# Poincaré inequality and Quantitative De Giorgi method for hypoelliptic
operators

Francesca Anceschi  Università Politecnica delle Marche: Dipartimento di
Ingegneria Industriale e Scienze Matematiche, Via Brecce Bianche 12 60131
Ancona, Italy<EMAIL_ADDRESS>,  Helge Dietert  Université Paris
Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75006 Paris, France
<EMAIL_ADDRESS>,  Jessica Guerand  Université de Montpellier,
499-554 Rue du Truel, 34090 Montpellier, France
<EMAIL_ADDRESS>,  Amélie Loher  University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, United Kingdom<EMAIL_ADDRESS>,
Clément Mouhot  University of Cambridge, Wilberforce Road, Cambridge CB3 0WA,
United Kingdom<EMAIL_ADDRESS>and  Annalaura Rebucci  Max Planck
Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig,
Germany<EMAIL_ADDRESS>
(Date: Preliminary version (v8-CM), )

###### Abstract.

We propose a systematic elementary approach based on trajectories to prove
Poincaré inequalities for hypoelliptic equations with an arbitrary number of
Hörmander commutators, both in the local and in the non-local case. Our method
generalises and simplifies the method introduced in [8] in the local case with
one commutator, and extended in [12] to the non-local case with one
commutator. It draws inspiration from the paper [15], although we use
different trajectories. We deduce Harnack inequalities and Hölder regularity
along the line of the De Giorgi method. Our results recover those in [2] in
the local case, and are new in the non-local case.

AL is supported by the Cambridge Trust. CM is supported by ERC grant MAFRAN

###### Contents

  1. 1 Introduction
    1. 1.1 The question at hand
    2. 1.2 Examples of equations
    3. 1.3 General assumptions (H)
    4. 1.4 Main result
    5. 1.5 Motivation
    6. 1.6 Strategy of proof
    7. 1.7 Outline
  2. 2 Construction of the trajectories
    1. 2.1 The base case
    2. 2.2 The general case
  3. 3 Proof of the Poincaré inequality
  4. 4 Proof of the Harnack inequality

##  1\. Introduction

###  1.1. The question at hand

This paper is devoted to the regularity theory of hypoelliptic equations with
rough coefficients. We consider equations with an arbitrary number of
commutators to obtain regularity in all directions, in the sense of Hörmander
[9], and whose diffusive part is either local (second-order) or non-local
(fractional order). Beyond their structural intrinsic importance, such
equations naturally appear in financial models (see Section 1.5).

We consider an unknown function $f=f(x,t)$ satisfying equations of the form

(1.1) $\partial_{t}f+{\mathcal{B}}f+{\mathcal{A}}^{*}{\mathcal{A}}f=0$

where ${\mathcal{B}}$ is a first-order differential operator and
${\mathcal{A}}$ is a derivative, or fractional derivative, operator in some of
the directions $x^{(0)}$ of $x$. We assume no regularity on the coefficients
of the diffusive part ${\mathcal{A}}$. Then provided that ${\mathcal{A}}$ and
${\mathcal{B}}$ satisfy the Hörmander commutator condition [9], one expects
the following _Poincaré inequality_ to hold for weak non-negative sub-
solutions:

$\int_{Q_{+}}\left(f-\frac{1}{|Q_{-}|}\int_{Q_{-}}f\right)_{+}\lesssim\int_{\Omega}|{\mathcal{A}}f|$

where $Q^{-},Q^{+}$ are two disjoint cylinders in the open domain $\Omega$ so
that $Q^{+}$ is a time-translation of $Q^{-}$ into the future. In this paper
we prove such an inequality with a quantitative control on the constant for a
large class of equations of the type (1.1). The main novelty is the method
itself, but the result is also new when $\mathcal{A}$ is non-local and more
than one commutator is involved.

The proof is based on the combination of a method introduced in [8] and the
explicit construction of appropriate new trajectories inspired partly from
[15]. The Poincaré inequality is a powerful integral way to measure the
oscillation of the sub-solution, and its use in this context goes back to [14,
13]. We deduce quantitatively the De Giorgi lemmas, the Hölder regularity for
weak solutions, and the (weak) Harnack inequality for sub-solutions following
the methodology recently developed in [8, 12].

###  1.2. Examples of equations

When $\kappa=1$, the prototypical local equation is the so-called _Kolmogorov
equation_

(1.2)
$\partial_{t}f+v\cdot\nabla_{x}f=\nabla_{v}\cdot\left[A\nabla_{v}f\right]$

on the unknown $f=f(x,v,t)$ depending on $x,v\in{\mathbb{R}}^{d}$ with
$d\in{\mathbb{N}}$, $t\in{\mathbb{R}}$, where the matrix $A=A(x,v,t)$ is
measurable symmetric so that, for some $\Lambda\geq 1$,

(1.3) $\forall\,(x,v,t)\in{\mathbb{R}}^{2d+1},\quad\Lambda^{-1}\leq
A(x,v,t)\leq\Lambda.$

When the number of commutators $\kappa\geq 2$, the prototypical local equation
is

(1.4)
$\partial_{t}f+x^{(\kappa-1)}\cdot\nabla_{x^{(\kappa)}}f+x^{(\kappa-2)}\cdot\nabla_{x^{(\kappa-1)}}f+\dots+v\cdot\nabla_{x^{(1)}}f=\nabla_{v}\cdot\left[A\nabla_{v}f\right]$

on the unknown $f=f(x^{(\kappa)},x^{(\kappa-1)},\dots,x^{(1)},v,t)$, depending
on $x^{(\kappa)},x^{(\kappa-1)},\dots,x^{(1)},v\in{\mathbb{R}}^{d}$,
$t\in{\mathbb{R}}$, where the matrix $A$ is measurable, symmetric and so that,
for some $\Lambda\geq 1$,

(1.5) $\displaystyle\forall\,(x^{(\kappa)},x^{(\kappa-1)},\dots,x^{(1)},v,t)$
$\displaystyle\in{\mathbb{R}}^{(\kappa+1)d+1},$ $\displaystyle\Lambda^{-1}\leq
A(x^{(\kappa)},x^{(\kappa-1)},\dots,x^{(1)},v,t)\leq\Lambda.$

We denote $v=x^{(0)}$ for notational consistency.

In the fractional (non-local) case and with $\kappa=1$ commutators, the
prototypical equation is the following fractional Kolmogorov equation

(1.6)
$\partial_{t}f+v\cdot\nabla_{x}f=\left(-\Delta_{v}\right)^{\frac{\beta}{2}}\cdot\left[a\left(-\Delta_{v}\right)^{\frac{\beta}{2}}f\right]$

on the unknown $f=f(x,v,t)$ depending on $x,v\in{\mathbb{R}}^{d}$ and
$t\in{\mathbb{R}}$, with $\beta\in(0,1)$ and a scalar function $a$ that is
measurable and so that, for some $\Lambda\geq 1$,

(1.7) $a(x,v,t)\in[\Lambda^{-1},\Lambda].$

When the number of commutators $\kappa\geq 2$, the prototypical non-local
equation is

(1.8)
$\partial_{t}f+x^{(\kappa-1)}\cdot\nabla_{x^{(\kappa)}}f+x^{(\kappa-2)}\cdot\nabla_{x^{(\kappa-1)}}f+\dots+v\cdot\nabla_{x^{(1)}}f=\left(-\Delta_{v}\right)^{\frac{\beta}{2}}\cdot\left[a\left(-\Delta_{v}\right)^{\frac{\beta}{2}}f\right]$

on the unknown $f$ which depends on
$x^{(\kappa)},x^{(\kappa-1)},\dots,x^{(1)},x^{(0)}\in{\mathbb{R}}^{d}$,
$t\in{\mathbb{R}}$, and where the function $a$ is measurable and so that, for
some $\Lambda\geq 1$,

(1.9)
$\forall\,(x^{(\kappa)},x^{(\kappa-1)},\dots,x^{(1)},v,t)\in{\mathbb{R}}^{(\kappa+1)d+1},\quad\Lambda^{-1}\leq
a(x^{(\kappa)},x^{(\kappa-1)},\dots,x^{(1)},v,t)\leq\Lambda.$

In fact, our results still apply when the fractional diffusion operator on the
right hand side of (1.6) and (1.8) is replaced by the more general nonlocal
operator

$\displaystyle\mathcal{L}f\left(x^{(\kappa)},\dots,x^{(1)},x^{(0)},t\right)$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\left[f\left(x^{(\kappa)},\dots,x^{(1)},w,t\right)-f\left(x^{(\kappa)},\dots,x^{(1)},x^{(0)},t\right)\right]K\left(x^{(\kappa)},\dots,x^{(1)},x^{(0)},w,t\right){\,\mathrm{d}}w$

where the kernel $K$ satisfies pointwise bounds

(1.10)
$\forall\,\left(x^{(\kappa)},\dots,x^{(1)},v,w,t\right)\in{\mathbb{R}}^{(\kappa+2)d+1},\\\
\Lambda^{-1}\big{|}v-w\big{|}^{-(d+2\beta)}\leq
K\left(x^{(\kappa)},\dots,x^{(1)},v,w,t\right)\leq\Lambda\big{|}v-w\big{|}^{-(d+2\beta)},$

and the symmetry

(1.11)
$\forall\,\left(x^{(\kappa)},\dots,x^{(1)},v,w,t\right)\in{\mathbb{R}}^{(\kappa+2)d+1},\\\
K\left(x^{(\kappa)},\dots,x^{(1)},v,w,t\right)=K\left(x^{(\kappa)},\dots,x^{(1)},w,v,t\right).$

###  1.3. General assumptions (H)

We denote the variable $z:=(x,t)\in{\mathbb{R}}^{N+1}$ with $t\in{\mathbb{R}}$
and $x=(x^{(\kappa)},x^{(1)},\dots,x^{(0)})$ with each
$x^{(i)}\in{\mathbb{R}}^{d_{i}}$, $d_{i}\in{\mathbb{N}}^{*}$ and
$i\in\\{0,\dots,\kappa\\}$. We assume that $d_{0}\geq d_{1}\geq\dots\geq
d_{\kappa}\geq 1$ and denote $N:=d_{0}+\cdots+d_{\kappa}$.

We consider (1.1) with operators ${\mathcal{A}}$ and ${\mathcal{B}}$ as
follows. The operator ${\mathcal{B}}$ is given by

${\mathcal{B}}:=({\mathsf{B}}x)\cdot\nabla_{x},$

where the $N\times N$ matrix ${\mathsf{B}}$ has the form ($\mathbb{O}$ denotes
a block matrix of zeros)

(1.12)
${\mathsf{B}}:=\begin{pmatrix}\mathbb{O}&{\mathsf{B}}_{\kappa}&\mathbb{O}&\cdots&\cdots&\mathbb{O}\\\
\vdots&\mathbb{O}&{\mathsf{B}}_{\kappa-1}&\mathbb{O}&\cdots&\mathbb{O}\\\
\vdots&\vdots&\mathbb{O}&\ddots&\mathbb{O}&\mathbb{O}\\\
\vdots&\vdots&\vdots&\ddots&{\mathsf{B}}_{2}&\mathbb{O}\\\
\vdots&\vdots&\vdots&\vdots&\mathbb{O}&{\mathsf{B}}_{1}\\\
\mathbb{O}&\mathbb{O}&\cdots&\cdots&\cdots&\mathbb{O}\end{pmatrix},$

with $d_{i}\times d_{i-1}$ blocks ${\mathsf{B}}_{i}$ which are constant
matrices of rank $d_{i}$.

The operator ${\mathcal{A}}$ is a linear operator acting only on the variable
$x^{(0)}$ and $\beta$-homogeneous in this variable with $\beta\in(0,1]$ and it
satisfies the following pointwise control from above and integral control from
below, given $\varphi\in C^{\infty}_{c}({\mathbb{R}}^{N})$:

(1.13)
$\|{\mathcal{A}}\varphi\|_{\infty}\lesssim\|\nabla_{(0)}\varphi\|_{L^{\infty}}^{\beta}\|\varphi\|_{L^{\infty}}^{1-\beta},\quad\int\left|{\mathcal{A}}\varphi\right|{\,\mathrm{d}}z\gtrsim\int\left|(-\Delta_{v})^{\beta/2}\varphi\right|{\,\mathrm{d}}z.$

The transport operator $\mathcal{T}:=\partial_{t}+{\mathcal{B}}$ naturally
defines the non-commutative group of _hyper-Galilean transformations_ , that
generalises the Galilean transformations:

(1.14)
$(\tilde{x},\tilde{t})\circ(x,t)=(x+\exp(t{\mathsf{B}})\tilde{x},t+\tilde{t}),\hskip
14.22636pt(x,t),\ (\tilde{x},\tilde{t})\in{\mathbb{R}}^{N+1}.$

This transformation leaves the operator $\mathcal{T}$ invariant:
$(\mathcal{T}f)(\tilde{z}\circ z)=\mathcal{T}[f(\tilde{z}\circ\cdot)](z)$.

