# Molecular Chirality Quantification: Tools and Benchmarks

Ethan Abraham<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA  Abraham
Nitzan  Department of Chemistry, University of Pennsylvania, Philadelphia,
Pennsylvania 19104, USA  School of Chemistry, Tel Aviv University, Tel Aviv
69978, Israel

###### Abstract

Molecular chirality has traditionally been viewed as a binary property where a
molecule is classified as either chiral or achiral, yet in the recent decades
mathematical methods for quantifying chirality have been explored. Here we use
toy molecular systems to systematically compare the performance of two state
of the art chirality measures (1) the Continuous Chirality Measure (CCM) and
(2) the Chirality Characteristic ($\chi$). We find that both methods exhibit
qualitatively similar behavior when applied to simple molecular systems such
as a four-site molecule or the polymer double-helix, but we show that the CCM
may be more suitable for evaluating the chirality of arbitrary molecules or
abstract structures such as normal vibrational modes. We discuss a range of
considerations for applying these methods to molecular systems in general, and
we provide links to user-friendly codes for both methods. We aim this paper to
serve as a concise resource for scientists attempting to familiarize
themselves with these chirality measures or attempting to implement chirality
measures in their own work.

##  I Introduction

Chirality is defined as the absence of mirror symmetry. Since symmetry
properties are binary in the sense that an object either possesses or does not
possess a particular symmetry element, molecular chirality has historically
been a qualitative subject. But with molecular chirality playing a central
role in a variety of optical, biological, and chemical phenomenal, methods for
quantifying chirality could have great utility for research that attempts to
treat chirality as a physical variable. Renewed interest in molecular
chirality and its quantification has grown with the surge of observations
associated with the chiral induced spin selectivity phenomenon.Naaman and
Waldeck (2012); Evers  _et al._ (2022)

An example of the utility of chirality measurement a recent study by Piaggi et
al.Piaggi  _et al._ (2023) used molecular dynamics simulations of a liquid
composed of a simple four-site molecular model to demonstrate the existence of
symmetry-breaking phase transitions from a supercritical racemic liquid, into
subcritical D-rich and L-rich liquids. These phases were observed via the
distribution of chirality values of the liquid’s molecular constituents, each
of which could assume an arbitrary conformation (and thus chirality) at any
point throughout the simulation. As such, the success of this study depended
on the quality and implementation of a chirality measurement tool.

Chirality measures have recently been applied in a wide variety of contexts
such as the analysis of protein structure in biology,Sidorova  _et al._ (2021,
2019); Baruch-Shpigler  _et al._ (2017) the implication of chiral fermions in
topological condensed matter phenomena,Wan  _et al._ (2011); Armitage, Mele,
and Vishwanath (2018) and the role of chirality in photoelectron
spectroscopy.Katsoulis  _et al._ (2022) In our own work, chirality measurement
has been used to explore the role of mechanical twist on the thermal
conductivity of polymer wires,Abraham  _et al._ (2023) and to study the
correlation of chiral normal vibrational modes to the chirality of the
underlying molecular structure.Abraham and Nitzan (2023)

Approaches to chirality quantification have fallen into two general
categories. The first is based on geometric overlap between a structure and
its enantiomer, where high overlap is indicative of low chirality, and larger
difference is indicative of higher chirality. This idea was first explored
several decades ago when Gilat and Schulman defined a dimensionless chirality
measure as one minus the fractional volume overlap of the two
enantiomers.Gilat and Schulman (1985) This method has since been refined and
generalized by Avnir et al.,Dryzun and Avnir (2011); Pinsky  _et al._ (2008a)
using the Continuous Symmetry Measure (CSM) which relies on vector inner
products instead of volume overlap.Pinsky  _et al._ (2008b) This is the basis
of the Continuous Chirality Measure (CCM), the first tool addressed in this
paper.Dryzun and Avnir (2011); Pinsky  _et al._ (2008a)

An alternative approach to chirality quantification is based on the scalar
triple product of sequential vectors, utilizing the fact that an ordered set
of three vectors is achiral if and only if they lie in the same plane, in
which case the scalar triple product measures zero volume. In all other cases
the magnitude of the scalar triple product is nonzero, achieving a maximum
when the three vectors are mutually orthogonal. Like the overlap-based method,
the idea of using the scalar triple product to measure chirality can be traced
back several decades,Damhus and Schaeffer (1983) but has been more recently
refined by Sidorova et al. in their application to biological
proteins.Sidorova  _et al._ (2019, 2021) A strength of this measure is that
the anti-commutativity of the scalar triple product means that opposite
enantiomers will have opposite signs. This is the basis of the Chirality
Characteristic ($\chi$), the second tool addressed in this paper. Though
incredibly useful for certain simple systems as well as systems with linear
connectivity (e.g. polymers), a weakness of the scalar triple product approach
is that it is in general difficult to assign a definite order of sequential
vectors rendering the measure ill-defined. With greater mathematical
sophistication, Harris et al. presented a more general way of defining the
scalar triple product based on multipole moments of the molecule’s density
function,Harris, Kamien, and Lubensky (1999) but here we address simpler
choices for the triple-product vectors at the expense of some generality.

Figure 1: Left: ball-and-stick cartoon of a unit turn of a right-handed two
polymer double helix, the one of the toy molecular system on which chirality
measures are evaluated in this work. Different chains are denoted by different
colors. Three sequential separation vectors are highlighted. Middle: Mirror
plane reflection of the same structure to generate its left handed enantiomer.
Right: overlay of the two enantiomers on a common center of mass origin.

Figure 1 provides intuition for both approaches. The left panel depicts a
right-handed double helix, which is one of the toy molecular system on which
we will evaluate the CCM and $\chi$ in this work. It can be seen from the
vectors in the left panel that the magnitude of the inner product
$(\textbf{v}_{i}\times\textbf{v}_{i+1})\cdot\textbf{v}_{i+2}$ reports the
helicity of the structure, while the sign of this quantity distinguishes
between the enantiomers. This is the essence of the scalar-triple-product-
based chirality measures such as the CCM. The middle panel shows the left-
handed enantiomer of the same helix, formed by a mirror reflection. The right
panel shows an overlay of both mirror images on a common center of mass
origin. For an achiral structure, the second structure could be rotated such
that the overlap is perfect. For a more chiral structures, the overlap will be
less perfect. This is the essence of the overlap-based chirality measures such
as the CCM.

