## Liouville-type theorems for the archetypal equation with rescaling

#### Leonid Bogachev (University of Leeds)

Frank Adams 2,

\( \newcommand{\PP}{\mathop{{\mathbb{P}}{}}\nolimits} \newcommand{\EE}{\mathop{{\mathbb{E}}{}}\nolimits} \newcommand{\const}{\mathrm{const}} \newcommand{\myp}{\mbox{$\:\!$}} \newcommand{\mypp}{\mbox{$\;\!$}} \newcommand{\myn}{\mbox{$\;\!\!$}} \newcommand{\mynn}{\mbox{$\:\!\!$}} \newcommand{\mynnn}{\mbox{$\!$}} \newcommand{\rd}{{\rm d}} \newcommand{\re}{{\rm e}} \newcommand{\ri}{{\rm i}} \newcommand{\RR}{{\mathbb R}} \newcommand{\CC}{{\mathbb C}} \newcommand{\NN}{{\mathbb N}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\QQ}{{\mathbb Q}} \)In this talk, we consider a linear functional-integral equation

\[

y(x)=\iint_{\mathbb{R}^2} y(a(x-b)) \,\mu(\rd a,\rd b),\qquad x\in

\mathbb{R},

\]

where \(\mu\) is a probability measure on \(\mathbb{R}^2\);

equivalently, \(y(x)=\EE\{y(\alpha(x-\beta))\}\), with random

\((\alpha,\beta)\) and \(\EE\) denoting expectation. This is a rich

source of various functional and functional-differential equations

with rescaling (hence the name `archetypal'), including the

integrated Cauchy equation \(y(x)=\EE\{y(x-\beta)\}\) (i.e.,

\(\alpha\equiv 1\)) and the functional-differential (`pantograph')

equation \(y'(x)+y(x) =\EE\{y(\alpha(x-\gamma))\}\).

Interpreting solutions \(y(x)\) as harmonic functions of the

associated Markov chain \((X_n)\), we discuss Liouville-type theorems

asserting that any bounded continuous solution is constant. The

results crucially depend on the criticality parameter

\(K:=\EE\{\ln\myn|\alpha|\}\); e.g., if \(K<0\) then a Liouville theorem

is always true, but the case \(K\ge0\) is more interesting (and

difficult). The proofs utilize the iterated equation

\(y(x)=\EE\{y(X_\tau)\myp|\myp X_0=x\}\) (with a suitable stopping

time \(\tau\)) due to Doob's optional stopping theorem applied to the

martingale \(y(X_n)\).

This is joint work with Gregory Derfel (Beer Sheva) and Stanislav

Molchanov (UNC-Charlotte).