The scaling properties of ${\mathcal{A}}$ and ${\mathcal{B}}$ naturally define
the _hypoelliptic dilations_ for $r>0$:

(1.15)
$\delta_{r}:=\textrm{diag}\left(r^{1+2\kappa\beta}\mathbb{I}_{d_{\kappa}},\dots,r^{1+2\beta}\mathbb{I}_{d_{1}},r\mathbb{I}_{d_{0}},r^{2\beta}\right)$

and we have the relation

$({\mathcal{T}}+{\mathcal{A}}^{*}{\mathcal{A}})[f\circ\delta_{r}]=r^{2\beta}\big{[}\left({\mathcal{T}}+{\mathcal{A}}^{*}{\mathcal{A}}\right)f\big{]}\circ\delta_{r}.$

These transformations allow to define the _hypoelliptic cylinders_

$\displaystyle
Q_{r}(\tilde{z}):=\tilde{z}\circ\left(\delta_{r}\left(Q_{1}\right)\right)\quad\text{
with }\quad Q_{1}:=B_{1}\times\ldots\times
B_{1}\times(-1,0]\subset{\mathbb{R}}^{N+1}.$

The (local) Kolmogorov equation (1.2) corresponds to $\kappa=1$,
$d_{0}=d_{1}=d$,
$(x^{(1)},x^{(0)})=(x,v)\in{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}$,
${\mathcal{A}}:=\sqrt{A}\nabla_{v}f$ with a $d\times d$-matrix $A=A(x,v,t)$
measurable symmetric so that $A\in[\Lambda^{-1},\Lambda]$, and the matrix

(1.16) ${\mathsf{B}}:=\begin{pmatrix}\mathbb{O}&\mathbb{I}_{d}\\\
\mathbb{O}&\mathbb{O}\end{pmatrix}.$

The (local) higher-order Kolmogorov equation (1.4) corresponds to $d_{i}=d$
for all $i$,
$(x^{(\kappa)},\dots,x^{(1)},x^{(0)})\in{\mathbb{R}}^{(\kappa+1)d}$,
${\mathcal{A}}:=\sqrt{A}\nabla_{(0)}f$ with a $d\times d$-matrix
$A=A(x^{(\kappa)},\dots,x^{(1)},x^{(0)},t)$ measurable symmetric so that
$A\in[\Lambda^{-1},\Lambda]$, and the matrix

${\mathsf{B}}:=\begin{pmatrix}\mathbb{O}&\mathbb{I}_{d}&\mathbb{O}&\ldots&\mathbb{O}\\\
\mathbb{O}&\mathbb{O}&\mathbb{I}_{d}&\mathbb{O}&\vdots\\\
\mathbb{O}&\vdots&\ddots&\ddots&\mathbb{O}\\\
\vdots&\vdots&\vdots&\ddots&\mathbb{I}_{d}\\\
\mathbb{O}&\mathbb{O}&\ldots&\ldots&\mathbb{O}\end{pmatrix}.$

The fractional Kolmogorov equations (1.6) and (1.8) are obtained along the
same line with ${\mathcal{A}}:=\sqrt{a}(-\Delta_{v})^{\beta/2}$ and the same
matrices ${\mathsf{B}}$.

###  1.4. Main result

Our first result is the Poincaré inequality:

######  Theorem 1.

Consider operators ${\mathcal{A}},{\mathcal{B}}$ satisfying the assumptions
(H). Let $R>0$ be sufficiently large, depending on the number of commutators
$\kappa$. Let $\Omega=B_{R}\times\dots\times B_{R}\times(-5,0)$ and suppose
$f$ is a weak non-negative sub-solution to (1.1) in $\Omega$. Then

(1.17)
$\int_{Q^{+}}\left(f-\frac{1}{|Q^{-}|}\int_{Q^{-}}f\right)_{+}{\,\mathrm{d}}z\lesssim\int_{\tilde{\Omega}}\big{|}{\mathcal{A}}f\big{|}{\,\mathrm{d}}z,$

where $Q^{+}:=Q_{1}\subset\Omega$ denotes the future cylinder, and
$Q^{-}=B_{1}\times\ldots\times B_{1}\times(-5,-4]\subset\Omega$ denotes the
past cylinder (see Figure 1). Moreover, if $\beta=1$, that is in the local
case, $\tilde{\Omega}=\Omega$, whereas, if $\beta\in(0,1)$, that is in the
non-local case, $\tilde{\Omega}={\mathbb{R}}^{d_{0}}\times
B_{R}\times\dots\times B_{R}\times(-5,0)$. The constant is universal and
depends on $\kappa$, $\beta$, $d_{0},\dots,d_{\kappa}$, ${\mathsf{B}}$, $R$
and $\Lambda$.

$\Omega$$Q^{+}$$Q^{-}$ Figure 1. The different cylinders in the statement in
Poincaré inequality (the time variable is represented vertically and upward).

######  Remarks 2.

  1. (1)

The radius $R$ of the ambient space $Q_{R}$ is large enough such that the
trajectories constructed in the proof (see (2.2)) do not exit $Q_{R}$. Since
all maps are affine, we know that $R<+\infty$. One could try to optimise the
size of the ambient space by optimising the control function used in the
construction of the trajectories.

  2. (2)

It is easy to include a source term $S$ in the right hand side of (1.1). Then
the $L^{1}$ norm of $S$ on $\Omega$ should appear on the right hand side of
(1.17).

  3. (3)

It is likely that the approach can be extended to the case where the matrix
${\mathsf{B}}$ has non-zero terms above the sub-diagonal, since they do not
break the commutator structure. This, however, would require additional
techniques, which we leave for further work.

  4. (4)

Under the same assumptions, it is straightforward to prove $L^{p}$ versions of
the Poincaré inequality (1.17) by the same method, for $p\in(1,+\infty)$:

$\int_{Q^{+}}\big{(}f-\langle
f\rangle_{Q^{-}}\big{)}_{+}^{p}{\,\mathrm{d}}z\lesssim\int_{\Omega}\big{|}{\mathcal{A}}f\big{|}^{p}{\,\mathrm{d}}z.$

In fact, it is even possible to obtain directly a gain of integrability inside
the Poincaré by our proof, see the related comments in Section 4.

  5. (5)

The left hand side of (1.17) could be mollified as follows

(1.18)
$\int_{\Omega}\left(f-\int_{\Omega}f\psi_{-}\right)\psi_{+}{\,\mathrm{d}}z\lesssim_{\psi_{\pm}}\int_{\Omega}\big{|}{\mathcal{A}}f\big{|}^{p}{\,\mathrm{d}}z$

where $\psi_{\pm}\in C^{\infty}_{c}(Q^{\pm})$ are non-negative and integrate
to one. This amounts to replacing respectively the indicator functions
$\chi_{f(\cdot)\geq\langle f\rangle_{Q^{-}}}\chi_{Q^{+}}$ and $\chi_{Q^{-}}$
by some mollified versions $\psi_{+}$ and $\psi_{-}$ in the proof.

  6. (6)

In such a mollified formulation, the Poincaré inequality in the local case
would follow from the following “wholespace hypoelliptic Bogovoskiǐ result”
(in the spirit of [5]): given $g=\psi_{+}-\psi_{-}$ with $\psi_{\pm}$ as
above, construct a vector field $\mathbf{F}:=(F_{T},F_{(0)})$ with $F_{T}$
one-dimensional and $F_{(0)}$ $d_{0}$-dimensional, so that

${\mathcal{T}}^{*}F_{T}-\nabla_{(0)}\cdot F_{(0)}=g\quad\text{ and }\quad
F_{T}\geq 0\quad\text{ and
}\quad\|\nabla_{(0)}F_{T}\|_{\infty}+\|F_{(0)}\|_{\infty}\lesssim_{g}1.$

This is an open question, but we believe this vector field to exist.

  7. (7)

It is also possible to prove the local case of our Poincaré inequality through
a “ _parabolic wholespace Bogovoskiǐ inequality_ ” in
$(x^{(\kappa)},\dots,x^{(1)},t)$ (excluding the $x^{(0)}$ variable, i.e. the
velocity in case of a Kolmogorov equation), by following the arguments in [7].
This parabolic wholespace Bogovoskiǐ inequality is an easy generalisation of
the Bogovoskiǐ inequality, where the explicit representation of Bogovoskiǐ is
used to show that the first coordinate of the vector field solving the
divergence problem is controlled below, due to the form of $g$ given in (6).

Our second main result is the weak Harnack inequality

######  Theorem 3 (Weak Harnack Inequality).

Let the operators ${\mathcal{A}},{\mathcal{B}}$ be such that ${\mathcal{B}}$
is given by (H) and ${\mathcal{A}}$ is more specifically given by

  * (i)

${\mathcal{A}}:=\sqrt{A}\nabla_{(0)}$ with $A\in[\Lambda^{-1},\Lambda]$
measurable symmetric matrix (local case),

  * (ii)

${\mathcal{A}}=\sqrt{a}(-\Delta_{v})^{\beta/2}$ with
$a\in[\Lambda^{-1},\Lambda]$ scalar measurable (non-local case).

Let $f$ be a weak super-solution to (1.1) on $Q_{1}$ with $0\leq f\leq 1$ in
$Q_{1}$ (local case) or on ${\mathbb{R}}^{d}\times
B_{1}^{\otimes\kappa}\times[-1,0]$ with $0\leq f\leq 1$ (non-local case). Then
there is $C>0$ and $\zeta>0$ depending on $\beta,N,\kappa,\Lambda$ such that
for $r_{0}>0$ sufficiently small the Weak Harnack inequality is satisfied:

(1.19)
$\Bigg{(}\int_{\tilde{Q}^{-}_{r_{0}/2}}f^{\zeta}(z){\,\mathrm{d}}z\Bigg{)}^{\frac{1}{\zeta}}\leq
C\inf_{Q_{r_{0}/2}}f,$

where
$\tilde{Q}_{\frac{r_{0}}{2}}^{-}:=Q_{\frac{r_{0}}{2}}\big{(}(0,\dots,0,-\frac{5}{2}r_{0}^{2\beta}+\frac{1}{2}\big{(}\frac{r_{0}}{2}\big{)}^{2\beta})\big{)}\in{\mathbb{R}}^{N+1}$
(the local case is $\beta=1$).

######  Remark 4.

  1. (1)

In the local case the strong Harnack follows immediately from the weak Harnack
inequality above and the gain of integrability (first lemma of De Giorgi). By
contrast, the strong Harnack inequality does not follow as immediately in the
non-local case, since the function values outside the domain affect the
solution inside the domain. It is however possible to adapt the recent work
[11], which proves the strong Harnack inequality in the non-local case with
one commutator, to our setting.

  2. (2)

It is standard to deduce the Hölder regularity from the weak Harnack
inequality. We do not repeat the argument, but it is identical for instance to
those in [8, 12].

###  1.5. Motivation

Kolmogorov equations naturally appear in the theory of stochastic processes:
equation (1.2) was indeed studied by Kolmogorov in 1934 [10] when $A$ is the
identity matrix to understand the “time-integrated Brownian motion”. The
equation is then indeed the Kolmogorov forward equation for the process

${\rm d}V_{t}={\rm d}W_{t},\quad{\rm d}X_{t}=V_{t}{\,\mathrm{d}}t.$

Kolmogorov obtained explicit formula for its fundamental solution in [10]
which inspired the seminal work [9]. Integrating twice in time the Brownian
motion would lead to equation (1.4) with $\kappa=2$ and $A$ the identity
matrix. Due to this connection to stochastic processes, several mathematical
models involving linear and nonlinear Kolmogorov type equations have also
appeared in finance [1, 3, 6], and in particular equations of the type (1.4)
appear in various models for pricing of path-dependent financial instruments,
e.g. [4]). For example the equation for $P=P(S,A,t)$

(1.20) $\partial_{t}P+\tfrac{1}{2}{\sigma}^{2}S^{2}\partial^{2}_{S}P+(\log
S)\partial_{A}P+r(S\partial_{S}P-P)=0,\qquad S>0,\,A,t\in\mathbb{R},$

arises in the Black and Scholes option pricing problem

$\begin{cases}{\rm d}S_{t}=\mu
S_{t}{\,\mathrm{d}}t+{\sigma}S_{t}{\,\mathrm{d}}W_{t},\\\ {\rm
d}A_{t}=S_{t}{\,\mathrm{d}}t,\end{cases}$

where ${\sigma}$ is the volatility of the stock price $S$, $\mu$ is the
interest rate of a riskless bond and $P=P(S,A,t)$ is the price of the Asian
option depending on the price of the stock $S$, the geometric average $A$ of
the past price and the time to maturity $t$. For a more exhaustive treatment
of the applications of Kolmogorov operators to finance and to stochastic
theory, we refer to the monograph [16] by Pascucci.

Let us close this paragraph by mentioning that analogously, in the non-local
case when $a=1$, or equivalently when $\mathcal{A}$ corresponds to the
fractional Laplacian, the stochastic process underlying this fractional
diffusion is a Lévy process.

###  1.6. Strategy of proof

The starting point is the method developed in [8] for proving the Poincaré
inequality using trajectories and deducing the second Lemma of De Giorgi and
the weak Harnack inequality.