In this paper, we evaluate these chirality measures on toy molecular systems
in order to compare their performance. In Sec. II we present the mathematical
formulas for these measures. In Sec. III, we discuss the application of such
measures using two simple molecular models. The first model is a linear, four-
site molecule, and the second is a polymer double helix. Our findings are
summarized in Sec. IV.

##  II Chirality Measures

Here we will provide two state state of the art tools to measure the chirality
of molecular systems. The two tools presented correspond to the two general
classes of chirality measures discussed above, with the first based on the
overlap of enantiomers and the second based on the scalar triple product.

Continuous Chirality Measure. Let
$\textbf{Q}=\\{\textbf{q}_{1},...,\textbf{q}_{n}\\}$ denote the $n$ atomic
coordinates of the molecule of interest relative the center of mass. Then the
CCM is given by

$CCM=1-\frac{\max_{\hat{\sigma},\hat{P}}\sum_{i=1}^{n}\textbf{q}_{i}\cdot(\frac{\hat{\mathds{1}}+\hat{\sigma}}{2})\hat{P}\textbf{q}_{i}}{\sum_{i=1}^{n}\lVert\textbf{q}_{i}\rVert^{2}},$
(1)

where $\hat{\mathds{1}}$ is the identity operator, $\hat{\sigma}$ is a
reflection operator, and $\hat{P}$ is a permutation operator
($\hat{P}\textbf{q}_{i}=\textbf{q}_{j}).$Dryzun and Avnir (2011) For
derivation and further discussion of the CCM, see Ref. (13) and (14). Naively,
the intuition for the form of Eq. 1 is that the numerator of the second term
reports the maximal inner product between the original structure Q and the
nearest achiral structure
$\frac{1}{2}(\hat{\mathds{1}}+\hat{\sigma})\hat{P}\textbf{Q}$ formed by
averaging the original molecule Q with its enantiomer
$\textbf{Q'}=\hat{\sigma}\hat{P}\textbf{Q}$. Although not enforced in all past
implementations of the CCM, we require that $\hat{P}\textbf{Q}=\textbf{Q}$,
meaning that $\hat{P}$ must be restricted to permutations that are invariant
with respect to groups of chemically equivalent atom. Thus, with the
denominator serving normalization purposes, the second term in Eq. 1 reports a
dimensionless measure of the similarity between Q and its nearest achiral
structure. The CCM thus returns 0 for an a achiral molecules and at most 1 for
chiral molecules. For discussion of how to maximize the second term over the
$\hat{\sigma},$ see Ref. (15). Maximization over the $\hat{P}$ is performed
computationally. For such calculations, we use the cosymlib open source
software released by the developers of the CCM to which a link is provided in
the data availability section below.

Chirality Characteristic. Let $\textbf{Q}=(\textbf{q}_{1},...,\textbf{q}_{n})$
denote an ordered set of $n$ atomic coordinates of the molecule of interest
relative the center of mass. Let
$\textbf{v}_{i}=\textbf{q}_{i+1}-\textbf{q}_{i}$ for $i=1,...,n-1.$ Then we
define the Chirality Characteristic as

$\chi=\frac{\sum_{i=1}^{n-3}(\textbf{v}_{i}\times\textbf{v}_{i+1})\cdot\textbf{v}_{i+2}}{\sum_{i=1}^{n-3}\|\textbf{v}_{i}\rVert\lVert\textbf{v}_{i+1}\rVert\lVert\textbf{v}_{i+2}\rVert}.$
(2)

This expression reports a normalized average of scalar triple products of
successive bond vectors throughout the molecule. Note that unlike for the CCM,
where the labeling of the $\textbf{q}_{i}$ could be arbitrary, here we see
that $\chi$ is dependent on the ordering of the $\textbf{q}_{i}$. Thus we see
that $\chi$ is natural to evaluate in molecules with linear connectivity but
difficult to define on molecules with branched structures. Note that in Ref.
(4), a different normalization method was used; we instead follow Ref. (3) in
using the normalizing denominator that ensures that the Chirality
Characteristic is dimensionless (ranging from $-1$ to 1). We note that unlike
the CCM which reports the same value for either enantiomer, the $\chi$ assigns
opposite sign to opposite enantiomers. In particular, right-handed helices
have positive $\chi$ values and left handed helices have negative $\chi$
values. We provide link to an R script used to compute this quantity in the
data availability below.

##  III Performance on Toy Molecular Systems

In this section we show results for evaluating these chirality measures on two
toy molecular systems. The first such system is the four-site molecular model,
and the second is the polyethelene double helix. Finally, we comment briefly
on how the measures perform on a single helix, as compared to a double helix.

Four Carbon System. Figure 2 shows the simple four-site atomic model that we
use to evaluate the aforementioned chirality measures. The atoms in Fig. 2 are
labeled as Carbons, but the identity of the atoms in the model do not affect
either chirality measurement provided that the atoms are identical. Assuming
all three bonds have length $l$, we note that the conformation of such a
structure is fully determined by three parameters. We orient our coordinate
system so that the origin is at the location of C2, C3 lies on the $y$-axis,
and C1 lies in the $xy$-plane. Then as indicated in Fig. 2, any conformation
is specified by three angles: $\theta_{1}=\angle$C1C2C3, $\theta_{2}~{}-$ the
angle made by the C3-C2 bond with the projection of the C3-C4 bond on the
$xy$-plane, and $\phi$ which is the the angle between the C3-C4 bond and the
$xy$-plane.

Figure 2: Schematic of a linearly connected four-carbon system. Holding bond-
lengths constant, all conformations can be specified by three parameters
$(\theta_{1},\theta_{2},\phi)$ where C2 and C3 are fixed, and angle
$\angle$C1C2C3 is confined to the $xy$-plane. We assume that all three bonds
have length $l$.

We note that for this simple system, $\chi$ has has a concise analytical
treatment
$\chi^{4\text{-site}}=l^{-3}(\textbf{r}_{12}\times\textbf{r}_{23})\cdot\textbf{r}_{34}=-$sin$\theta_{1}$sin$\phi$,
while the CCM lends itself to no such concise interpretation.

