathcal {G}_-}$ (see Assumption 7). By Assumption 8, the ‘tweaked’ BPHZ character $f^{\xi } \circ (f^{\varepsilon })^{-1} \circ g^{\varepsilon }$ is an element of $f^{\xi }\circ \mathcal {H}$ for any $\varepsilon>0$ , and using the fact that $f^{\xi } \circ (f^{\varepsilon })^{-1} \to \mathbf {1}^*$ as $\varepsilon \to 0$ concludes the proof.