The strategy to obtain a Poincaré inequality for the local case with one
commutator in [8] can be summarised as follows. The first step is to mollify
the characteristic function of $Q^{-}$, by adding an error term

$\displaystyle\int_{Q^{+}}\left(f-\langle
f\rangle_{Q^{-}}\right)_{+}{\,\mathrm{d}}z\leq$ $\displaystyle\int_{z_{+}\in
Q^{+}}\left(\int_{z_{-}\in
Q^{-}}\left[f(z_{+})-f(z_{-})\right]\varphi_{\varepsilon}(w,y){\,\mathrm{d}}z_{-}\right)_{+}{\,\mathrm{d}}z_{+}$
$\displaystyle+\varepsilon^{2d}\|f\|_{L^{2}(Q_{R})}$

with $0\leq\varphi_{\varepsilon}\leq 1$ smooth, $\varepsilon>0$,
$z_{+}=(x,v,t)\in Q^{+}$ and $z_{-}=(y,w,s)\in Q^{-}$. The second step
consists in constructing piecewise affine trajectories following the two
vector fields $\mathcal{T}:=\partial_{t}+v\cdot\nabla_{x}$ and $\nabla_{v}$ to
connect any $z_{+}\in Q^{+}$ with any $z_{-}\in Q^{-}$:

$\displaystyle(x,v,t)\underset{\nabla_{v}}{\longrightarrow}\left(x,\frac{x-y}{t-s},t\right)\underset{{\mathcal{T}}}{\longrightarrow}\left(y,\frac{x-y}{t-s},s\right)\underset{\nabla_{v}}{\longrightarrow}(y,w,s).$

Note here that a positive time gap between the two cylinders $Q^{-}$ and
$Q^{+}$ is used to make sure the intermediate velocity $(x-y)/(t-s)$ remains
bounded. We then write $f(z_{+})-f(z_{-})$ as an integral along the chosen
trajectory and use ${\mathcal{T}}f+{\mathcal{A}}^{*}{\mathcal{A}}f\leq 0$:

$\displaystyle\int_{Q^{-}}\left[f(z_{+})-f(z_{-})\right]\varphi_{\varepsilon}(z_{-}){\,\mathrm{d}}z_{-}$
$\displaystyle\lesssim\int_{Q^{-}}\int_{\text{trajectory}}{\mathcal{T}}f(\cdots)\varphi_{\varepsilon}(w,y){\,\mathrm{d}}s{\,\mathrm{d}}z_{-}$
$\displaystyle\quad+\int_{Q^{-}}\int_{\text{trajectory}}\nabla_{v}f(\cdots)\varphi_{\varepsilon}(w,y){\,\mathrm{d}}s{\,\mathrm{d}}z_{-}$
$\displaystyle\lesssim\int_{Q^{-}}\int_{\text{trajectory}}\left[\nabla_{v}\cdot\left(A\nabla_{v}f\right)\right](\cdots)\varphi_{\varepsilon}(w,y){\,\mathrm{d}}s{\,\mathrm{d}}z_{-}$
$\displaystyle\quad+\int_{Q^{-}}\int_{\text{trajectory}}\nabla_{v}f(\cdots)\varphi_{\varepsilon}(w,y){\,\mathrm{d}}s{\,\mathrm{d}}z_{-}.$

We then want to integrate by parts the $v$-divergence in the past variable
$z_{-}$. This integration by parts degenerates near the future point $z_{+}$,
since $z_{+}$ does not depend on $z_{-}$, and it produces a non-integrable
singularity. In order to overcome this singularity, the paper [8] introduced
an additional small fourth sub-trajectory along $\nabla_{x}$:

$\displaystyle(v,x,t)$
$\displaystyle\underset{\nabla_{x}}{\longrightarrow}(v,x+\varepsilon
w,t)\underset{\nabla_{v}}{\longrightarrow}\left(\frac{x+\varepsilon
w-y}{t-s},x+\varepsilon w,t\right)$
$\displaystyle\underset{{\mathcal{T}}}{\longrightarrow}\left(\frac{x+\varepsilon
w-y}{t-s},y,s\right)\underset{\nabla_{v}}{\longrightarrow}(w,y,s).$

This sub-trajectory “noises” the future position variable with the past
velocity. This produces a second error term of the form
$\varepsilon^{\sigma}\|f\|_{L^{1}_{t,v}W^{\sigma,1}_{x}}$, which can, however,
be controlled for $\varepsilon$ small by the integral regularity
$L^{1}_{t,v}W^{\sigma,1}_{x}$ (which can be established for sub-solutions to
the Kolmogorov equation, for a small $\sigma>0$). Finally it yields

$\int_{Q^{+}}\big{(}f-\langle
f\rangle_{Q^{-}}\big{)}_{+}{\,\mathrm{d}}z\lesssim
C_{\varepsilon}\int_{Q_{R}}\big{|}S\big{|}{\,\mathrm{d}}z+\int_{Q_{R}}\big{|}\nabla_{v}f\big{|}{\,\mathrm{d}}z+\varepsilon^{\sigma}\|f\|_{L^{1}_{t,v}W^{\sigma,1}_{x}(Q_{R})}+\varepsilon^{2d}\|f\|_{L^{2}(Q_{R})}$

for a constant $C_{\varepsilon}>0$ depending on $\varepsilon>0$. The two
additional errors weakens the Poincaré inequality, but the latter is shown to
be sufficient for implementing the next steps of the De Giorgi theory in [8].

In the present article, we improve the method by simplifying the proof and
removing the two error terms in the inequality. Moreover, the gain of
integrability can directly be extracted from the proof of the Poincaré
inequality. The first novel idea is to use three cylinders $Q^{-}$, $Q^{0}$
and $Q^{+}$, and mollify the characteristic function only in the intermediate
cylinder $Q^{0}$, see Figure 2.

$\Omega$$Q^{+}$$Q^{0}$$Q^{-}$$\Gamma_{+}$$\Gamma_{-}$$z_{+}$$z_{0}$$z_{-}$
Figure 2. Construction of the trajectories. The curve $\Gamma_{+}$ connects
any point $z_{+}\in Q^{+}$ to some intermediate point $z_{0}\in Q^{0}$,
whereas the curve $\Gamma_{-}$ connects any point $z_{-}\in Q^{-}$ to some
intermediate point $z_{0}\in Q^{0}$.

It is easy to deduce from the triangle inequality

$\displaystyle\fint_{Q^{+}}\left[f(z_{+})-\langle
f\rangle_{Q^{-}}\right]_{+}{\,\mathrm{d}}z_{+}\leq$
$\displaystyle\fint_{Q^{+}}\left(\fint_{Q^{0}}\left[f(z_{+})-f(z_{0})\right]\varphi_{\varepsilon}(v_{0},x_{0}){\,\mathrm{d}}z_{0}\right)_{+}{\,\mathrm{d}}z_{+}$
(1.21)
$\displaystyle+\fint_{Q^{-}}\left(\fint_{Q^{0}}\left[f(z_{0})-f(z_{-})\right]\varphi_{\varepsilon}(v_{0},x_{0}){\,\mathrm{d}}z_{0}\right)_{+}{\,\mathrm{d}}z_{-}$

for $\varphi_{\varepsilon}\geq 0$ smooth, with mass $1$, where
$z_{0}=(x_{0},v_{0},t_{0})\in Q^{0},z_{+}\in Q^{+},z_{-}\in Q^{-}$, and where
we have denoted by $\fint_{Q}:=|Q|^{-1}\int_{Q}$ the normalized integral. We
then construct trajectories to connect $z_{+}$ to $z_{0}$, and $z_{-}$ to
$z_{0}$. This removes the first error term we had introduced in relation to
the mollification $\varphi_{\varepsilon}$.

The second novel idea is to connect the points by a curved trajectory, rather
than a piecewise affine one, by solving a control problem with a well-chosen
forcing (i.e. control function), in order to reduce the order of the
singularity of the integration by parts near $z_{\pm}$. More precisely, the
first idea is to consider trajectories whose speed diverges at $z_{\pm}$ but
with an integrable divergence so that the trajectories remain bounded. This
idea is inspired from the reading of [15], but we propose simpler and more
systematic trajectories. The second important idea is to use $\kappa+1$
linearly independent control power functions with such well-balanced diverging
behaviour at $z_{\pm}$ in order to connect all points between the cylinders,
thanks to the Hörmander commutator condition.

###  1.7. Outline

In Section 2 we provide the explicit construction of the trajectories
connecting points in the future to points in the past. These trajectories are
then used in Section 3 to prove the Poincaré inequality; they allow to
estimate the $L^{1}$ norm in the future of the difference between $f$ and the
past average of $f$, or in other words, the left hand side of (1.17). In
Section 4, we prove the Harnack inequalities on the basis of this Poincaré
inequality and deduce the Hölder regularity.

##  2\. Construction of the trajectories

###  2.1. The base case

Let us first consider the case with one commutator ($\kappa=1$),
$d_{1}=d_{0}=d$ and ${\mathsf{B}}_{1}=\text{Id}$. Given three points $z_{+}\in
Q^{+},z_{0}\in Q^{0}$ and $z_{-}\in Q^{-}$, we want to construct two paths
$s\rightarrow\Gamma_{+}(s)$ and $s\rightarrow\Gamma_{-}(s)$ for $s\in[0,1]$
such that (see Figures 2 and 3)

$\displaystyle\Gamma_{+}(s)=\left(X^{(1)}_{+}(s),X^{(0)}_{+}(s),T_{+}(s)\right)^{T},\qquad\Gamma_{+}(0)=z_{+},\qquad\Gamma_{+}(1)=z_{0},$
$\displaystyle\Gamma_{-}(s)=\left(X^{(1)}_{-}(s),X^{(0)}_{-}(s),T_{-}(s)\right)^{T},\qquad\Gamma_{-}(0)=z_{-},\qquad\Gamma_{-}(1)=z_{0},$

where

(2.1)
$\displaystyle\begin{cases}{\frac{\mathrm{d}}{\mathrm{d}s}}X^{(0)}_{\pm}(s)=\mathsf{m}_{\pm}^{(0)}g_{0}^{\prime\prime}(s)+\mathsf{m}^{(1)}_{\pm}g_{1}^{\prime\prime}(s),\\\\[5.69054pt]
{\frac{\mathrm{d}}{\mathrm{d}s}}X^{(1)}_{\pm}(s)=\delta_{\pm}X^{(0)}_{\pm}(s),\\\\[5.69054pt]
{\frac{\mathrm{d}}{\mathrm{d}s}}T_{\pm}(s)=\delta_{\pm},\end{cases}$

where $\delta_{\pm}:=t_{0}-t_{\pm}$ and the control functions $g_{i}\in
C^{2}((0,1])$ with $g_{i}(0)=g_{i}^{\prime}(0)=0$ for $i=0,1$, and
$\mathsf{m}^{(i)}_{\pm}\in{\mathbb{R}}^{d}$ for $i=0,1$.

-$\nabla_{(0)}$$\nabla_{(0)}$${\mathcal{T}}$$\dot{\Gamma}^{+}(s)$$x^{(0)}$$t$$x^{(1)}$$\Gamma_{+}(0)$$\Gamma_{+}(1)$ Figure 3. Construction of the trajectories. The curve $\Gamma_{+}$ connects any point $\Gamma_{+}(0)=z_{+}\in Q^{+}$ to some intermediate point $\Gamma_{+}(1)=z_{0}\in Q^{0}$ along the vector fields ${\mathcal{T}}$p and $\nabla_{(0)}$.

Solving the differential equations yields

(2.2)
$\displaystyle\begin{cases}X^{(0)}_{\pm}(s)=x^{(0)}_{\pm}+\mathsf{m}_{\pm}^{(0)}g_{0}^{\prime}(s)+\mathsf{m}^{(1)}_{\pm}g_{1}^{\prime}(s),\\\\[5.69054pt]
X^{(1)}_{\pm}(s)=x^{(1)}_{\pm}+s\delta_{\pm}x^{(0)}_{\pm}+\delta_{\pm}\left[\mathsf{m}_{\pm}^{(0)}g_{0}(s)+\mathsf{m}^{(1)}_{\pm}g_{1}(s)\right],\\\\[5.69054pt]
T_{\pm}(s)=st_{0}+(1-s)t_{\pm}.\end{cases}$

Let us denote $\text{{\bf 1}}_{d}$ the $d$-vector of $1$’s, $\text{{\bf
0}}_{d}$ the $d$-vector of $0$’s, $\text{Id}_{d}$ the $d\times d$ identity
matrix, $\mathbb{O}$ a zero block of arbitrary size, and

$\displaystyle\mathsf{M}_{\pm}:=\begin{pmatrix}\mathsf{m}^{(0)}_{\pm}\\\
\mathsf{m}^{(1)}_{\pm}\end{pmatrix},\quad{\mathsf{W}}(s):=\begin{pmatrix}g_{0}^{\prime}(s)\text{Id}_{d}&g_{1}^{\prime}(s)\text{Id}_{d}\\\
g_{0}(s)\text{Id}_{d}&g_{1}(s)\text{Id}_{d}\end{pmatrix},$
$\displaystyle{\mathsf{W}}_{\pm}^{\delta}(s):=\begin{pmatrix}\text{Id}_{d}&\mathbb{O}\\\
\mathbb{O}&\delta_{\pm}\text{Id}_{d}\end{pmatrix}{\mathsf{W}}(s),\quad\mathsf{Y}_{\pm}=\begin{pmatrix}\mathsf{y}^{(0)}_{\pm}\\\
\mathsf{y}^{(1)}_{\pm}\end{pmatrix}:=\begin{pmatrix}x^{(0)}_{0}-x^{(0)}_{\pm}\\\
x^{(1)}_{0}-x^{(1)}_{\pm}-\delta_{\pm}x^{(0)}_{\pm}\end{pmatrix}.$

The boundary conditions $\Gamma_{\pm}(1)=z_{0}$ impose

(2.3)
$\displaystyle{\mathsf{W}}_{\pm}^{\delta}(1)\mathsf{M}_{\pm}=\mathsf{Y}_{\pm}\quad\Longrightarrow\quad\mathsf{M}_{\pm}={\mathsf{W}}_{\pm}^{\delta}(1)^{-1}\mathsf{Y}_{\pm},$

provided that the Wronskian matrix ${\mathsf{W}}(1)$ is invertible at $s=1$.
We deduce

$\displaystyle\begin{pmatrix}X^{(0)}_{\pm}(s)\\\
X^{(1)}_{\pm}(s)\end{pmatrix}={\mathsf{W}}_{\pm}^{\delta}(s)\mathsf{M}_{\pm}+\begin{pmatrix}\text{Id}_{d}&\mathbb{O}\\\
\delta_{\pm}s\text{Id}_{d}&\text{Id}_{d}\end{pmatrix}\begin{pmatrix}x_{\pm}^{(0)}\\\
x_{\pm}^{(1)}\end{pmatrix}.$