Figure 3 shows the results for evaluating both the CCM and $\chi$ for various
choices $(\theta_{1},\theta_{2},\phi).$ In the top panels $\theta_{1}$ and
$\theta_{2}$ are fixed while $\phi$ varies, while in the bottom panels
$\theta_{2}$ and $\phi$ are fixed while $\theta_{1}$ is allowed to vary. Note
that structures examined based on the angles chosen return negative $\chi$
values. We are therefore required to compare the CCM to $-\chi$ (or $|\chi|$)
in order to make such a comparison meaningful. Since the CCM reports positive
values for both enantiomers, such considerations are moot.
Apart from the quantitative results, which can serve as benchmark values for
future chirality calculations, we note several qualitative observations:

Figure 3: Evaluation of the Continuous Chirality measure (CCM) and Chirality
Characteristic ($\chi$) on the four-carbon system for selected values
$(\theta_{1},\theta_{2},\phi)$. The left panels show the CCM results while the
right panels show the results for (negative) $\chi$. In the top panels, $\phi$
varies as the independent variable while the values
$\theta_{1}=\theta_{2}=\theta$ are held fixed at $60^{\circ}$ (green),
$90^{\circ}$ (blue), and $112^{\circ}$ (red). For the bottom panels,
$\theta_{1}$ is allowed to vary as the independent variable while
$\theta_{2}=112^{\circ}$ is fixed, and also $\phi$ is held fixed at
$20^{\circ}$ (orange), $60^{\circ}$ (indigo), and $120^{\circ}$ (green).

> (a) If $\phi$ or $\theta_{1}$ are equal to $0^{\circ}$ or $180^{\circ}$, the
> structure is coplanar and therefore achiral. Both measures correctly return
> zero chirality for such cases, as indicated by the endpoints of each line in
> Fig. 3.
> (b) As the independent angle varies from $0^{\circ}$ to $180^{\circ},$ in
> all cases the chirality magnitude reaches one local and global maximum.
> However, while for $\chi$ this maximum occurs exactly halfway through the
> angle’s domain (at $90^{\circ}$), for the CCM the maximum occurs at
> different points depending on the other two angles.
> (c) For $\chi$, the lines for $\chi=60^{\circ}$ and $\chi=120^{\circ}$
> coincide perfectly because the sines of both angles are equal. This is not
> the case for the CCM.

Finally, we note the importance of restricting the permutation operator
$\hat{P}$ to permutations of chemically equivalent elements as mentioned in
Sec. II. In this system, C1 and C4 are chemically equivalent and so are C2 and
C3. Note that although all atoms are identical, the atoms C1 an C3 are not
chemically equivalent. If the permutation operator permuted C1 with C3, it can
be seen that $\hat{\sigma}\hat{P}\textbf{Q}$ may not be an enantiomer of
$\textbf{Q},$ and the CCM would no longer be as chemically meaningful.

Polymer Double Helix. The second system on which we evaluate the CCM and
$\chi$ is the polymer double helix. Figure 4a shows a unit helical turn of
such a helix, which is the structure structure on which we evaluate the
chirality measures here. Such a helix can be characterized by the helix radius
$a$ and the pitch $\rho$, as well as the number of polymer units per turn of
the helix $n$.

Figure 4: (a) Ball-and-stick cartoon of a unit turn of a right-handed
polyethelene (PE) double helix, where different chains are denoted by
different colors. The double-helix is charecterized by the radius $a$ and the
pitch $\rho$ as well as the number of polymer units per helical turn $n$. (b)
Snapshots of a model double-helical polyethylene (PE) wire containing various
levels of twist. (c,d,e) Estimated helical parameters (radius $a$, pitch
$\rho$, polymer units per turn $n$) as a function of twist for the PE model.
Note that the the length of the polymer is held constant at its natural
untwisted length. Twist is reported as the fraction of maximal twist
$N_{T}/N_{T}^{\text{max}}$ as defined in Ref. 10.

We take a model polyethylene (PE) molecule used in prior work (see Ref. (10))
in which we performed molecular dynamics (MD) simulations to generate PE
double helices with various levels of twist. A United Atom model is used which
coarse-grains each CH2 monomer into a single interaction site. As discussed in
Ref. (10), the level of twist can be characterized by the fraction of the
maximal twist that can be attained before bonds begin to break, which we
denote $N_{T}/N_{T}^{\text{max}}$. Figure 4b shows snapshots of such a
simulated double-helix for various such levels of twist, when the polymer is
held at its natural untwisted length as described in Ref. (10). We observed
that the helical radius $a$ decreases with the level of twist, as shown in
Fig. 4c. The helical pitch $\rho$ and the number of polymer units per helical
turn also decrease with twist as expected (see Fig. 4d-e). However, we note
from inspecting Fig. 4b that the polyethelene model does not form a perfect
helix. Specifically, for the lower levels of twist, zig-zag substructures are
observed that which deviate from an ideal helical curve (see also Ref. (10)).
We therefore compare our results to that of an ideal helix fitted to the
helical parameters observed in the PE model. The position coordinates for the
atoms relative to the center of mass for such an ideal single helix are given
by

$\textbf{q}_{k}=\left(a\text{cos}\left(\frac{2\pi
k}{n}-\pi\right),a\text{sin}\left(\frac{2\pi
k}{n}-\pi\right),\frac{k\rho}{n}\right)^{T}$ (3)

for $k=1,...,n$, and a double helix can easily be formed by adding another
such single-helix rotated by $\pi$ about the $z$-axis.

Figure 5: The CCM (left panel) and $\chi$ value (right panel) of a unit
helical turn for a polymer right-handed double-helix as a function of twist,
reported as the PE model’s fraction of maximal twist before bonds begin to
break $N_{T}/N_{T}^{\text{max}}$. Results for the PE model double helix (blue
triangles) are compared to results for a corresponding ideal double helix (red
circles), where the latter is computed by fitting the mathematical equation of
a helix to the parameters shown in Fig. 3. For the PE model, statistical
uncertainties (between arbitrary samplings from simulations) are shown when
larger than the symbol sizes. For the idealized helix, such statistics are not
relevant.

