Abstract
We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the \(\Phi ^4_3\) Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of \(3\)-dimensional ferromagnets near their critical temperature.
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Acknowledgments
I am very grateful to M. Gubinelli and to H. Weber for our numerous discussions on quantum field theory, renormalisation, rough paths, paraproducts, Hopf algebras, etc. These discussions were of enormous help in clarifying the concepts presented in this article. Many other people provided valuable input that helped shaping the theory. In particular, I would like to mention A. Debussche, B. Driver, P. Friz, J. Jones, D. Kelly, X.-M. Li, M. Lewin, T. Lyons, J. Maas, K. Matetski, J.-C. Mourrat, N. Pillai, D. Simon, T. Souganidis, J. Unterberger, and L. Zambotti. Special thanks are due to L. Zambotti for pointing out a mistake in an earlier version of the definition of the class of regularity structures considered in Sect. 8. Financial support was kindly provided by the Royal Society through a Wolfson Research Merit Award and by the Leverhulme Trust through a Philip Leverhulme Prize.
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Appendices
Appendix A: A generalised Taylor formula
Classically, Taylor’s formula for functions on \(\mathbf{R}^d\) is obtained by applying the one-dimensional formula to the function obtained by evaluating the original function on a line connecting the start and endpoints. This however does not yield the “right” formula if one is interested in obtaining the correct scaling behaviour when applying it to functions with inhomogeneous scalings. In this section, we provide a version of Taylor’s formula with a remainder term having the correct scaling behaviour for any non-trivial scaling \(\mathfrak {s}\) of \(\mathbf{R}^d\). Although it is hard to believe that this formula isn’t known (see [15] for some formulae with a very similar flavour) it seems difficult to find it in the literature in the form stated here. Furthermore, it is of course very easy to prove, so we provide a complete proof.
In order to formulate our result, we introduce the following kernels on \(\mathbf{R}\):
For \(\ell = 0\), we extend this in a natural way by setting \(\mu _0(x,dy) = \delta _x(dy)\). With these notations at hand, any multiindex \(k \in \mathbf{N}^d\) gives rise to a kernel \({\mathcal Q}^k\) on \(\mathbf{R}^d\) by
where we define
where we defined the quantity
Note that, in any case, one has the identity \(\mu _i^k(z, \mathbf{R}) = {z^{k_i}\over k_i !}\), so that
Recall furthermore that \(\mathbf{N}^d\) is endowed with a natural partial order by saying that \(k \le \ell \) if \(k_i \le \ell _i\) for every \(i \in \{1,\ldots ,d\}\). Given \(k \in \mathbf{N}^d\), we use the shorthand \(k_< = \{\ell \ne k\,:\, \ell \le k\}\).
Proposition 11.1
Let \(A \subset \mathbf{N}^d\) be such that \(k\in A \Rightarrow k_{<} \subset A\) and define \(\partial A = \{k \not \in A\,:\, k - e_{\mathfrak {m}(k)} \in A\}\). Then, the identity
holds for every smooth function \(f\) on \(\mathbf{R}^d\).
Proof
The case \(A = \{0\}\) is straightforward to verify “by hand”. Note then that, for every set \(A\) as in the statement, one can find a sequence \(\{A_n\}\) of sets such as in the statement with \(A_a = \{0\}\), \(A_{|A|} = A\), and \(A_{n+1} = A_n \cup \{k_n\}\) for some \(k_n \in \partial A_n\). It is therefore sufficient to show that if (11.2) holds for some set \(A\), then it also holds for \(\bar{A} = A \cup \{\ell \}\) for any \(\ell \in \partial A\).
