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).

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