Figure 5 shows results for evaluating the CCM and $\chi$ on both the PE model
and analogous ideal helix as a function of the level of twist
$N_{T}/N_{T}^{\text{max}}$ (where $N_{T}^{\text{max}}$ from the PE model is
used also for the ideal helix). The qualitative similarity between the
behavior of the two measures is striking. An important observation is that
while the CCM behaves similarly on the ideal and molecular models, $\chi$
deviates at moderate levels of twist. We attribute the reduced $\chi$ value in
the PE model to the zig-zag substructure, which diminishes as maximal twist is
approached.Abraham  _et al._ (2023)

It is interesting to inquire regarding what the plots in Fig. 5 would look
like if analogous single helices were used instead of a double helix. It can
be easily be guessed that the $\chi$ values would be unaffected because the
normalization in Eq. 2 would counteract the decreased number of terms. The
effect on the CCM is more difficult to predict. We found that for the ideal
helix the plot was qualitatively similar but with magnitudes of roughly 1/3
the double-helical counterpart (result not shown). Similarly, if the measures
were evaluated on sections of the double helix of lengths different than
$\rho$, the $\chi$ value would be unaffected, but the CCM would vary as
discussed in Ref. (11).

Figure 6: Density of normal modes binned by CCM value (left panels) and by
$\chi$ value (right panels) for length $N=98$ left-handed double-helical
polyethylene wires that are (top) untwisted and (bottom) twisted to 8 twists
(corresponding to $N_{T}/N_{T}^{\text{max}}=0.47$). Note that the area under
the curve for all four plots are equal, but the $y$-dimension of the right
panels are scaled differently compered to the left panels in order to
accommodate the sharp peak at $\chi=0$ for 0 twist.

##  IV Performance on Abstract Structures: Normal Modes

It is also worthwhile to compare the performance of both chirality measures on
more abstract structures such as molecular normal vibrational modes. Indeed,
recent suggestions that chiral phonons may be implicated in the chiral induced
spin selectivity (CISS) effectsKim  _et al._ (2023) have focused interest on
the connection between the chirality of molecular vibrational modes and that
of the underlying molecular structure. We computed the normal vibrational
modes for PE double helices using the model described above, using chains of
length $N=98$ where $N_{T}^{\text{max}}\approx 17$, as discussed in Ref. (10).
We use $\textbf{c}_{k,i}$ to denote the mass weighted displacement vector of
the $i$th atom in normal mode $k$. On such structures, the definitions of the
CCM and $\chi$ have not been well-established, and some creativity is required
in order to choose the most natural definition. We follow Ref. (11) in
defining the CCM of normal mode $k$ by taking
$\textbf{q}_{i}=\textbf{c}_{k,i}$ in Eq. 1 and enforcing
$\hat{P}=\hat{\mathds{1}}.$ To our knowledge, defining $\chi$ on normal modes
has never been done before, and we find that the simplest approach is to take
$\textbf{v}_{i}=\textbf{c}_{k,i}$ in Eq. 2

Figure 6 shows the results of such calculation, binning the normal mode
spectrum of the model PE (left-handed) double helix by each chirality measure.
By comparing the top and bottom panels, we see how twist (i.e. chirality of
the underlying molecular structure) affects the chirality of the normal mode
spectrum as measured by the CCM (left) and $\chi$ (right). Note that the plots
have different $x$-axis domains because $\chi$ reports negative values for
left-handed enantiomers. There are two observations we find noteworthy:

> (a) In the twisted structure compared to the untwisted structure, both the
> CCM and $\chi$, report that modes tend to have higher chirality values
> magnitudes.
> (b) For the CCM, the peak of the distribution shifts with twist, while for
> $\chi$, the peak remains at zero while the distribution broadens.

Overall, the data in Fig. 6 indicates that chirality of molecular normal modes
is correlated with the chirality of the underlying molecular structures. It
may be surprising that that the distribution for $\chi$, which classifies
modes by helical handedness, remains centered at zero for the twisted
structure despite the left-handed asymmetry of the underlying structure. We
have found upon further analysis that the handedness of the helix does
introduce asymmetries in the normal mode spectrum, which is the subject of
Ref. (11).

##  V Conclusion

We have discussed two measures of molecular chirality, the Continuous
Chirality Measure and the Chirality Characteristic, and their realization when
applied to simple model molecular systems. We find that the behavior of both
measures is qualitatively similar as structural parameters of the molecules
are varied. We conclude by listing technical considerations for choosing
between the two measures and applying these measures to molecular systems:

> (a) The Chirality Characteristic is only well defined on structures with
> linear connectivity. On branched structures, the CCM might be a better
> choice. Further consideration of bond ordering is needed to apply $\chi$ to
> such structures.
> (b) The CCM returns the same value for either enantiomer while $\chi$
> assigns opposite sign to opposite enantiomers. For this reason, the CCM
> alone might be difficult to correlate with physical properties that are
> enantiomer-dependent.
> (c) The CCM may be more effective than $\chi$ in measuring the chirality of
> abstract quantities such as normal vibrational modes.
> (d) When applying the CCM, care should be taken to restrict the permutation
> operator $\hat{P}$ such that the original structure can be accurately
> compared to its enantiomer without mixing atoms that are not chemically
> equivalent.

The wide variety of contexts to which the chirality measures addressed in this
work can be applied, as well as the physical meanings of these measures, are
still to be explored. We hope that this work serves as a useful guide for
future efforts to measure the chirality of molecular systems.

## Acknowledgements

The research of A.N. is supported by the Air Force Office of Scientific
Research under award number FA9550-23-1-0368 and the University of
Pennsylvania. E.A. aknowledges the support of the the University of
Pennsylvania (GfFMUR Grant). The authors are grateful to and Claudia Climent
for technical assistance and support.

## Data Availability

The software released by Avnir et al. to compute the CCM is available at at:
https://cosymlib.readthedocs.io/en/pere_tutorial/. Our R script to compute
$\chi$ can be found at: https://github.com/eabes23/chi_char/. The data to
generate the twisted polyethylene double helices can be found at
https://github.com/eabes23/polymer_twist/, in accordance with Ref. (10).