Using (2.3) we obtain

$\displaystyle\begin{pmatrix}X^{(0)}_{\pm}(s)\\\
X^{(1)}_{\pm}(s)\end{pmatrix}$
$\displaystyle={\mathsf{W}}_{\pm}^{\delta}(s){\mathsf{W}}_{\pm}^{\delta}(1)^{-1}\mathsf{Y}_{\pm}+\begin{pmatrix}\text{Id}_{d}&\mathbb{O}\\\
\delta_{\pm}s\text{Id}_{d}&\text{Id}_{d}\end{pmatrix}\begin{pmatrix}x_{\pm}^{(0)}\\\
x_{\pm}^{(1)}\end{pmatrix}$
$\displaystyle=W^{\delta}_{\pm}(s)\big{[}W^{\delta}_{\pm}(1)\big{]}^{-1}\begin{pmatrix}x_{0}^{(0)}\\\
x_{0}^{(1)}\end{pmatrix}$
$\displaystyle\quad-W^{\delta}_{\pm}(s)\big{[}W^{\delta}_{\pm}(1)\big{]}^{-1}\begin{pmatrix}x^{(0)}_{\pm}\\\
x^{(1)}_{\pm}+\delta_{\pm}x^{(0)}_{\pm}\end{pmatrix}+\begin{pmatrix}\text{Id}_{d}&\mathbb{O}\\\
\delta_{\pm}s\text{Id}_{d}&\text{Id}_{d}\end{pmatrix}\begin{pmatrix}x_{\pm}^{(0)}\\\
x_{\pm}^{(1)}\end{pmatrix}$
$\displaystyle=\mathfrak{A}^{s}\begin{pmatrix}x_{0}^{(0)}\\\
x_{0}^{(1)}\end{pmatrix}+\mathfrak{B}^{s}=\mathfrak{A}^{s}x_{0}+\mathfrak{B}^{s}=:\Phi^{s}_{\pm}(x_{0})$

is an affine function with matrix
$\mathfrak{A}^{s}:={\mathsf{W}}_{\pm}^{\delta}(s)[{\mathsf{W}}_{\pm}^{\delta}(1)]^{-1}$
and a vector $\mathfrak{B}^{s}$ that depend only on $s$ and $x_{\pm}$. We
still have to prove that the matrix $\mathfrak{A}^{s}$ is invertible for
$s\in(0,1]$. If so, given $s\in(0,1]$, the derivative along the first variable
$X^{(0)}_{\pm}(s)$ of the inverse is

$\nabla_{(0)}\left(\Phi_{\pm}^{s}\right)^{-1}={\mathsf{W}}^{\delta}_{\pm}(1)\big{[}{\mathsf{W}}^{\delta}_{\pm}(s)\big{]}^{-1}\left(\begin{matrix}\text{{\bf
1}}_{d}\\\ \text{{\bf 0}}_{d}\end{matrix}\right).$

We choose $g_{i}(s):=(1+\alpha_{0})^{-1}(2+\alpha_{0})^{-1}s^{2+\alpha_{0}}$,
$i=0,1$, with $\alpha_{0},\alpha_{1}\in(-1,0)$ and
$\alpha_{0}\not=\alpha_{1}$. Then the Wronskian matrix is invertible for all
$s\not=0$:

(2.4) $\displaystyle\det{\mathsf{W}}(s)$
$\displaystyle=\left(g_{0}^{\prime}(s)g_{1}(s)-g_{0}(s)g_{1}^{\prime}(s)\right)^{d}$
$\displaystyle=\frac{(\alpha_{0}-\alpha_{1})^{d}}{(1+\alpha_{0})^{d}(1+\alpha_{1})^{d}(2+\alpha_{0})^{d}(2+\alpha_{1})^{d}}s^{(3+\alpha_{0}+\alpha_{1})d}\not=0.$

Our choice of two linearly independent control functions $g_{0}$ and $g_{1}$
is guided by ensuring the Wronskian matrix to be invertible. Moreover,

$\displaystyle{\mathsf{W}}_{\pm}^{\delta}(1):=\begin{pmatrix}\frac{\text{Id}_{d}}{(1+\alpha_{0})}&\frac{\text{Id}_{d}}{(1+\alpha_{1})}\\\
\frac{\delta_{\pm}\text{Id}_{d}}{(1+\alpha_{0})(2+\alpha_{0})}&\frac{\delta_{\pm}\text{Id}_{d}}{(1+\alpha_{1})(2+\alpha_{1})}\end{pmatrix},$
$\displaystyle{\mathsf{W}}_{\pm}^{\delta}(s)^{-1}:=\frac{1}{\delta_{\pm}\det{\mathsf{W}}(s)}\begin{pmatrix}\frac{\delta_{\pm}s^{2+\alpha_{1}}\text{Id}_{d}}{(1+\alpha_{1})(2+\alpha_{1})}&\frac{-s^{1+\alpha_{1}}\text{Id}_{d}}{(1+\alpha_{1})}\\\
\frac{-\delta_{\pm}s^{(2+\alpha_{0})}\text{Id}_{d}}{(1+\alpha_{0})(2+\alpha_{0})}&\frac{s^{1+\alpha_{0}}\text{Id}_{d}}{(1+\alpha_{0})}\end{pmatrix},$

which finally yields

$\displaystyle\nabla_{(0)}\left(\Phi_{\pm}^{s}\right)^{-1}$
$\displaystyle=W^{\delta}_{\pm}(1)\big{[}W^{\delta}_{\pm}(s)\big{]}^{-1}\begin{pmatrix}\text{{\bf
1}}_{d}\\\ \text{{\bf 0}}_{d}\end{pmatrix}$
$\displaystyle=\frac{1}{(\alpha_{0}-\alpha_{1})}\begin{pmatrix}(2+\alpha_{0})s^{-1-\alpha_{0}}-(2+\alpha_{1})s^{-1-\alpha_{1}}\text{{\bf
1}}_{d}\\\
\delta_{\pm}\left(s^{-1-\alpha_{0}}-s^{-1-\alpha_{1}}\right)\text{{\bf
1}}_{d}\end{pmatrix}$
$\displaystyle=O(s^{-1-\alpha_{0}})+O(s^{-1-\alpha_{1}}),$

which remains integrable for $\alpha_{0},\alpha_{1}\in(-1,0)$ with
$\alpha_{0}\neq\alpha_{1}$. Observe also that our choice of control functions
$g_{0}$ and $g_{1}$ implies that all their derivatives up to order two are
integrable on $s\in[0,1]$, which implies that the trajectories are bounded
(with integrable tangent vector field).

###  2.2. The general case

Let us now consider the general case with $\kappa\geq 2$ commutators, general
dimensions $d_{0}\geq d_{1}\geq\dots\geq d_{\kappa}\geq 1$ and surjective
$d_{i}\times d_{i-1}$ matrices ${\mathsf{B}}_{i}$. Consider for $s\in[0,1]$
the paths

$\displaystyle\Gamma_{+}(s)=\left(X^{(\kappa)}_{+}(s),\dots,X^{(0)}_{+}(s),T_{+}(s)\right)^{T},\qquad\Gamma_{+}(0)=z_{+},\qquad\Gamma_{+}(1)=z_{0},$
$\displaystyle\Gamma_{-}(s)=\left(X^{(\kappa)}_{-}(s),\dots,X^{(0)}_{-}(s),T_{-}(s)\right)^{T},\qquad\Gamma_{-}(0)=z_{-},\qquad\Gamma_{-}(1)=z_{0},$

and the following control problem

(2.5)
$\displaystyle\begin{cases}{\frac{\mathrm{d}}{\mathrm{d}s}}X^{(0)}_{\pm}(s)=\sum_{i=0}^{\kappa}\mathsf{m}_{\pm}^{(i)}g_{i}^{(\kappa+1)}(s),\\\\[5.69054pt]
{\frac{\mathrm{d}}{\mathrm{d}s}}X^{(1)}_{\pm}(s)=\delta_{\pm}{\mathsf{B}}_{1}X^{(0)}_{\pm}(s),\\\\[5.69054pt]
\quad\vdots\\\\[5.69054pt]
{\frac{\mathrm{d}}{\mathrm{d}s}}X^{(\kappa)}_{\pm}(s)=\delta_{\pm}{\mathsf{B}}_{\kappa}X^{(\kappa-1)}_{\pm}(s),\\\\[5.69054pt]
{\frac{\mathrm{d}}{\mathrm{d}s}}T_{\pm}(s)=\delta_{\pm}\end{cases}$

with $\delta_{\pm}:=t_{0}-t_{\pm}$ and the control functions $g_{i}\in
C^{\kappa+1}((0,1])$ so that
$g_{i}(0)=g_{i}^{\prime}(0)=\dots=g_{i}^{(\kappa)}(0)=0$ for
$i=0,1,\dots,\kappa$, and $\mathsf{m}^{(i)}_{\pm}\in{\mathbb{R}}^{d_{0}}$ for
$i=0,\dots,\kappa$. Then

(2.6)
$\displaystyle\begin{cases}X^{(0)}_{\pm}(s)=x^{(0)}_{\pm}+\sum_{i=0}^{\kappa}\mathsf{m}_{\pm}^{(i)}g_{i}^{(\kappa)}(s),\\\\[5.69054pt]
X^{(1)}_{\pm}(s)=x^{(1)}_{\pm}+s\delta_{\pm}\tilde{\mathsf{B}}_{1,1}x^{(0)}_{\pm}+\delta_{\pm}\left[\sum_{i=0}^{\kappa}\tilde{\mathsf{B}}_{1,1}\mathsf{m}_{\pm}^{(i)}g_{i}^{(\kappa-1)}(s)\right],\\\\[5.69054pt]
X^{(2)}_{\pm}(s)=x^{(2)}_{\pm}+s\delta_{\pm}\tilde{\mathsf{B}}_{2,2}x^{(1)}_{\pm}+\frac{(s\delta_{\pm})^{2}}{2}\tilde{\mathsf{B}}_{2,1}x^{(0)}_{\pm}+\delta_{\pm}^{2}\left[\sum_{i=0}^{\kappa}\tilde{\mathsf{B}}_{2,1}\mathsf{m}_{\pm}^{(i)}g_{i}^{(\kappa-2)}(s)\right],\\\\[5.69054pt]
\quad\vdots\\\\[5.69054pt]
X^{(\kappa)}_{\pm}(s)=x_{\pm}^{(\kappa)}+\sum_{i=1}^{\kappa}\frac{(s\delta_{\pm})^{i}}{i!}\tilde{\mathsf{B}}_{\kappa,\kappa-i+1}x^{(\kappa-i)}_{\pm}+\delta_{\pm}^{\kappa}\left[\sum_{i=0}^{\kappa}\tilde{\mathsf{B}}_{\kappa,1}\mathsf{m}_{\pm}^{(i)}g_{i}(s)\right],\\\\[5.69054pt]
T_{\pm}(s)=st_{0}+(1-s)t_{\pm}\end{cases}$

where
$\tilde{\mathsf{B}}_{i,j}:={\mathsf{B}}_{i}{\mathsf{B}}_{i-1}\cdots{\mathsf{B}}_{j}$
for $1\leq j\leq i$, which is a $d_{i}\times d_{j-1}$ block matrix. We denote

(2.7) $\displaystyle\mathsf{M}_{\pm}:=\begin{pmatrix}\mathsf{m}^{(0)}_{\pm}\\\
\vdots\\\
\mathsf{m}^{(\kappa)}_{\pm}\end{pmatrix},\quad{\mathsf{W}}(s):=\begin{pmatrix}g_{0}^{(\kappa)}(s)\text{Id}_{d_{0}}&\dots&g_{\kappa}^{(\kappa)}(s)\text{Id}_{d_{0}}\\\
\vdots&\vdots&\vdots\\\
g_{0}(s)\text{Id}_{d_{0}}&\dots&g_{\kappa}(s)\text{Id}_{d_{0}}\end{pmatrix},$
(2.8)
$\displaystyle{\mathsf{W}}_{\pm}^{\delta}(s):=\begin{pmatrix}\text{Id}_{d_{0}}&\mathbb{O}&\cdots&\cdots&\mathbb{O}\\\
\mathbb{O}&\tilde{\mathsf{B}}_{1,1}\delta_{\pm}&\mathbb{O}&\cdots&\mathbb{O}\\\
\vdots&\mathbb{O}&\tilde{\mathsf{B}}_{2,2}\delta_{\pm}^{2}&\ddots&\mathbb{O}\\\
\vdots&\vdots&\ddots&\ddots&\mathbb{O}\\\
\mathbb{O}&\cdots&\cdots&\mathbb{O}&\tilde{\mathsf{B}}_{\kappa,\kappa}\delta_{\pm}^{\kappa}\end{pmatrix}W(s),$
(2.9) $\displaystyle\mathsf{Y}_{\pm}=\begin{pmatrix}\mathsf{y}^{(0)}_{\pm}\\\
\vdots\\\ \vdots\\\
\mathsf{y}^{(\kappa)}_{\pm}\end{pmatrix}:=\begin{pmatrix}\mathsf{x}_{0}^{(0)}\\\
\vdots\\\ \vdots\\\
\mathsf{x}_{0}^{(\kappa)}\end{pmatrix}-\mathsf{T}_{\pm}^{s}\begin{pmatrix}\mathsf{x}^{(0)}_{\pm}\\\
\vdots\\\ \vdots\\\ \mathsf{x}^{(\kappa)}_{\pm}\end{pmatrix},$ (2.10)
$\displaystyle{\mathsf{T}}_{\pm}^{s}:=\begin{pmatrix}\text{Id}_{d_{0}}&\mathbb{O}&\cdots&\cdots&\mathbb{O}\\\
(s\delta_{\pm})\tilde{\mathsf{B}}_{1,1}&\hbox{Id}_{d_{1}}&\mathbb{O}&\cdots&\mathbb{O}\\\
\vdots&(s\delta_{\pm})\tilde{\mathsf{B}}_{2,2}&\hbox{Id}_{d_{2}}&\ddots&\mathbb{O}\\\
\vdots&\vdots&\ddots&\ddots&\mathbb{O}\\\
\frac{(s\delta_{\pm})^{\kappa}}{\kappa!}\tilde{\mathsf{B}}_{\kappa,1}&\cdots&\cdots&(s\delta_{\pm})\tilde{\mathsf{B}}_{\kappa,\kappa}&\hbox{Id}_{d_{\kappa}}\end{pmatrix}.$