Assume from now on that (11.2) holds for some \(A\) and we choose some \(\ell \in \partial A\). Inserting the first-order Taylor expansion (i.e. (11.2) with \(A = \{0\}\)) into the term involving \(D^\ell f\) and using (11.1), we then obtain the identity
It is straightforward to check that one has the identities
valid for every \(m,n \ge 0\). As a consequence, it follows from the definition of \({\mathcal Q}^k\) that one has the identity
The claim now follows from the fact that, by definition, \(\partial \bar{A}\) is precisely given by \((\partial A \setminus \{\ell \}) \cup \{\ell + e_i \,:\, i \le \mathfrak {m}(\ell )\}\). \(\square \)
Appendix B: Symbolic index
In this appendix, we collect the most used symbols of the article, together with their meaning and the page where they were first introduced.
Symbol | Meaning |
---|---|
\(\mathbf {1}\) | Unit element in \(T\) |
\(\mathbf {1}^*\) | Projection onto \(\mathbf {1}\) |
\(|\cdot |_\mathfrak {s}\) | Scaled degree of a multiindex |
\(\Vert \cdot \Vert _\mathfrak {s}\) | Scaled distance on \(\mathbf{R}^d\) |
\(|\!|\!|\cdot |\!|\!|_{\gamma ;\mathfrak {K}}\) | “Norm” of a model / modelled distribution |
\(\circ \) | Dual of \(\Delta ^+\) |
\(\star \) | Generic product on \(T\) |
\(*\) | Convolution of distributions on \(\mathbf{R}^d\) |
\(A\) | Set of possible homogeneities for \(T\) |
\({\mathcal {A}}\) | Antipode of \({\mathcal {H}}_+\) |
\({\mathrm {Alg}}({\mathcal {C}})\) | Subalgebra of \({\mathcal {H}}_+\) determined by \({\mathcal {C}}\) |
\(\beta \) | Regularity improvement of \(K\) |
\({\mathcal {C}}^\alpha _\mathfrak {s}\) | \(\alpha \)-Hölder continuous functions with scaling \(\mathfrak {s}\) |
Symbol | Meaning |
---|---|
\(\fancyscript{D}\) | Abstract gradient |
\({\mathcal {D}}^\gamma \) | Modelled distributions of regularity \(\gamma \) |
\({\mathcal {D}}_P^{\gamma ,\eta }\) | Singular modelled distributions of regularity \(\gamma \) |
\(\Delta \) | Comodule structure of \({\mathcal {H}}\) over \({\mathcal {H}}_+\) |
\(\Delta ^+\) | Coproduct in \({\mathcal {H}}_+\) |
\(\Delta ^{\!M}\) | Action of \(M\) on \(\Pi \) |
\({\hat{\Delta }}^{\!M}\) | Action of \(M\) on \(\Gamma \) |
\(F\) | Nonlinearity of the SPDE under consideration |
\(F_x\) | Factor in \(\Gamma _{xy} = F_x^{-1} F_y\) |
\({\mathcal {F}}\) | All formal expressions for the model space |
\({\mathcal {F}}_+\) | All formal expressions representing Taylor coefficients |
\({\mathcal {F}}_F\) | Subset of \({\mathcal {F}}\) generated by \(F\) |
\({\mathcal {F}}_F^+\) | Subset of \({\mathcal {F}}_+\) generated by \(F\) |
\(G\) | Structure group |
\(\Gamma _g\) | Action of \({\mathcal {H}}_+^*\) onto \({\mathcal {H}}\) |
\({\mathcal {H}}_+^*\) | Dual of \({\mathcal {H}}_+\) |
\({\mathcal {H}}_+\) | Linear span of \({\mathcal {F}}_+\) |
\({\mathcal {H}}_F^+\) | Linear span of \({\mathcal {F}}_F^+\) |
\({\mathcal {H}}\) | Linear span of \({\mathcal {F}}\) |
\({\mathcal {I}}\) | Abstract integration map for \(K\) |
\({\mathcal {I}}_k\) | Abstract integration map for \(D^k K\) |