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11institutetext: University of Warwick, UK, 11email<EMAIL_ADDRESS>22institutetext: University of Warwick, UK, 22email<EMAIL_ADDRESS>
# Discretisation of regularity structures

Dirk Erhard1 and Martin Hairer2

###### Abstract

We introduce a general framework allowing to apply the theory of regularity
structures to discretisations of stochastic PDEs. The approach pursued in this
article is that we do not focus on any one specific discretisation procedure.
Instead, we assume that we are given a scale $\varepsilon>0$ and a “black box”
describing the behaviour of our discretised objects at scales below
$\varepsilon$.

###### Contents

  1. 1 Introduction
    1. 1.1 Discussion
    2. 1.2 Strategy
    3. 1.3 Structure of the article
    4. 1.4 Notation
  2. 2 A framework for discretisations

##  1 Introduction

The theory of regularity structures is a framework developed by the second
author in [Regularity] which allows to renormalise stochastic PDEs of the form

${\mathcal{L}}u=F(u,\nabla u,\xi),$ (1)

that are ill-posed in the classical sense. Here, ${\mathcal{L}}$ is typically
a parabolic differential operator, $\xi$ is a very irregular random input and
$F$ is some local non-linearity. The naïve approach to study well-posedness of
(1) would be to consider a sequence of smooth approximations

${\mathcal{L}}u_{\varepsilon}=F(u_{\varepsilon},\nabla
u_{\varepsilon},\xi_{\varepsilon}),$ (2)

and simply declare the limit of this sequence to be the solution to (1).
However, in many cases it turns out that the sequence $u_{\varepsilon}$ either
does not possess a limit or the limit is trivial. A way to circumvent this is
to renormalise, which can be interpreted as a kind of “recentering” of the
equation, see [BHZalg].

This amounts to allowing some of the constants appearing in the definition of
$F$ appearing in the right hand side of (2) to be $\varepsilon$-dependent and
more specifically to diverge as $\varepsilon$ tends to zero. The theory of
regularity structures is very successful when applied to approximations
$u_{\varepsilon}$ that are functions defined on $\mathbb{R}^{d}$, it however
presently does not in general apply to discrete approximations of stochastic
PDEs or to equations where the operator ${\mathcal{L}}$ itself is perturbed by
a higher-order term, in such a way that its scaling properties are different
at small scales (think for instance of ${\mathcal{L}}$ given by
$\partial_{t}-\Delta$ and
${\mathcal{L}}_{\varepsilon}=\partial_{t}-\Delta+\varepsilon^{2}\Delta^{2}$).
In the present article, we develop a general framework that is able to deal
with these cases. The guiding principle is a separation of scales, i.e., above
a certain scale $\varepsilon$ we show that the theory of regularity structures
still applies, whereas scales below $\varepsilon$ are treated as a black box.
As a result, our framework is flexible and does not rely on any specific
discretisation procedure. More precisely the results in the article show that,
given a stochastic PDE such that the theory of regularity structures can be
applied to renormalise it, our present framework can be used to treat a large
class of natural discretisations for it.

###  1.1 Discussion

Let us first comment on our motivation for this work as well as its relation
to some previous work. One of the guiding questions in the study of particle
systems is the question of universality, i.e., what are the characterising
features for a class of particle systems that converge to the same limit under
appropriate rescaling and how can one prove this convergence? The bulk of the
literature is concerned with scaling limits of a fixed system which
necessarily implies that the resulting limit is itself a scale-invariant
object. The present article however is motivated by the situation where one
considers a family of systems indexed by some parameter and one simultaneously
tunes this parameter as one rescales the system. In this way, one typically
obtains scaling limits that are not scale invariant themselves. One of the
insights of the theory presented in [Regularity] is that they are however
locally described by linear combinations of objects that are scale-invariant,
but with different scaling exponents.

So far, the state of the art for answering such questions (in the second case
where the limiting object can, at least formally, be described by a singular
stochastic PDE) is to heavily rely on special features of the model(s) under
consideration. In the type of situation of interest to us, some standard
techniques consist of

###### Claim 1

extending the discrete equation to an equation defined on $\mathbb{R}^{d}$,
see for instance the articles of Gubinelli and Perkowski [KPZreloaded],
Mourrat and Weber [MourratWeber], Shen and Weber [ShenWeber], Zhu and Zhu
[ZhuZhuNS, ZhuZhuPhi];

linearising the problem via a Hopf-Cole transformation, see for example the
articles of Bertini and Giacomin [MR1462228], Corwin and Shen [CorwinShen],
Corwin, Shen and Tsai [CorwinShenTsai], Corwin and Tsai [CorwinTsai], Dembo
and Tsai [DemboTsai], Labbé [Labbe] in the setting of the KPZ equation;

transforming the equation into a martingale problem, see for instance the
series of articles by Gonçalvez and Jara [GoncalvesJaraKPZ], Gubinelli and
Jara [GubinelliJara], Gubinelli and Perkowski [GubinelliPerkowskienergy], in
which the concept of energy solution is developed to study the stochastic
Burgers / KPZ equation.

When applied in the correct context these techniques can be very powerful. The
drawback however is that they are often quite sensitive to small perturbations
of the model. Another problem of analytical techniques like regularity
structures or paraproducts is that while they provide a rather clean and
general-purpose toolbox in the continuum, their extensions to the discrete
setting require purpose-built modifications that are much less reusable. It is
therefore desirable to have a robust theory at hand which is insensitive to
the details of the underlying discrete setting, and it is the goal of the
present article to make a first attempt at developing such a theory.

Another motivation to introduce the framework developed here stems from the
fact that a common way to derive properties of a stochastic PDE is to
approximate it by discrete systems for which the desired property holds and
then show that it remains stable under passage to the limit. This methodology
was for instance successfully applied by Hairer and Matetski
[MatetskiDiscrete] to prove that the $\Phi_{3}^{4}$-measure built in
[GlimmJaffe, Fel74] is invariant for the $\Phi_{3}^{4}$-equation. In the
specific case where the solution to the stochastic PDE at hand is a function
of time, [MatetskiDiscrete] developed a framework adapting the theory of
regularity structures to allow for certain spatial discretisations.