Note that the matrix ${\mathsf{W}}(s)$ is the Wronskian of the family of
functions $(g_{i})_{i=0}^{\kappa}$ and is invertible when they are the
linearly independent solutions to a $(\kappa+1)$-order linear ODE. The
boundary conditions $\Gamma_{\pm}(1)=z_{0}$ impose

$\displaystyle{\mathsf{W}}_{\pm}^{\delta}(1)\mathsf{M}_{\pm}=\mathsf{Y}_{\pm}\quad\Longrightarrow\quad\mathsf{M}_{\pm}={\mathsf{W}}_{\pm}^{\delta}(1)^{-1}\mathsf{Y}_{\pm},$

provided that the Wronskian matrix ${\mathsf{W}}(1)$ is invertible at $s=1$,
and by using the standard pseudo-inverse of the surjective matrices
$\tilde{\mathsf{B}}_{i}$ (which is zero on the orthogonal of the range). We
deduce that

$\displaystyle\begin{pmatrix}X^{(0)}_{\pm}(s)\\\ \ \vdots\\\
X^{(\kappa)}_{\pm}(s)\end{pmatrix}$
$\displaystyle={\mathsf{W}}_{\pm}^{\delta}(s)\big{[}{\mathsf{W}}_{\pm}^{\delta}(1)\big{]}^{-1}\begin{pmatrix}x_{0}^{(0)}\\\
\vdots\\\ x_{0}^{(\kappa)}\end{pmatrix}$
$\displaystyle\quad-{\mathsf{W}}_{\pm}^{\delta}(s)\big{[}{\mathsf{W}}_{\pm}^{\delta}(1)\big{]}^{-1}\begin{pmatrix}x_{\pm}^{(0)}\\\
\vdots\\\
\sum_{i=0}^{\kappa}\frac{\delta_{\pm}^{i}}{i!}x_{\pm}^{(\kappa-i)}\end{pmatrix}+\begin{pmatrix}x^{(0)}_{\pm}\\\
\vdots\\\
\sum_{i=0}^{\kappa}\frac{(s\delta_{\pm})^{i}}{i!}x^{(\kappa-i)}_{\pm}\end{pmatrix}$
$\displaystyle={\mathsf{W}}_{\pm}^{\delta}(s)\big{[}{\mathsf{W}}_{\pm}^{\delta}(1)\big{]}^{-1}x_{0}+\left(\hbox{Id}-{\mathsf{W}}_{\pm}^{\delta}(s)\big{[}{\mathsf{W}}_{\pm}^{\delta}(1)\big{]}^{-1}\right){\mathsf{T}}_{\pm}^{s}x_{\pm}$
(2.11)
$\displaystyle=\mathfrak{A}^{s}x_{0}+\mathfrak{B}^{s}=:\Phi^{s}_{\pm}(x_{0})$

is an affine function with matrix
$\mathfrak{A}^{s}:={\mathsf{W}}_{\pm}^{\delta}(s)[{\mathsf{W}}_{\pm}^{\delta}(1)]^{-1}$
and a vector $\mathfrak{B}^{s}$ that depend only on $s$ and $x_{\pm}$. We
still have to prove that the matrix $\mathfrak{A}^{s}$ is invertible for
$s\in(0,1]$. If so, given $s\in(0,1]$, the derivative along the first variable
$X^{(0)}_{\pm}(s)$ of the inverse is

(2.12)
$\nabla_{(0)}\left(\Phi_{\pm}^{s}\right)^{-1}={\mathsf{W}}_{\pm}^{\delta}(1)\big{[}{\mathsf{W}}_{\pm}^{\delta}(s)\big{]}^{-1}\begin{pmatrix}\text{{\bf
1}}_{d_{0}}\\\ \text{{\bf 0}}_{d_{1}}\\\ \vdots\\\ \text{{\bf
0}}_{d_{\kappa}}\end{pmatrix}.$

We finally choose the control functions

(2.13)
$g_{i}(s):=\frac{s^{1+\kappa+\alpha_{i}}}{(1+\alpha_{i})(2+\alpha_{i})\cdots(1+\kappa+\alpha_{i})}$

with $\alpha_{i}\in(-1,0)$ pairwise distinct for $i=0,\dots,\kappa$. Then,
when $s\neq 0$, the Wronskian matrix is invertible with the precise
$s$-behaviour given by the following lemma:

######  Lemma 5.

Consider the $(\kappa+1)\times(\kappa+1)$ matrix
${\mathsf{W}}:=(s^{1+\alpha_{j}+i}/p_{i,j})_{i,j=0}^{\kappa}$ with
$p_{i,j}:=(1+\alpha_{j})\cdots(1+i+\alpha_{j})$. Then its determinant is given
by

(2.14) $\displaystyle\det{\mathsf{W}}(s)$
$\displaystyle\quad={\mathsf{G}}_{\kappa}[\alpha_{0},\dots,\alpha_{\kappa}]\left(\frac{\prod_{i,j=0|i<j}^{\kappa}(\alpha_{i}-\alpha_{j})^{d_{0}}}{\prod_{i,j=0}^{\kappa}(1+i+\alpha_{j})^{d_{0}}}\right)s^{d_{0}\frac{(\kappa+1)(\kappa+2)}{2}+d_{0}\sum_{i=0}^{\kappa}\alpha_{i}},$

where ${\mathsf{G}}_{\kappa}[\alpha_{0},\dots,\alpha_{\kappa}]\neq 0$ for all
families of $(\alpha_{i})_{i=0}^{\kappa}$ pairwise distinct.

###### Proof of Lemma 5.

The proof is done by induction on $\kappa\geq 1$. The calculation (2.4) of the
previous subsection shows that ${\mathsf{G}}_{1}[\alpha_{0},\alpha_{1}]=1$ in
the case $\kappa=1$. Then at step $\kappa\geq 2$ one performs an expansion of
the determinant along the last row to get

${\mathsf{G}}_{\kappa}[\alpha_{0},\dots,\alpha_{\kappa}]=\sum_{j=0}^{\kappa}\frac{\prod_{i=0|i\neq
j}^{\kappa}|\alpha_{i}-\alpha_{j}|^{d_{0}}}{p_{\kappa,j}^{d_{0}}}{\mathsf{G}}_{\kappa-1}\left[(\alpha_{i})_{i=0|i\not=j}^{\kappa}\right].$

This proves that ${\mathsf{G}}_{\kappa-1}\neq 0$ implies
${\mathsf{G}}_{\kappa}\neq 0$ for $(\alpha_{i})_{i=0}^{\kappa}$ pairwise
distinct, and concludes the proof. ∎

By Lemma 5, we thus find

${\mathsf{W}}(s)^{-1}:=\frac{1}{\det{\mathsf{W}}(s)}\text{Comatrix}{\mathsf{W}}(s).$

Going back to the calculation of (2.12), and combining (2.14) and the
$s$-scaling of the entries of the cofactor matrix, we get

(2.15)
$\nabla_{(0)}\left(\Phi^{s}_{\pm}\right)^{-1}={\mathsf{W}}_{\pm}^{\delta}(1)\big{[}{\mathsf{W}}_{\pm}^{\delta}(s)\big{]}^{-1}\begin{pmatrix}\text{{\bf
1}}_{d_{0}}\\\ \text{{\bf 0}}_{d_{1}}\\\ \vdots\\\ \text{{\bf
0}}_{d_{\kappa}}\end{pmatrix}=\sum_{i=0}^{\kappa}O(s^{-1-\alpha_{i}}),$

which is integrable for $\alpha_{i}\in(-1,0)$. Observe also that our choice of
control functions $g_{i}$, $i=1,\dots,\kappa$, implies that all their
derivatives up to order $\kappa+1$ are integrable on $s\in[0,1]$, which
implies that the trajectories are bounded (with integrable tangent vector
field).

##  3\. Proof of the Poincaré inequality

We consider the three cylinders $Q^{+},Q^{-},Q^{0}\subset\Omega$ as in Figure
2, operators $\mathcal{A}$ and $\mathcal{B}$ that satisfy (H), and a
subsolution $f$ of (1.1). Let $\varphi\in C_{c}^{\infty}({\mathbb{R}}^{N})$ be
a non-negative function in the first $N$ variables
$(x^{(\kappa)},\dots,x^{(0)})$ (excluding the time component) with compact
support in any time-slice of $Q^{0}$ and such that
$\fint_{Q^{0}}\varphi{\,\mathrm{d}}x=1$. Then

(3.1) $\displaystyle\fint_{Q^{+}}\Big{(}f(z_{+})-\langle
f\rangle_{Q^{-}}\Big{)}_{+}{\,\mathrm{d}}z_{+}$
$\displaystyle\leq\fint_{Q^{+}}\fint_{Q^{-}}\Big{(}f(z_{+})-f(z_{-})\Big{)}_{+}{\,\mathrm{d}}z_{-}{\,\mathrm{d}}z_{+}$
$\displaystyle\leq\fint_{Q^{+}}\fint_{Q^{-}}\Big{(}f(z_{+})-\langle
f\varphi\rangle_{Q^{0}}\Big{)}_{+}{\,\mathrm{d}}z_{-}{\,\mathrm{d}}z_{+}$
$\displaystyle\quad+\fint_{Q^{+}}\fint_{Q^{-}}\Big{(}\langle
f\varphi\rangle_{Q^{0}}-f(z_{-})\Big{)}_{+}{\,\mathrm{d}}z_{-}{\,\mathrm{d}}z_{+}$
$\displaystyle\leq\fint_{Q^{+}}\Bigg{\\{}\underbrace{\fint_{Q^{0}}\big{(}f(z_{+})-f(z_{0})\big{)}\varphi(x_{0}){\,\mathrm{d}}z_{0}}_{=:{\mathcal{I}}^{+}}\Bigg{\\}}_{+}{\,\mathrm{d}}z_{+}$
$\displaystyle\quad+\fint_{Q^{-}}\Bigg{\\{}\underbrace{\fint_{Q^{0}}\big{(}f(z_{0})-f(z_{-})\big{)}\varphi(x_{0}){\,\mathrm{d}}z_{0}}_{=:{\mathcal{I}}^{-}}\Bigg{\\}}_{+}{\,\mathrm{d}}z_{-}.$

where $z_{+}=(x_{+},t_{+}),z_{-}=(x_{-},t_{-})$ and $z_{0}=(x_{0},t_{0})$.
Note that $t_{-}<t_{0}<t_{+}$. We now use the trajectories constructed in
Section 2 to estimate the right hand side.

Using the chain rule, (1.1), (2.5) with the choice (2.13) we get

(3.2) $\displaystyle{\mathcal{I}}^{\pm}$
$\displaystyle=\pm\fint_{Q^{0}}\Big{(}f(z_{\pm})-f(z_{0})\Big{)}\,\varphi(x_{0}){\,\mathrm{d}}z_{0}$
$\displaystyle=\mp\fint_{Q^{0}}\int_{0}^{1}\frac{{\,\mathrm{d}}}{{\,\mathrm{d}}s}f\big{(}\Gamma_{\pm}(s)\big{)}\,\varphi(x_{0}){\,\mathrm{d}}s{\,\mathrm{d}}z_{0}$
$\displaystyle=\mp\delta_{\pm}\fint_{Q^{0}}\int_{0}^{1}\left(\mathcal{T}f\right)\big{(}\Gamma_{\pm}(s)\big{)}\,\varphi(x_{0}){\,\mathrm{d}}s{\,\mathrm{d}}z_{0}$
$\displaystyle\qquad\mp\fint_{Q^{0}}\int_{0}^{1}\left({\frac{\mathrm{d}}{\mathrm{d}s}}X^{(0)}(s)\right)\cdot\left(\nabla_{(0)}f\right)(\Gamma_{\pm}(s))\,\varphi(x_{0}){\,\mathrm{d}}s{\,\mathrm{d}}z_{0}$
$\displaystyle\leq\underbrace{\pm\delta_{\pm}\fint_{Q^{0}}\int_{0}^{1}\left[\left(\mathcal{A}^{*}\mathcal{A}\right)f\right]\big{(}\Gamma_{\pm}(s)\big{)}\,\varphi(x_{0}){\,\mathrm{d}}s{\,\mathrm{d}}z_{0}}_{=:{\mathcal{I}}^{\pm}_{1}}$
$\displaystyle\qquad\underbrace{\mp\fint_{Q^{0}}\int_{0}^{1}\sum_{i=0}^{\kappa}s^{\alpha_{i}}\mathsf{m}_{\pm}^{(i)}\cdot\left(\nabla_{(0)}f\right)(\Gamma_{\pm}(s))\,\varphi(x_{0}){\,\mathrm{d}}s{\,\mathrm{d}}z_{0}}_{=:{\mathcal{I}}^{\pm}_{2}}.$

Note that the only difference between the two terms ${\mathcal{I}}^{+}$ and
${\mathcal{I}}^{-}$ is the role of $z_{0}$: in the former case it is the past
variable, in the latter it is the future variable.

In the following computations to estimate $\mathcal{I}_{1}^{\pm}$ and
$\mathcal{I}_{2}^{\pm}$, the local and the non-local case can be treated
similarly, upon replacing the integration domain over $Q^{0}$ by
${\mathbb{R}}^{d_{0}}\times Q^{0}_{v}$ in the non-local case, where
$Q^{0}_{v}\subset{\mathbb{R}}^{N+1-d_{0}}$ is the domain of the cylinder
$Q^{0}$ for the variables $(x^{(1)},\dots,x^{(\kappa)},t)$, that is we slice
out the $x^{(0)}$ variable.

