2012/01/18 by Jürgen Kampf, Günter Last, Ilya Molchanov · 4 citations
Mathematics · Biochemistry, Genetics and Molecular Biology · #Point processes and geometric inequalities #Markov Chains and Monte Carlo Methods #Diffusion and Search Dynamics
paper · pdf · doi:10.1090/s0002-9939-2012-11128-1
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha element-of left-parenthesis 1 comma 2 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <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">α ∈ (1,2]</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="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an <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 Rd</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-valued symmetric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">α</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable Lévy process starting at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We consider the closure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript t"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">St</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the path described by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the interval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma t right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,t]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and its convex hull <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript t"> <mml:semantics> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Zt</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The first result of this paper provides a formula for certain mean mixed volumes of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript t"> <mml:semantics> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Zt</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and in particular for the expected first intrinsic volume of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript t"> <mml:semantics> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Zt</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The second result deals with the asymptotics of the expected volume of the stable sausage <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript t Baseline plus upper B"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Zt+B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an arbitrary convex body with interior points) as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t→ 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For this we assume that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has independent components.