###  1.2 Strategy

The main idea to accommodate a large class of discretisations of stochastic
PDEs is to consider the behaviour below scale $\varepsilon$ as encoded in a
“black box”. In order to describe this, our main ingredient is a sequence of
linear spaces ${\mathcal{X}}_{\varepsilon}$ that can be viewed as subspaces of
${\mathcal{D}}^{\prime}(\mathbb{R}^{d})$, the space of distributions, and that
possess a natural family of (extended) seminorms. We then work with stochastic
PDEs of the type (2) such that
$u_{\varepsilon}\in{\mathcal{X}}_{\varepsilon}$. Examples for
${\mathcal{X}}_{\varepsilon}$ include ${\mathcal{D}}^{\prime}(\mathbb{R}^{d})$
(thus recovering the original setting in [Regularity]), but also the space of
functions defined on a discrete grid of mesh size $\varepsilon$, or simply
some space of smooth functions. To cast this into the framework of regularity
structures we first work with an abstract version of $u_{\varepsilon}$, i.e.,
we write $u_{\varepsilon}$ as a generalised Taylor expansion. Here, the
monomials may represent the classical Taylor monomials or they may represent
abstract expressions that are functions of $\xi$. The control of these
monomials typically amount to the control of explicit stochastic objects and
heavily depend on the choice of the equation at hand.

The “Taylor coefficients” can be thought of as the “derivatives” of an
abstract version of a ${\mathcal{C}}^{\gamma}$-function and they are given by
the solution of an abstract fixed point problem. We denote the class of
abstract functions just described by ${\mathcal{D}}_{\varepsilon}^{\gamma}$.
The subindex $\varepsilon$ indicates that the way regularity is measured above
and below scale $\varepsilon$ may differ. Indeed, above scale $\varepsilon$
regularity is measured as in the continuous setting [Regularity], whereas
below scale $\varepsilon$, besides some natural constraints, the way to
measure regularity is not further determined. The idea of introducing
${\mathcal{D}}^{\gamma}$-spaces that depend on a parameter $\varepsilon$ is
not new and already appeared in the works [KPZJeremy, Phi4Weijun]. In a sense
the current article generalises some of the ideas developed there.

Note that the solution to (2) is a random space-time function / distribution,
whereas the approach outlined above gives rise to an abstract “modelled
distribution” $f\in{\mathcal{D}}_{\varepsilon}^{\gamma}$. To link the abstract
object with a concrete object we construct a “reconstruction map”
${\mathcal{R}}^{\varepsilon}$. Unlike in [Regularity], we do not think of
${\mathcal{R}}^{\varepsilon}f$ as a distribution, but rather as an element of
${\mathcal{X}}_{\varepsilon}$, which could for example represent a space of
functions defined on a discrete grid at scale $\varepsilon$. The way to
construct ${\mathcal{R}}^{\varepsilon}$ is to _postulate_ the existence of a
map ${\mathcal{R}}^{\varepsilon}$ satisfying certain estimates on scales below
$\varepsilon$ and to then show that analogous estimates automatically hold on
all larger scales. Note that there may be several candidates for
${\mathcal{R}}^{\varepsilon}$ all satisfying the same quantitative estimates
on small scales, so that in our context the reconstruction map is in general
not uniquely defined. To proceed, we further need to define operations on
${\mathcal{D}}_{\varepsilon}^{\gamma}$ in order to actually construct abstract
solutions to (2). In particular we define an abstract notion of convolution
$\mathcal{K}_{\gamma}^{\varepsilon}$ against the Green function
$K^{\varepsilon}$ of ${\mathcal{L}}$ in (2) (which is an operator mapping into
${\mathcal{X}}_{\varepsilon}$!) and we show that
$\mathcal{K}_{\gamma}^{\varepsilon}$ satisfies a certain Schauder estimate. It
turns out that unlike in the continuous setting [Regularity] it is not
necessarily true that the convolution operator
$\mathcal{K}_{\gamma}^{\varepsilon}$ can be defined in a natural way so that
it intertwines with ${\mathcal{R}}^{\varepsilon}$ in the sense that

${\mathcal{R}}^{\varepsilon}\mathcal{K}_{\gamma}^{\varepsilon}=K^{\varepsilon}{\mathcal{R}}^{\varepsilon}.$
(3)

We however argue that in many cases one can enforce (3) by tweaking the
definition of $\mathcal{K}_{\gamma}^{\varepsilon}$. So far the strategy
outlined above appears to only allow us to describe solutions to (2), but, as
shown in [Regularity] and [BHZalg], the encoding of renormalisation procedures
in the theory of regularity structures is of purely algebraic nature and does
not depend on any specific discretisation procedure, see also Remark 1.1
below.

We finish with some concluding remarks that comment on how one does apply the
theory developed in this article.

###### Remark 1.1

To apply the theory developed in this article to a concrete problem, an
important ingredient is the construction of a suitable regularity structure
and a suitable renormalised model. In the usual (continuous) case, a framework
was built in [Regularity, Sec. 8], and further refined in [BHZalg, Ajay] that
automatises this construction. To perform an analogous construction in the
present context, one then needs an algebra structure on
${\mathcal{X}}_{\varepsilon}$, as well as a representation of the Taylor
polynomials, which is often the case. The general analytical results of [Ajay]
however fall out the scope of this article and would have to be adapted.