We now integrate by parts the terms ${\mathcal{I}}^{\pm}_{1}$ after a change
of variables $x_{0}\mapsto y:=\Phi^{s}_{\pm}(x_{0})$ for $s,t_{0}$ fixed
($\Phi^{s}_{\pm}$ is the affine map defined in (2.11), which is invertible for
$s\not=0$):

$\displaystyle{\mathcal{I}}_{1}^{\pm}$
$\displaystyle=\pm\delta_{\pm}\fint_{Q^{0}}\int_{0}^{1}\left[\left(\mathcal{A}^{*}\mathcal{A}\right)f\right]\big{(}\Gamma_{\pm}(s)\big{)}\,\varphi(x_{0}){\,\mathrm{d}}s{\,\mathrm{d}}z_{0}$
$\displaystyle=\pm\delta_{\pm}\fint_{Q^{0}}\int_{0}^{1}\left[\left(\mathcal{A}^{*}\mathcal{A}\right)f\right]\left(\Phi^{s}_{\pm}(x_{0}),st_{0}+(1-s)t_{\pm}\right)\,\varphi(x_{0}){\,\mathrm{d}}s{\,\mathrm{d}}x_{0}{\,\mathrm{d}}t_{0}$
$\displaystyle=\pm\frac{\delta_{\pm}}{|Q^{0}|}\int_{(\Phi^{s}_{\pm}\otimes\text{Id})(Q^{0})}\int_{0}^{1}\left[\left(\mathcal{A}^{*}\mathcal{A}\right)f\right]\left(y,st_{0}+(1-s)t_{\pm}\right)\,\varphi\left(\left(\Phi^{s}_{\pm}\right)^{-1}(y)\right)\frac{{\,\mathrm{d}}s{\,\mathrm{d}}y{\,\mathrm{d}}t_{0}}{\left|\det\mathfrak{A}^{s}\right|}$
$\displaystyle=\pm\frac{\delta_{\pm}}{|Q^{0}|}\int_{(\Phi^{s}_{\pm}\otimes\text{Id})(Q^{0})}\int_{0}^{1}\left[\mathcal{A}f\right]\left(y,st_{0}+(1-s)t_{\pm}\right)\,{\mathcal{A}}\left[\varphi\left(\left(\Phi^{s}_{\pm}\right)^{-1}(y)\right)\right]\frac{{\,\mathrm{d}}s{\,\mathrm{d}}y{\,\mathrm{d}}t_{0}}{\left|\det\mathfrak{A}^{s}\right|}$
$\displaystyle=\pm\frac{\delta_{\pm}}{|Q^{0}|}\int_{Q^{0}}\int_{0}^{1}\left[\mathcal{A}f\right]\left(\Gamma_{\pm}(s)\right)\,\left\\{{\mathcal{A}}\left[\varphi\left(\left(\Phi^{s}_{\pm}\right)^{-1}(y)\right)\right]\right\\}_{|y=\Phi^{s}_{\pm}(x_{0})}{\,\mathrm{d}}s{\,\mathrm{d}}x_{0}{\,\mathrm{d}}t_{0}.$

We then use the first bound in (1.13) on the operator $\mathcal{A}$ in
assumption (H):

$\displaystyle\left|\left\\{{\mathcal{A}}\left[\varphi\left(\left(\Phi^{s}_{\pm}\right)^{-1}(y)\right)\right]\right\\}_{|y=(\Phi^{s}_{\pm})^{-1}(x_{0})}\right|$
$\displaystyle\lesssim_{\varphi}\left\|\nabla_{(0)}\left(\Phi^{s}_{\pm}\right)^{-1}\right\|_{\infty}^{\beta}$
$\displaystyle\lesssim_{\varphi}\left(\sum_{i=0}^{\kappa}s^{-1-\alpha_{i}}\right)^{\beta}\lesssim_{\varphi}\left(\sum_{i=0}^{\kappa}s^{-\beta-\beta\alpha_{i}}\right)$

and deduce finally

${\mathcal{I}}_{1}^{\pm}\lesssim|\delta_{\pm}|\fint_{Q^{0}}\int_{0}^{1}\left|\left[\mathcal{A}f\right]\left(\Gamma_{\pm}(s)\right)\right|\left(\sum_{i=0}^{\kappa}s^{-\beta-\beta\alpha_{i}}\right){\,\mathrm{d}}s{\,\mathrm{d}}x_{0}{\,\mathrm{d}}t_{0}.$

Now let us turn to ${\mathcal{I}}_{2}^{\pm}$ in (3.2). We find, using the same
change of variables,

$\displaystyle{\mathcal{I}}_{2}^{\pm}$
$\displaystyle=\mp\int_{0}^{1}\fint_{Q^{0}}\sum_{i=0}^{\kappa}s^{\alpha_{i}}\mathsf{m}_{\pm}^{(i)}\cdot\nabla_{(0)}f(\Gamma_{\pm}(s))\varphi(x_{0}){\,\mathrm{d}}s{\,\mathrm{d}}x_{0}{\,\mathrm{d}}t_{0}$
$\displaystyle=\mp\frac{1}{|Q^{0}|}\sum_{i=0}^{\kappa}\int_{0}^{1}\int_{(\Phi^{s}_{\pm}\otimes\text{Id})(Q^{0})}s^{\alpha_{i}}\mathsf{m}_{\pm}^{(i)}\cdot\nabla_{(0)}f\left(y,st_{0}+(1-s)t_{\pm}\right)$
$\displaystyle\hskip
227.62204pt\times\varphi\left(\left(\Phi^{s}_{\pm}\right)^{-1}(y)\right)\frac{{\,\mathrm{d}}s{\,\mathrm{d}}y{\,\mathrm{d}}t_{0}}{\left|\det\mathfrak{A}^{s}\right|}$
$\displaystyle=\mp\frac{1}{|Q^{0}|}\sum_{i=0}^{\kappa}\int_{0}^{1}\int_{(\Phi^{s}_{\pm}\otimes\text{Id})(Q^{0})}s^{\alpha_{i}}\mathsf{m}_{\pm}^{(i)}\cdot\nabla_{(0)}(-\Delta_{(0)})^{-\frac{1-\beta}{2}}f\left(y,st_{0}+(1-s)t_{\pm}\right)$
$\displaystyle\hskip
170.71652pt\times(-\Delta_{(0)})^{\frac{1-\beta}{2}}\left\\{\varphi\left[\left(\Phi_{\pm}^{s}\right)^{-1}(y)\right]\right\\}\frac{{\,\mathrm{d}}s{\,\mathrm{d}}y{\,\mathrm{d}}t_{0}}{\left|\det\mathfrak{A}^{s}\right|}$
$\displaystyle=\mp\frac{1}{|Q^{0}|}\sum_{i=0}^{\kappa}\int_{0}^{1}\int_{(\Phi^{s}_{\pm}\otimes\text{Id})(Q^{0})}s^{\alpha_{i}}\mathsf{m}_{\pm}^{(i)}\cdot\left(\nabla_{(0)}(-\Delta_{(0)})^{-\frac{1-\beta}{2}}f\right)\left(\Gamma_{\pm}(s)\right)$
$\displaystyle\hskip
99.58464pt\times\left\\{(-\Delta_{(0)})^{\frac{1-\beta}{2}}\left\\{\varphi\left[\left(\Phi_{\pm}^{s}\right)^{-1}(y)\right]\right\\}\right\\}_{|y=(\Phi^{s}_{\pm})^{-1}(x_{0})}{\,\mathrm{d}}s{\,\mathrm{d}}x_{0}{\,\mathrm{d}}t_{0}.$

We then observe that

$\left|(-\Delta_{(0)})^{\frac{1-\beta}{2}}\left\\{\varphi\left[\left(\Phi_{\pm}^{s}\right)^{-1}(y)\right]\right\\}\right|\lesssim\left\|\nabla_{(0)}\left(\Phi^{s}_{\pm}\right)^{-1}\right\|_{\infty}^{1-\beta}\lesssim\left(\sum_{j=0}^{\kappa}s^{-1-\alpha_{j}}\right)^{1-\beta}$

and we deduce

${\mathcal{I}}_{2}^{\pm}\leq\frac{1}{|Q^{0}|}\int_{0}^{1}\int_{Q^{0}}\left(\sum_{i,j=0}^{\kappa}s^{\beta-1+\alpha_{i}\beta+(\alpha_{j}-\alpha_{i})}\right)\left|\left(\nabla_{(0)}(-\Delta_{(0)})^{-\frac{1-\beta}{2}}f\right)\left(\Gamma_{\pm}(s)\right)\right|{\,\mathrm{d}}s{\,\mathrm{d}}x_{0}{\,\mathrm{d}}t_{0}.$

It follows from (3.1) and (3.2) that we are left with estimating

${\mathcal{J}}:=\fint_{Q^{\pm}}\big{\\{}{\mathcal{I}}^{\pm}_{1}\big{\\}}_{+}{\,\mathrm{d}}z_{\pm}+\fint_{Q^{\pm}}\big{\\{}{\mathcal{I}}^{\pm}_{2}\big{\\}}_{+}{\,\mathrm{d}}z_{\pm}.$

The previous estimates imply

(3.3) $\displaystyle{\mathcal{J}}$
$\displaystyle\lesssim\int_{Q^{\pm}}\int_{0}^{1}\int_{Q^{0}}\left(\sum_{i=0}^{\kappa}\frac{1}{s^{(\alpha_{i}+1)\beta}}\right)\left|\left({\mathcal{A}}f\right)(\Gamma_{\pm}(s))\right|{\,\mathrm{d}}z_{\pm}{\,\mathrm{d}}s{\,\mathrm{d}}z_{0}$
$\displaystyle+\int_{Q^{\pm}}\int_{0}^{1}\int_{Q^{0}}\left(\sum_{i,j=0}^{\kappa}s^{(\beta-1)(1+\alpha_{i})+\alpha_{j}}\right)\left|\left(\nabla_{(0)}(-\Delta_{(0)})^{-\frac{1-\beta}{2}}f\right)\left(\Gamma_{\pm}(s)\right)\right|{\,\mathrm{d}}z_{\pm}{\,\mathrm{d}}s{\,\mathrm{d}}z_{0}.$

In order to balance optimally the singularities we then choose the
$\alpha_{i}$’s all close to $(2\beta)^{-1}-1\in(-1,0)$ but pairwise distinct.
Then (3.3) simplifies into, for some $\varepsilon$ as small as wanted,

${\mathcal{J}}\lesssim\int_{Q^{\pm}}\int_{0}^{1}\int_{Q^{0}}s^{-\frac{1}{2}-\varepsilon}\left(\left|\left({\mathcal{A}}f\right)(\Gamma_{\pm}(s))\right|+\left|\left(\nabla_{(0)}(-\Delta_{(0)})^{-\frac{1-\beta}{2}}f\right)\left(\Gamma_{\pm}(s)\right)\right|\right){\,\mathrm{d}}z_{\pm}{\,\mathrm{d}}s{\,\mathrm{d}}z_{0}.$

We are now in a position to use not only the intermediate variable $z_{0}$,
but also the future/past variables $z_{\pm}$ for a change of variables, since
we are now integrating in both $Q^{0}$ and $Q_{\pm}$. Note that it was not
possible to use the integration in $Q_{\pm}$ before because of the positive
value around the $Q^{0}$ integral. Note also that, by adopting the mollified
formulation (1.18) and by keeping the test functions $\psi_{\pm}$ until the
integral ${\mathcal{J}}$ above, one could use the variable $z_{0}$ to
integrate the test function in the non-singular region $s\in[s_{0},1]$, keep
an $L^{\infty}$ control of the test function in the singular region, and
optimise $s_{0}$ in order to deduce a gain of integrability.

We split ${\mathcal{J}}$ as follows, given $s_{0}\in(0,1)$,

$\displaystyle\int_{Q^{\pm}}\int_{Q^{0}}\int_{0}^{1}s^{-\frac{1}{2}-\varepsilon}\Big{(}\cdots\Big{)}{\,\mathrm{d}}s{\,\mathrm{d}}z_{0}{\,\mathrm{d}}z_{\pm}$
$\displaystyle=\underbrace{\int_{Q^{\pm}}\int_{Q^{0}}\int_{0}^{s_{0}}s^{-\frac{1}{2}-\varepsilon}\Big{(}\cdots\Big{)}{\,\mathrm{d}}s{\,\mathrm{d}}z_{0}{\,\mathrm{d}}z_{\pm}}_{=:{\mathcal{J}}^{\pm}_{1}}+\underbrace{\int_{Q^{\pm}}\int_{Q^{0}}\int_{s_{0}}^{1}s^{-\frac{1}{2}-\varepsilon}\Big{(}\cdots\Big{)}{\,\mathrm{d}}s{\,\mathrm{d}}z_{0}{\,\mathrm{d}}z_{\pm}}_{=:{\mathcal{J}}^{\pm}_{2}}.$

The two changes of variables on each part are represented in Figure 4.

-$\nabla_{(0)}$$\nabla_{(0)}$${\mathcal{T}}$$\dot{\Gamma}^{+}(s)$$\Omega$$Q^{+}$$Q^{0}$$Q^{-}$$\Gamma_{+}$$\Gamma_{-}$$\Psi_{+}$$\Psi_{-}$$\Phi_{+}$$\Phi_{-}$$z_{+}$$z_{0}$$z_{-}$ Figure 4. The change of variables that we use in the proof for some fixed $s\in(0,1)$. For $s\in(0,s_{0})$ we use $\Psi_{\pm}$ that map $z_{\pm}$ onto $\Gamma_{\pm}$, whereas for $s\in(s_{0},1)$ we use $\Phi_{\pm}$ that map $z_{0}$ onto $\Gamma_{\pm}$.