\({\mathcal {J}}(x)\) | Taylor expansion of \(K * \Pi _x\) |
\({\mathcal {J}}_k \tau \) | Abstract placeholder for Taylor coefficients of \(\Pi _x \tau \) |
\(\mathfrak {K}\) | Generic compact set in \(\mathbf{R}^d\) |
\(K\) | Truncated Green’s function of \({\mathcal {L}}\) |
\(K_n\) | Contribution of \(K\) at scale \(2^{-n}\) |
\(K_{n,xy}^\alpha \) | Remainder of Taylor expansion of \(K_n\) |
\({\mathcal {K}},{\mathcal {K}}_\gamma \) | Operator such that \({\mathcal {R}}{\mathcal {K}}f = K* {\mathcal {R}}f\) |
\({\mathcal {L}}\) | Linearisation of the SPDE under consideration |
\(\Lambda _n^\mathfrak {s}\) | Diadic grid at level \(n\) for scaling \(\mathfrak {s}\) |
\({\mathcal {M}}\) | Multiplication operator on \({\mathcal {H}}_+\) |
\(M\) | Renormalisation map |
\({\hat{M}}\) | Action of \(M\) on \(f\) |
\(M_g\) | Action of \(\fancyscript{S}\) on \(T\) |
\(\mathfrak {M}_F\) | Basic building blocks of \(F\) |
\(\fancyscript{M}_{\fancyscript{T}}\) | All models for \({\fancyscript{T}}\) |
\(\fancyscript{M}_F\) | All admissible models associated to \(F\) |
\({\mathcal {N}}_\gamma \) | Operator such that \({\mathcal {K}}_\gamma = {\mathcal {I}}+ {\mathcal {J}}+ {\mathcal {N}}_\gamma \) |
\((\varvec{\Pi },f)\) | Alternative representation of an admissible model |
\((\varvec{\Pi }^M,f^M)\) | Renormalised model |
\((\Pi ,\Gamma )\) | Model for a regularity structure |
\(\varphi _x^n\) | Scaling function at level \(n\) around \(x\) |
\(\psi _x^n\) | Wavelet at level \(n\) around \(x\) |
\(P\) | Time \(0\) hyperplane |
\({\mathcal {P}}_F\) | Formal expressions required to represent \(\partial u\) |
\({\mathcal Q}_\alpha \) | Projection onto \(T_\alpha \) |
\({\mathcal Q}_\alpha ^-\) | Projection onto \(T_\alpha ^-\) |
Symbol | Meaning |
---|---|
\(\varvec{R}^+\) | Restriction to positive times |
\(R\) | Smooth function such that “\(G = K + R\)” |
\(R_\gamma \) | Convolution by \(R\) on \({\mathcal {D}}^\gamma \) |
\({\mathcal {R}}\) | Reconstruction operator |
\(\mathfrak {R}\) | Renormalisation group |
\(\mathfrak {s}\) | Scaling of \(\mathbf{R}^d\) |
\({\mathcal {S}}_{\mathfrak {s},x}^\delta \) | Scaling by \(\delta \) around \(x\) |
\({\mathcal {S}}_P^\delta \) | Scaling by \(\delta \) in directions normal to \(P\) |
\(\fancyscript{S}\) | Discrete symmetry group |
\(T\) | Model space |
\(T_\alpha \) | Elements of \(T\) of homogeneity \(\alpha \) |
\(T_\alpha ^+\) | Elements of \(T\) of homogeneity \(\alpha \) and higher |
\(T_\alpha ^-\) | Elements of \(T\) of homogeneity strictly less than \(\alpha \) |
\(T_g\) | Action of \(\fancyscript{S}\) on \(\mathbf{R}^d\) |
\({\bar{T}}\) | Abstract Taylor polynomials in \(T\) |
\({\fancyscript{T}}\) | Generic regularity structure \({\fancyscript{T}}= (A,T,G)\) |
\({\mathcal T}_\beta \) | Classical Taylor expansion of order \(\beta \) |
\({\mathcal {U}}_F\) | Formal expressions required to represent \(u\) |
\(V,W\) | Generic sector of \(T\) |
\(X^k\) | Abstract symbol representing Taylor monomials |
\(\Xi \) | Abstract symbol for the noise |