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

#### Leonid Bogachev (University of Leeds)

$$\newcommand{\PP}{\mathop{{\mathbb{P}}{}}\nolimits} \newcommand{\EE}{\mathop{{\mathbb{E}}{}}\nolimits} \newcommand{\const}{\mathrm{const}} \newcommand{\myp}{\mbox{\:\!}} \newcommand{\mypp}{\mbox{\;\!}} \newcommand{\myn}{\mbox{\;\!\!}} \newcommand{\mynn}{\mbox{\:\!\!}} \newcommand{\mynnn}{\mbox{\!}} \newcommand{\rd}{{\rm d}} \newcommand{\re}{{\rm e}} \newcommand{\ri}{{\rm i}} \newcommand{\RR}{{\mathbb R}} \newcommand{\CC}{{\mathbb C}} \newcommand{\NN}{{\mathbb N}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\QQ}{{\mathbb Q}}$$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).