The part ${\mathcal{J}}^{\pm}_{2}$ is controlled by using the same change of
variables $\Phi^{s}_{\pm}$ as before:

$\displaystyle{\mathcal{J}}^{\pm}_{2}$
$\displaystyle=\int_{Q^{0}}\int_{s_{0}}^{1}s^{-\frac{1}{2}-\varepsilon}\left|\left(\nabla_{(0)}(-\Delta_{(0)})^{-\frac{1-\beta}{2}}f\right)\left(y,st_{0}+(1-s)t_{\pm}\right)\right|\frac{{\,\mathrm{d}}s{\,\mathrm{d}}y{\,\mathrm{d}}t_{0}}{\left|\det\mathfrak{A}^{s}\right|}$
$\displaystyle\lesssim_{s_{0}}\int_{\tilde{\Omega}}\left|(-\Delta_{(0)})^{\frac{\beta}{2}}f(z)\right|{\,\mathrm{d}}z\lesssim\int_{\tilde{\Omega}}\left|{\mathcal{A}}f(z)\right|{\,\mathrm{d}}z,$

where we have used first that the integral in $s$ avoids the singularity at
$0$, and second assumption (1.13) in (H). To control the part
${\mathcal{J}}^{\pm}_{1}$ we parametrise $\Gamma_{\pm}(s)$ by the $z_{\pm}$
coordinates, for $s\in[0,s_{0}]$ and $t_{0},t_{\pm}$ all fixed:

$(x_{\pm},t_{\pm})\to\Gamma_{\pm}(s)=(\bar{\Gamma}^{s}_{\pm},t_{0}s+(1-s)t_{\pm})=:(\Psi^{s}_{\pm}(x_{\pm}),t_{0}s+(1-s)t_{\pm})$

Since $s$ does not approach $1$, we can prove that this change of variables in
not singular. The mappings $\Psi_{\pm}^{s}$ are determined by solving (2.11)
for $x_{\pm}$ instead of $x_{0}$. It yields
$\Psi_{\pm}^{s}x_{\pm}=\mathfrak{a}^{s}_{\pm}x_{\pm}+\mathfrak{b}^{s}_{\pm}$
with the matrix $\mathfrak{a}^{s}_{\pm}$ and vector $\mathfrak{b}^{s}_{\pm}$
given by

$\mathfrak{a}^{s}_{\pm}:=\left(\hbox{Id}-{\mathsf{W}}_{\pm}^{\delta}(s)\big{[}{\mathsf{W}}_{\pm}^{\delta}(1)\big{]}^{-1}\right){\mathsf{T}}_{\pm}^{s},\quad\mathfrak{b}^{s}_{\pm}:={\mathsf{W}}_{\pm}^{\delta}(s)\big{[}{\mathsf{W}}_{\pm}^{\delta}(1)\big{]}^{-1}x_{0}$

which depend only on $s$ and $x_{0}$. Since we have proved in Lemma 5 that
${\mathsf{W}}(s)={\mathcal{O}}\left(s^{d_{0}\frac{(\kappa+1)(\kappa+2)}{2}+d_{0}\sum_{i=0}^{\kappa}\alpha_{i}}\right)$
it follows that
${\mathsf{W}}^{\delta}_{\pm}(s)={\mathcal{O}}\left(s^{d_{0}\frac{(\kappa+1)(\kappa+2)}{2}+d_{0}\sum_{i=0}^{\kappa}\alpha_{i}}\right)$
and thus ${\mathsf{W}}^{\delta}_{\pm}(s)$ goes to zero as $s\to 0$. Since
${\mathsf{T}}^{\delta}_{\pm}$ is a lower triangular matrix with unitary
diagonal it is invertible for all $s$, and finally $\mathfrak{a}^{s}_{\pm}$ is
invertible with uniformly bounded inverse on $s\in[0,s_{0}]$ for $s_{0}>0$
small enough. We apply this change of variables to estimate
${\mathcal{J}}_{1}^{\pm}$:

${\mathcal{J}}^{\pm}_{1}=\int_{Q^{0}}\int_{0}^{s_{0}}s^{-\frac{1}{2}-\varepsilon}\left|\left({\mathcal{A}}f\right)\left(y,st_{0}+(1-s)t_{\pm}\right)\right|\frac{{\,\mathrm{d}}s{\,\mathrm{d}}y{\,\mathrm{d}}t_{0}}{\left|\det\mathfrak{a}^{s}\right|}\lesssim_{s_{0}}\int_{\tilde{\Omega}}\left|{\mathcal{A}}f(z)\right|{\,\mathrm{d}}z$

which concludes the proof. Note that in the local case $\tilde{\Omega}$
coincides with the solution domain $\Omega$, whereas in the non-local case,
the domain for the first $d_{0}$ variables of $\tilde{\Omega}$ coincides with
the whole space ${\mathbb{R}}^{d_{0}}$, so that
$\tilde{\Omega}={\mathbb{R}}^{d_{0}}\times B_{R}\times\dots\times
B_{R}\times(-5,0)$.

##  4\. Proof of the Harnack inequality

In this section we briefly explain how to adapt the approach of [8] for
deducing the intermediate value of De Giorgi from the Poincaré inequality we
have proved. This approach was developed in the local case with one commutator
and with a weaker version of the Poincaré inequality. A non-local extension,
with one commutator, was developed in [12]. This approach can be described as
a “trajectory viewpoint” on the De Giorgi theory, and it is fully
quantitative. The first De Giorgi lemma writes in its simplest form

######  Lemma 6 (First Lemma of De Giorgi).

Let the operators ${\mathcal{A}},{\mathcal{B}}$ be such that ${\mathcal{B}}$
is given by (H) and ${\mathcal{A}}$ is more specifically given by

  * (i)

${\mathcal{A}}:=\sqrt{A}\nabla_{(0)}$ with $A\in[\Lambda^{-1},\Lambda]$
measurable symmetric matrix (local case),

  * (ii)

${\mathcal{A}}=\sqrt{a}(-\Delta_{v})^{\beta/2}$ with
$a\in[\Lambda^{-1},\Lambda]$ scalar measurable (non-local case).

Let $f$ be a weak sub-solution to (1.1) on $Q_{1}$ with $0\leq f$ in $Q_{1}$
(local case) or on ${\mathbb{R}}^{d}\times B_{1}^{\otimes\kappa}\times[-1,0]$
with $0\leq f\leq 1$ (non-local case). Then there exist $\varepsilon>0$ and
$r\in(0,1)$ so that $\int_{Q_{1}}f^{2}\leq\varepsilon$ implies $f\leq 1/2$ on
$Q_{r}$.

Note that in the local case, a stronger statement holds where the $L^{\infty}$
norm of $f$ on $Q_{r}$ is controlled by a constant (depending on
${\mathcal{A}}$, ${\mathcal{B}}$ and $r$) times the $L^{2}$ norm of $f$ on
$Q_{1}$. In the non-local case, variants of this stronger statement exist if
$f$ is not only a sub-solution but also a super-solution; we refer to [11].
The proof is based on an energy estimate and the gain of integrability of the
fundamental solution of the Kolmogorov equation, local or non-local. This is
classical and we refer to [8, 12, 2] in the local case and in the non-local
with one commutator. The non-local case with more than one commutator is
easily obtained by scaling arguments or by calculating the fundamental
solutions in Fourier.

The second De Giorgi Lemma, also known as the intermediate value Theorem,
writes

######  Lemma 7 (Second Lemma of De Giorgi).

Let the operators ${\mathcal{A}},{\mathcal{B}}$ be such that ${\mathcal{B}}$
is given by (H) and ${\mathcal{A}}$ is more specifically given by

  * (i)

${\mathcal{A}}:=\sqrt{A}\nabla_{(0)}$ with $A\in[\Lambda^{-1},\Lambda]$
measurable symmetric matrix (local case),

  * (ii)

${\mathcal{A}}=\sqrt{a}(-\Delta_{v})^{\beta/2}$ with
$a\in[\Lambda^{-1},\Lambda]$ scalar measurable (non-local case).

Let $f$ be a weak solution to (1.1) with $0\leq f\leq 1$ on $Q_{1}$ (local
case) or on ${\mathbb{R}}^{d}\times B_{1}^{\otimes\kappa}\times[-3,0]$ (non-
local case). Let $\delta_{1},\delta_{2}\in(0,1)$. Then there are constants
$r_{0}>0$ and $\nu\in(0,1)$ and $\theta\in(0,1)$ such that whenever

$|\\{f\leq 0\\}\cap Q_{r_{0}}^{-}|\geq\delta_{1}|Q_{r_{0}}^{-}|\qquad\textrm{
and }\qquad|\\{f\geq 1-\theta\\}\cap Q_{r_{0}}|\geq\delta_{2}|Q_{r_{0}}|,$

there holds

$\big{|}\\{0<f<1-\theta\\}\cap Q_{1/2}\big{|}\geq\nu|Q_{1/2}|$

in the local case, or

$\big{|}\\{0<f<1-\theta\\}\cap
B_{(\frac{1}{2})^{1+\kappa\beta}}\times\cdots\times
B_{(\frac{1}{2})^{1+\beta}}\times[-3,0]\big{|}\geq\nu|Q_{1/2}|$

in the non-local case.

The proof that this lemma follows from our Poincaré inequality and the first
lemma of De Giorgi by using the argument of [8, Subsection 3.2] in the local
case (in fact the proof is simpler since we do not have the error terms in the
Poincaré inequality that were present in [8]), and it follows from [12,
Subsection 5.2] in the non-local case (note that we have removed the
mollifications $F_{i}$ of the cutoff functions in the statement of [12] since
they can be used in the proof but removed from the statement straightforwardly
by making the border region small enough). These proofs are done in the case
of one commutator but they extend straightforwardly to the case of several
commutators because they do not depend on the transport part
${\mathcal{T}}=\partial_{t}+{\mathcal{B}}$ of the equation, but only on
${\mathcal{A}}$ and the Poincaré inequality and first lemma of De Giorgi.

With the first and second De Giorgi lemma at hand, one can deduce the
“measure-to-pointwise estimate” and the weak Harnack inequality by following
[8] in the local case, and [12] in the non-local case. These proofs are
independent of the equation and only depend on the previously established
functional inequalities. The only minor change is that the kinetic cylinders
have a slightly more complicated scaling on the variables
$(x^{(\kappa)},\dots,x^{(0)},t)$ but the technical changes needed are,
although slightly tedious, clear.

## References

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# Information-theoretic language of proteinoid gels: Boolean gates and QR
codes

Saksham Sharma  Cambridge Centre for Physical Biology, Cambridge;
Unconventional Computing Laboratory, UWE Bristol, UK  Adnan Mahmud  Department
of Chemical Engineering, Cambridge, Philippa Fawcett Drive, Cambridge CB3 0AS,
UK; Zuse Institute Berlin, Takustraße 7 14195, Germany  Giuseppe Tarabella
Institute of Materials for Electronic and Magnetism, National Research Council
(IMEM-CNR), Parma (Italy)  Panagiotis Mougoyannis  Andrew Adamatzky
Unconventional Computing Laboratory, UWE Bristol, UK

###### Abstract

Proteinoids are soft matter fluidic systems formed by heating and cooling of a
mixture of different poly(amino acids) based polymers. Such systems are being
shown previously to exhibit an oscillatory electrical activity inside, which
can be treated as the core ingredient in their information carrying and
transmission capacity. With an aim to build analog computers out of soft
matter fluidic systems in future, this work attempts to invent a new
information-theoretic language, in the form of two-dimensional Quick Response
(QR) codes. This language is, effectively, a digital representation of the
analog signals shown by the proteinoids. We use two different experimental
techniques: (i) a voltage-sensitive dye and (ii) a pair of differential
electrodes, to record the analog signals. The analog signals are digitally
approximatied (synthesised) by sampling the analog signals into a series of
discrete values, which are then converted into binary representations. We have
shown the AND-OR-NOT-XOR-NOR-NAND-XNOR gate representation of the digitally
sampled signal of proteinoids. Additional encoding schemes are applied to
convert the binary code identified above to a two-dimensional QR code. As a
result, the QR code becomes a digital, unique marker of a given proteinoid
network. We show that it is possible to retrieve the analog signal from the QR
code by scanning the QR code using a mobile phone. Our work shows that soft
matter fluidic systems, such as proteinoids, can have a fundamental
informatiom-theoretic language, unique to their internal information
transmission properties (electrical activity in this case) - such a language
can be made universal and accessible to everyone using 2D QR codes, which can
digitally encode their internal properties and give an option to recover the
original signal when required. On a more fundamental note, this study
identifies the techniques of approximating continuum properties of soft matter
fluidic systems using a series representation of gates and QR codes, which are
a piece-wise digital representation, and thus one step closer to programming
the fluids using information-theoretic methods, as suggested almost a decade
ago by Tao’s fluid program.

††preprint: APS/123-QED

Proteinoids carry information in the form of oscillatory electrical (analog)
signals. This oscillatory information can be interpreted as a core ingredient
in architecting algorithms such that the proteinoid ensembles act as an analog
computer in solving varieties of computational tasks. Though not shown in
proteinoids, there are many fluidic soft matter systems that have been shown
to have such algorithmic capabilities. In 1950s, Ishiguro developed a machine
to convert hydrographic and meteorological data into voltage and current
signals in order to electronically model storm surges and tidal phenomenon,
using wave-form generators and oscilloscopes [1].

