Some theorems on Feller processes: Transience, local times and ultracontractivity

Abstract

We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for Lévy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a by-product, we obtain for stable-like processes (in the sense of R. Bass) on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with smooth variable index <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha left-parenthesis x right-parenthesis element-of left-parenthesis 0 comma 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha (x)\in (0,2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a transience criterion in terms of the exponent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d equals 1"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf Underscript x element-of double-struck upper R Endscripts alpha left-parenthesis x right-parenthesis element-of left-parenthesis 1 comma 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="true" form="prefix">inf</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:munder> <mml:mi>α</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\inf _{x\in \mathbb {R}} \alpha (x)\in (1,2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the stable-like process has local times.