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• W1981419709 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X equals left-parenthesis upper X Subscript t Baseline comma upper P Superscript x Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>P</mml:mi> <mml:mi>x</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>X = ({X_t},,{P^x})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a right Markov process and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an excessive measure for <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>. Associated with the pair <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper X comma m right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(X,,m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a stationary strong Markov process <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y Subscript t Baseline comma upper Q Subscript m Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>({Y_t},,{Q_m})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with random times of birth and death, with the same transition function as <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>, and with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as one dimensional distribution. We use <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y Subscript t Baseline comma upper Q Subscript m Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>({Y_t},,{Q_m})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to study the cone of excessive measures for <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>. A weak order is defined on this cone: an excessive measure <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi> <mml:semantics> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:annotation encoding=application/x-tex>xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is weakly dominated by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=m> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding=application/x-tex>m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if there is a suitable homogeneous random measure <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=kappa> <mml:semantics> <mml:mi>κ<!-- κ --></mml:mi> <mml:annotation encoding=application/x-tex>kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y Subscript t Baseline comma upper Q Subscript xi Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>ξ<!-- ξ --></mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>({Y_t},,{Q_xi })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained by birthing <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y Subscript t Baseline comma upper Q Subscript m Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>({Y_t},,{Q_m})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, birth in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-bracket t comma t plus d t right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[t,,t + dt]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> occurring at rate <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=kappa left-parenthesis d t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>κ<!-- κ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>kappa (dt)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Random measures such as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=kappa> <mml:semantics> <mml:mi>κ<!-- κ --></mml:mi> <mml:annotation encoding=application/x-tex>kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are studied through the use of Palm measures. We also develop aspects of the general theory of processes over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y Subscript t Baseline comma upper Q Subscript m Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>({Y_t},,{Q_m})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, including the moderate Markov property of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper Y Subscript t Baseline comma upper Q Subscript m Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>({Y_t},,{Q_m})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when the arrow of time is reversed. Applications to balayage and capacity are suggested." @default.
• W1981419709 created "2016-06-24" @default.
• W1981419709 creator A5055561415 @default.
• W1981419709 date "1987-01-01" @default.
• W1981419709 modified "2023-10-18" @default.
• W1981419709 title "Homogeneous random measures and a weak order for the excessive measures of a Markov process" @default.
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• W1981419709 doi "https://doi.org/10.1090/s0002-9947-1987-0902778-5" @default.
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