###### Remark 1.2

It was shown in [Regularity, Thm 10.7] (see also [NonGaussShen, Ajay]) that in
order to obtain convergence of a sequence of models to a limiting model it is
enough to obtain suitable bounds on sufficiently high moments for its terms of
negative homogeneity. An important ingredient of the proof is the recursive
definition of the regularity structure and the model, provided by the
algebraic framework alluded to above. This is the same in the current case.
Another ingredient is the identity $\Pi_{z}\tau={\mathcal{R}}f_{\tau,z}$,
where $f_{\tau,z}(y)=\Gamma_{yz}\tau-\tau$ for any $\ell\in A$ and
$\tau\in{\mathcal{T}}_{\ell}$ with positive homogeneity and
$f_{\tau,z}(y)=\Gamma_{yz}\tau$ in case $\tau$ has negative homogeneity.
Provided that the norm
$|\\!|\\!|\cdot|\\!|\\!|_{\gamma;\mathfrak{K};\varepsilon}$ of Definition 2.1
below is chosen such that
$|\\!|\\!|z\mapsto\Gamma_{zz^{\prime}}^{\varepsilon}\tau|\\!|\\!|_{\ell;\mathfrak{K};\varepsilon}=0$
for any $\ell\in A$, any $\tau\in{\mathcal{T}}_{\ell}$ and any
$z^{\prime}\in\mathbb{R}^{d}$, the aforementioned identity follows from
Assumption LABEL:a:rec. A further ingredient is the characterisation of the
spaces of distributions under consideration by a wavelet basis. On scales
larger than $\varepsilon$ a similar characterisation holds in the current
case, see Definition LABEL:def:model below. On small scales, the situation
depends on the choice of seminorms on ${\mathcal{X}}_{\varepsilon}$ introduced
in Definition 2.1, but, in many cases of interest, we expect to have a choice
of these seminorms which makes the corresponding bounds simple to verify.
Finally, the last ingredient is the extension theorem [Regularity, Thm 5.14].
The analogous statement in the present setup is the content of Theorem
LABEL:thm:extension. Thus, under the appropriate assumptions [Regularity, Thm
10.7] also holds in the framework constructed in this article.

###  1.3 Structure of the article

In Section 2 we develop the framework we will work with, in particular we
introduce the sequence of spaces ${\mathcal{X}}_{\varepsilon}$, we define the
notion of discrete models and the
${\mathcal{D}}_{\varepsilon}^{\gamma}$-spaces. In Sections LABEL:S3–LABEL:S6
we explain the main operations on these spaces.

###  1.4 Notation

Throughout this work
$\mathfrak{s}=(\mathfrak{s}_{1},\ldots,\mathfrak{s}_{d})\in\mathbb{N}_{\geq
1}^{d}$ denotes a scaling of $\mathbb{R}^{d}$ and we associate to it the
metric $d_{\mathfrak{s}}$ on $\mathbb{R}^{d}$ given by

$d_{\mathfrak{s}}(y,z)=\mathop{\mathrm{sup}}_{i\in\\{1,\ldots,d\\}}|y_{i}-z_{i}|^{1/\mathfrak{s}_{i}}.$
(4)

We sometimes use the notation $\|y-z\|_{\mathfrak{s}}$ in place of
$d_{\mathfrak{s}}(y,z)$. Moreover, we let
$|\mathfrak{s}|=\mathfrak{s}_{1}+\cdots+\mathfrak{s}_{d}$ and for a multiindex
$k$ we use the notation $|k|_{\mathfrak{s}}=\sum_{i}\mathfrak{s}_{i}k_{i}$.
Given a set $B\subset\mathbb{R}^{d}$ and $z\in\mathbb{R}^{d}$ we denote the
distance of $z$ to $B$ with respect to the metric $d_{\mathfrak{s}}$ by
$d_{\mathfrak{s}}(z,B)$. Given $\delta>0$ and
$\varphi:\mathbb{R}^{d}\to\mathbb{R}$ we set
${\mathcal{S}}_{\mathfrak{s}}^{\delta}(z_{1},\ldots,z_{d})=(\delta^{-\mathfrak{s}_{1}}z_{1},\ldots,\delta^{-\mathfrak{s}_{d}}z_{d})$
and
$({\mathcal{S}}_{\mathfrak{s},z}^{\delta}\varphi)(y)=\delta^{-|\mathfrak{s}|}\varphi({\mathcal{S}}_{\mathfrak{s}}^{\delta}(y-z))$.
We also use occasionally the notation $\varphi_{z}^{\delta}$. Moreover, we
also write $\varphi_{z}^{n}$ in place of $\varphi_{z}^{2^{-n}}$. For a compact
subset $\mathfrak{K}$ of $\mathbb{R}^{d}$ (also written as
$\mathfrak{K}\Subset\mathbb{R}^{d}$) we denote its $1$-fattening and
$2$-fattening by $\bar{\mathfrak{K}}$ and $\bar{\bar{\mathfrak{K}}}$,
respectively. We further denote the $d_{\mathfrak{s}}$-ball of radius $\delta$
around $z\in\mathbb{R}^{d}$ by ${\mathcal{B}}_{\mathfrak{s}}(z,\delta)$.

We occasionally use the notation $[\eta]$ to denote the support of the
function $\eta$. In this article we use various notions of norms, seminorms
and metrics on various spaces. To improve readability we inserted a norm index
in the appendix listing all these distances.

Given a distribution $\xi$ and a test function $\varphi$, we use
interchangeably the notations $\xi(\varphi)$, $\langle\xi,\varphi\rangle$, and
$\int\varphi(x)\,\xi(dx)$ for the corresponding pairing.

### Acknowledgements

MH gratefully acknowledges support by the Leverhulme Trust and by an ERC
consolidator grant.

##  2 A framework for discretisations

Convention: From now on we assume that we are given a scaling $\mathfrak{s}$
of $\mathbb{R}^{d}$ and a regularity structure
$\mathscr{T}=(A,{\mathcal{T}},G)$ containing the canonical
$\mathfrak{s}$-scaled polynomials in $d$ indeterminates. We also fix a value
$r>|\mathop{\mathrm{min}}A\,|$ and we denote by $\Phi$ the set of all
functions $\varphi\in{\mathcal{C}}^{r}$ with
$\|\varphi\|_{{\mathcal{C}}^{r}}\leq 1$ such that
$\mathop{\mathrm{supp}}\nolimits\varphi\subset{\mathcal{B}}_{\mathfrak{s}}(0,1)$
and we simply say that “$\varphi$ is a test function” whenever
$\varphi\in\Phi$. Given an element $\tau\in{\mathcal{T}}=\bigoplus_{\alpha\in
A}{\mathcal{T}}_{\alpha}$, we write $\|\tau\|_{\alpha}$ for the norm of its
component in ${\mathcal{T}}_{\alpha}$.

We build a general framework for allowing for discretisations of models and
their convergence to continuous limiting models. Our construction relies
crucially on the following notion.

###### Definition 2.1

A discretisation for the regularity structure $\mathscr{T}$ and scaling
$\mathfrak{s}$ on $\mathbb{R}^{d}$ consists of the following data.