Water waves in a bucket was shown to solve the XOR problem and perform speech
recognition using a machine called Liquid State Machine [2]. In another
related work, Faraday waves observed on the surface of water were accessed as
information-processing substrates by using them to design timers [3].
Recently, shallow water waves, given by Korteweg-de Vries (KdV) equation, are
shown to have solutions that can encode information and aid in designing
artificial neural networks of the form of echo state networks [4, 5]. More
recently, the nonlinear oscillation of gas bubbles inside water were shown to
be integrable with the linear readout of an ESN algorithm, and thus, a bubble-
based computing system can be used to forecast different chaotic time series
[6]. Another experimental validation using solitary waves travelling on the
surface of a flowing liquid film using reservoir computing systems was
performed here [7].

Figure 1: (a) Voltage versus time signal of the oscillatory electrical
activity inside proteinoids measured using (i) differential electrodes and
(ii) voltage sensitive dyes. (b) Table of the encoding scheme used for Boolean
gates. (c) Table for the encoding scheme used for the proteinoid type.

One crucial step in implementing such soft matter fluidic algorithms is that
the appropriate continuum parameter in the problem concerned, is sampled at
equidistant time instances. The prediction using reservoir computing (RC)
algorithms, thereafter, is only 111This can be seen as a physical version of
the Church-Turing thesis: while the original Church-Turing thesis argues that
an algorithm exists for what is effectively calculable using pen-and-paper
method, in the context of physical problem, it can be interpreted as follows:
the physical systems which are governed by underlying analytical (pen-and-
paper) solutions can perform Turing-type computation of arbitrary precision.
possible if there is an analytical solution to the equation concerned that can
be replaced by the reservoir computing network. As a result, this technique is
still limited to physical problems where the underlying solution is known. The
algorithmic capabilities of nonlinear waves, such as solitons and KdV waves,
in the form of their Turing computability is rigorously established [9].
However, this is not the case for the Navier-Stokes equations which describe
dynamics of arbitrary soft matter fluidic systems, because of the lack of
general analytical solution [10] that can aid in Turing-computability and a
built-in recipe for algorithm-generation capability [11] for the problem
concerned.

By choosing to not much focus on what is fundamentally permissible and what is
not, to meet the industry standards of analog computing, it might be
beneficial to increase the power of soft fluidic computing algorithms. One way
of doing this is dispersing microparticles inside the fluid. Few previously
explored application in this direction are the microparticle-embedded fluids
[12, 13, 14]. When such model systems are actuated with electromagnetic
radiations of suitable wavelength, the frequency of response to the actuation
increases manifolds, making the system amenable to large computing power
requirements.

With the above-said pipeline in mind, we show in the present work, that
arbitrary analog signals exhibited by the electrical property of poly(amino
acid) ensembles can be used to construct digital fingerprint of the system in
the format of QR codes, which can be used to access and retrieve back the
original analog signals upon scanning. To the best of our knowledge, this has
not been attempted before, and we believe that this marks the first step
towards developing microparticle-embedded soft matter analog computers in the
future.

Figure 2: Projecting proteinoid solutions into QR codes (a) Subjecting the
proteinoids in the test tubes to illumination through black, white, and black-
white light to get the oscillatory reponse signal. (b) Converting the
oscillatory analog response signal to a digital signal by extracting the logic
gates from the analog signal. (c) Converting the series of logic gates in
decimal format to a binary format which is then encoded in a QR code. The
solid line represents an analog signal and dotted line represents a digital
signal in processing.

##  I Protocol for measuring electrical activity inside proteinoids

The proteinoids are prepared using the following apparatus [15]: powdered form
of amino acids; Tri-block heater; 3-hole round bottom flasks; nitrogen gas;
magnetic stirrer; cellophane membrane for dialysis; and a water bath. 1.5
grams each of L-Phenylalanine, L-Aspartic acid, L-Histidine, L-Glutamic acid
and L-Lysine is mixed and heated in the 3-hole round bottom flask at 290∘C
initially. The temperature is step-wise increased by 10∘ count, until fumes
start to emerge out. The exhaust in the fumehood (with N2 gas as the inlet) is
started to release the fumes out. To begin with, the powder undergoes a color
transformation from white to green, followed by a morphological transition
resulting in the formation of a sequence of simmering microspheres. After
ceasing the heating process, the remaining material solidifies and is
subsequently extracted and allowed to cool for a duration of thirty minutes.
The collected residue is then placed within the Slide-A-Lyzer mini dialysis
apparatus, with 10,000 molecular weight cut-off and employs water as the
dialysate. Dialysis is carried out continuously over a period of five days
until the residue comprises minute ensembles of microspheres. The residue
derived from the dialysis membrane is subjected to half an hour of heating in
a vacuum oven, facilitating the evaporation of the dialysate (water) from the
sample. Subsequently, the sample is scrutinized utilizing a transmission
electron microscope.

Subdermal iridium-coated stainless steel electrodes and potential-sensitive
ANEP voltage-sensitive dyes were used to record the electrical activity.
During the recording process, the data logger (ADC-24, Pico Technology, UK)
operated at its maximum capacity (600 data points per second). This rate of
data collection allowed for a comprehensive understanding of the electrical
activity exhibited by the proteinoids. Furthermore, the data logger stored and
saved the average value obtained from these measurements. This approach
provided a concise representation of the recorded electrical activity,
facilitating subsequent analysis and interpretation of the experimental
results.

##  II Boolean gate representation of the electrical activity data

There are five different kinds of proteinoids which are being assessed in the
data analysis shown in the present work. These proteinoids are, namely:
L-Glu:L-Phe:L-His; L-Glu:L-Phe; L-Phe:L-Lys; and L-Asp. The electricaly
activity recorded using the iridium-coated stainless steel electrodes and
voltage-sensitive dyes identifies that the periods median of these proteinoids
are distributed, fairly uniformly, at around 3500 seconds, except L-Phe which
is at 2200 seconds. Subsequently, fast fourier transform (FFT) is performed to
decomposed the time-varying signal into its frequency counterpart; further
details of the analysis are available here [16]. A typical signal recorded is
shown in Fig. 1(a). The next step is to select a window of a period of 3000
seconds and identify underlying spikes in that window and rewrite it in the
form of Boolean gates. The procedure on how to implement this has been
previously discussed here [17]. Once this procedure is successfully
implemented, the spikes in the window are usually classified into one of the
several Boolean gates enlisted in Fig. 3(b). These Boolean gates are denoted
by $g_{i}\,(1\leq i\leq 8)$, such that, $g_{i}\in\\{2,4,6\\}$. For example,
$g_{1}\equiv\textrm{OR gate}\equiv 44$, $g_{2}\equiv\textrm{NOT gate}\equiv
22$, $g_{3}\equiv\textrm{XOR gate}\equiv 24$ and so on. It is to be carefully
noted that if none of the Boolean gates in a given window, then the symbolic
notation for “No gate” is 26. Similar to how the gates are named using the
strings formed from symbols $\\{2,4,6\\}$, it is possible to encode the type
of proteinoid using a different set of symbols. Let us denote proteinoids as
$p_{i}\,(1\leq i\leq 5)$, such that, $p_{i}\in\\{3,5,7\\}$. To give an
example, $p_{1}$ denotes the proteinoid with composition L-Glu:L-Phe:L-His and
is denoted by “33”, $p_{4}$ denotes proteinoid with composition L-Phe and the
symbol “55”.

Figure 3: QR codes generated for the proteinoids by characterising them based
on the set of input lights: 01 (black and black-white); 10 (white and black
light); 11 (black and white light).

##  III Boolean gate to QR code conversion of electrical activity

In the previous section, we discussed the schemes adopted to encode different
types of proteinoids and the corresponding Boolean gates. Using these schemes,
in this section, we will characterise proteinoids and the signals hidden, and
divide them in three categories:

Figure 4: QR codes generated for the proteinoids by characterising them based
on the set of proteinoid types: 33 (L-Glu:L-Phe:L-His), 35 (L-Glu:L-Phe), 37
(L-Phe:Lys), 55 (L-Phe), and 57 (L-Asp). Figure 5: QR codes generated for the
proteinoids by characterising them based on the type of Boolean gates: 42
(AND), 44 (OR), 22 (NOT), 24 (XOR), 64 (NAND), 66 (XNOR), 62 (NOR), and 26 (no
gate)

  1. 1.

Characterisation based on the input lights: There are three different set of
input lights that illuminate proteinoids so as to produce time-varying voltage
signals. If the proteinoids are illuminated with black and black-white light
periodically, it is encoded as “01”. If the proteinoids, on the other hand,
are illuminated with white and black light periodically, then it is encoded as
“10”. Lastly, when the periodic illumination is done using black and white
light, then it is be encoded as “11”. Based on these primary encoding schemes,
the window carrying time-varying signal is converted to a series of Boolean
gates for each of the proteinoid types. To given an example, the QR code in
Fig. 3(a) when converted in the decimal format reads as:
$``334422642626242635442264262624263744262666\\\
42262655442664262624267744266426262426^{\prime\prime}$. While this may seem to
be a large number difficult to decrypt, a close look reveals that the string
begins with $p_{i}\in\\{3,5,7\\}$ followed by a series of eight Boolean gates
$g_{i}\in\\{2,4,6\\}$, and again followed by a new proteinoid $p_{i+1}$
accompanied by the next seven Boolean gates next to it. Through the same
procedure, the proteinoid types and corresponding Boolean gates are encoded in
QR codes for the periodic light of the type $\\{10\\}$ and $\\{11\\}$.

  2. 2.

Characterisation based on proteinoid types: It is also possible to
characterise the signals in terms of the type of proteinoids. Given that there
are five different kinds of proteinoids, $p_{i}\in\\{3,5,7\\}$, it means that
there are five different kinds of QR codes possible to be encrypted, as shown
in Fig. 4. To given an example, the decimal format of the binary code encoded
in Fig. 4(a) is: $10442264262624261144226426262426014422642626\\\ 2426$. It is
apparent that the binary code starts with “10” which is the description of
input light condition (white and light, periodically) followed by logic gates
$g_{i}$. This is then followed by “11” which corresponds to periodic black and
white illumination and again followed by series of logic gates, and so on.
Using this procedure, the Boolean gates corresponding to the pattern of light
illuminated is encoded for each proteinoid $p_{i}\in\\{3,5,7\\}$ in the QR
code that is shown in Fig. 4.

  3. 3.

Characterisation based on type of Boolean gates: Lastly, it is possible to
categorise the proteinoids based on the type of Boolean gates. Since there are
eight type of logic gates, there are eight different type of QR codes shown in
Fig. 5. To take an example, consider Fig. 5(a), where the binary code encoded,
when converted into decimal format, yields: $10370137$. This means that when
concerning AND gate, the proteinoid of the type $p_{3}$, i.e., L-Phe:Lys,
exhibits such a gate when the lights are illuminated corresponding to “10”
(white and black) and “01” (black and black-white) encoding.

##  IV Projecting proteinoid solutions onto QR codes

The process of projecting proteinoid solutions (a continuum matter) onto QR
codes, is effectively, a pipeline for analog-to-digital (ADC) conversion. As
shown in the schematic (Fig. 2), this involves taking proteinoids $p_{i}(1\leq
i\leq 5)$ and illumating them with black, white, or white-black lights to
yield an analog response which is discretised further to yield logic gates. By
following the notation introduced in Fig. 1, the sequence of logic gates, thus
obtained, are converted to QR codes using a suitable code available at
request. A web-interface to convert the string to a QR code (or vice-versa) is
available in our website.

##  V Pipeline for digital encoding of soft matter fluidic systems

Given that the oscillatory electrical activity inside the proteinoids is
converted successfully into a piecewise discretised information (in the format
of QR codes), it is an obvious next question to ask: what is the purpose of
such a task? As originally alluded in the starting of this article, we aim to
discard the continuum description of properties of a gel (proteinoids in this
case) and sample their discrete values in order to provide an information-
theoretic description. This constitutes the first step, in a supposed pipeline
presented in Fig. LABEL:fig:fig5, involving the measurement of continuum
properties of soft matter fluidic systems and representing them in an analog
fashion (such as, while recording in an oscilloscope). Once the measurement is
performed, the next step is to select the window and sample data from that
window (after performing fourier transform and related data conversion
techniques), in order to convert the analog signal to a piecewise data which
can be further mapped from a decimal format to a binary format. This step is
called discretisation. In the present case, QR code formulation of discretised
continuum data is an exemplary case to showcase this step. The obvious next
step is to scan the QR code, which is indeed another form of measurement, or
more precisely, an ante-measurement, where the idea is to obtain the binary
format of the continuum data back from the QR code description of continuum
properties of soft matter fluidic systems. It is to be noted that in most of
the cases where the data is not extraordinarily large, it is possible to
retrieve the original discretised data to 100% efficiency. After the data is
retrieved, it is possible through appropriate gluing techniques to construct
the original continuum level description of the soft matter system from the
binary data. This retrieval step is the most important and technique-dependent
step that should be undertaken carefully. More on this will be discussed
later.

##  VI Conclusion

The present article presents a novel information-theoretic language of
proteinoid gels, which are shown to have electrical oscillatory activity,
amenable to a binary description using the language of Boolean gates and two-
dimensional QR codes. We believe that the present work will guide the
researchers in the field of proto-cognition, polymers, and microfluidics to
consider QR codes and Boolean gates as a potential universal language of
quantitative description of the behaviour of soft matter fluidic systems.

##  VII Future directions and perspective

One of the emerging perspective from this article is developing a pipeline
(using idea of QR codes) to abridge the gap between continuum and discrete
descriptions of soft matter fluidic systems. This is particularly useful when
spiking activity of proteinoids is to be made universal and a unique marker of
the proteinoid gel. The next step after such a representation is to regard the
spiking activity using the typology of Izhikevich neurons. Such a
characterisation will help us to throw some light on the “neuroscience of
proteinoids”, where the learning-cognitive behaviour of proteinoids is that of
primal level, as often alluded with the term “proto-cognition”.

## Code

The GUI used to encode and decode the QR codes for proteinoids is available
here and is written solely by A.M.

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