###### Claim 2

A collection of linear spaces ${\mathcal{X}}_{\varepsilon}$ with
$\varepsilon\in(0,1]$ endowed with inclusion maps
$\iota_{\varepsilon}\colon{\mathcal{X}}_{\varepsilon}\hookrightarrow{\mathcal{D}}^{\prime}(\mathbb{R}^{d})$,
so that elements of ${\mathcal{X}}_{\varepsilon}$ can be interpreted as
distributions.

Each ${\mathcal{X}}_{\varepsilon}$ admits a family of extended seminorms
$\|\cdot\|_{\alpha;\mathfrak{K}_{\varepsilon};z;\varepsilon}$ (i.e., we do
_not_ exclude the possibility that
$\|f\|_{\alpha;\mathfrak{K}_{\varepsilon};z;\varepsilon}=\infty$ for some
$f\in{\mathcal{X}}_{\varepsilon}$) with $\alpha\in\mathbb{R}$, the
$\mathfrak{K}_{\varepsilon}$ are compact subsets of $\mathbb{R}^{d}$ of
diameter at most $2\varepsilon$ and $z\in\mathbb{R}^{d}$. We moreover assume
that these seminorms are local in the sense that if
$f,g\in{\mathcal{X}}_{\varepsilon}$ and
$(\iota_{\varepsilon}f)(\varphi)=(\iota_{\varepsilon}g)(\varphi)$ for every
$\varphi\in{\mathcal{C}}^{r}$ supported in $\mathfrak{K}_{\varepsilon}$, then
$\|f-g\|_{\alpha;\mathfrak{K}_{\varepsilon};z;\varepsilon}=0$.

Uniformly over all $f\in{\mathcal{X}}_{\varepsilon}$, $z\in\mathbb{R}^{d}$,
$\alpha\in\mathbb{R}$, and $\varphi\in\Phi$, one has the bound

$|(\iota_{\varepsilon}f)(\varphi_{z}^{\varepsilon})|\lesssim\varepsilon^{\alpha}\|f\|_{\alpha;[\varphi_{z}^{\varepsilon}];z;\varepsilon}.$
(5)

For any function $\Gamma\colon\mathbb{R}^{d}\times\mathbb{R}^{d}\to G$, there
exists a family of extended seminorms
$|\\!|\\!|\cdot|\\!|\\!|_{\gamma;\mathfrak{K};\varepsilon}$ on the space of
functions $f:\mathbb{R}^{d}\to{\mathcal{T}}_{<\gamma}$ with
$\gamma\in\mathbb{R}$, $\mathfrak{K}\Subset\mathbb{R}^{d}$ and
$\varepsilon\in(0,1]$. For the same set of indices, and given two functions
$\Gamma_{1},\Gamma_{2}:\mathbb{R}^{d}\times\mathbb{R}^{d}\to G$, there is a
family of seminorms
$|\\!|\\!|\cdot;\cdot|\\!|\\!|_{\gamma;\mathfrak{K};\varepsilon}$ on the space
of pairs $(f,g)$, with $f,g:\mathbb{R}^{d}\to{\mathcal{T}}_{<\gamma}$. Both
families of seminorms are assumed to depend only on the values of $f$ (and $g$
respectively) in a neighbourhood of size $\mathbf{c}\varepsilon$ around
$\mathfrak{K}$, for $\mathbf{c}\geq 0$ a fixed constant.

###### Remark 2.2

The above way of introducing the seminorms in the fourth item may be a bit
misleading. As we will see in the examples below, they indeed depend on the
functions $\Gamma,\Gamma_{1}$, and $\Gamma_{2}$ introduced above, so that the
correct way of denoting them would be
$|\\!|\\!|\cdot|\\!|\\!|_{\gamma;\Gamma;\mathfrak{K};\varepsilon}$ and
$|\\!|\\!|\cdot;\cdot|\\!|\\!|_{\gamma;\Gamma_{1},\Gamma_{2},\mathfrak{K};\varepsilon}$.
However, for the sake of readability we omit these additional indices.

###### Remark 2.3

Assume that in the fourth item in the above definition one has
$\Gamma_{1}=\Gamma_{2}$. In practice one then often has the relation
$|\\!|\\!|f;g|\\!|\\!|_{\gamma;\mathfrak{K};\varepsilon}=|\\!|\\!|f-g|\\!|\\!|_{\gamma;\mathfrak{K};\varepsilon}$.
See for instance the four examples introduced further below, where this is
indeed the case.

To illustrate the setup described above, we mention the following four
examples to which we will refer frequently in this work. We leave it as an
exercise to verify that these are indeed examples of discretisations in the
above sense. In all examples below $\mathfrak{K}_{\varepsilon}$ denotes a
compact subset of $\mathbb{R}^{d}$ with diameter bounded by $2\varepsilon$.

  * 1.

The purely discrete case. Let $\Lambda_{\varepsilon}^{d}\subset\mathbb{R}^{d}$
be a locally finite set such that, for some constants $c,C\in(0,1]$, it holds
that for every $z\in\Lambda_{\varepsilon}^{d}$, $\mathop{\mathrm{inf}}_{z\neq
z^{\prime}\in\Lambda_{\varepsilon}^{d}}\|z-z^{\prime}\|_{\mathfrak{s}}\in[c\varepsilon,C\varepsilon]$.
We then set
${\mathcal{X}}_{\varepsilon}=\mathbb{R}^{\Lambda_{\varepsilon}^{d}}$ with

$\bigl{(}\iota_{\varepsilon}f\bigr{)}(\varphi)=\varepsilon^{|\mathfrak{s}|}\sum_{z\in\Lambda_{\varepsilon}^{d}}f(z)\,\varphi(z)\;\hbox{,}\quad\|f\|_{\alpha;\mathfrak{K}_{\varepsilon};z;\varepsilon}=\varepsilon^{-\alpha}\mathop{\mathrm{sup}}_{y\in\mathfrak{K}_{\varepsilon}\cap\Lambda_{\varepsilon}^{d}}|f(y)|\;.{\hbox{\rm(6)}\kern
5.69046pt\kern-5.69046pt}q:discrXeps{=}e5{6}$