Nearly Hyperharmonic Functions are Infima of Excessive Functions

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10439109" target="_blank" >RIV/00216208:11320/20:10439109 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=OiaBIdHCmq" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=OiaBIdHCmq</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s10959-019-00927-8" target="_blank" >10.1007/s10959-019-00927-8</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Nearly Hyperharmonic Functions are Infima of Excessive Functions

Popis výsledku v původním jazyce

Let X be a Hunt process on a locally compact space X such that the set epsilon(X) of its Borel measurable excessive functions separates points, every function in epsilon(X) is the supremum of its continuous minorants in epsilon(X), and there are strictly positive continuous functions v, w is an element of epsilon(X) such that v/w vanishes at infinity. A numerical function u = 0 on X is said to be nearly hyperharmonic, if integral* u omicron X-tau V dP(x) &lt;= u(x) for every x is an element of X and every relatively compact open neighborhood V of x, where tau(V) denotes the exit time of V. For every such function u, its lower semicontinuous regularization (u) over cap is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that u = inf{w is an element of epsilon(X) : w &gt;= u} for every Borel measurable nearly hyperharmonic function on X. Principal novelties of our approach are the following: 1. A quick reduction to the special case, where starting at x is an element of X with u(x) &lt; infinity the expected number of times the process X visits the set of points y. X, where &lt;(u)over cap&gt;(y) := lim inf(z) -&gt; y u(z) &lt; u(y), is finite. 2. The consequent use of (only) the strong Markov property. 3. The proof of the equality integral u d mu = inf{integral w d mu: w is an element of epsilon(X), w &gt;= u} not only for measures mu satisfying integral w d mu &lt; infinity for some excessive majorant w of u, but also for all finite measures. At the end, the measurability assumption on u is weakened considerably.

Název v anglickém jazyce

Nearly Hyperharmonic Functions are Infima of Excessive Functions

Popis výsledku anglicky

Let X be a Hunt process on a locally compact space X such that the set epsilon(X) of its Borel measurable excessive functions separates points, every function in epsilon(X) is the supremum of its continuous minorants in epsilon(X), and there are strictly positive continuous functions v, w is an element of epsilon(X) such that v/w vanishes at infinity. A numerical function u = 0 on X is said to be nearly hyperharmonic, if integral* u omicron X-tau V dP(x) &lt;= u(x) for every x is an element of X and every relatively compact open neighborhood V of x, where tau(V) denotes the exit time of V. For every such function u, its lower semicontinuous regularization (u) over cap is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that u = inf{w is an element of epsilon(X) : w &gt;= u} for every Borel measurable nearly hyperharmonic function on X. Principal novelties of our approach are the following: 1. A quick reduction to the special case, where starting at x is an element of X with u(x) &lt; infinity the expected number of times the process X visits the set of points y. X, where &lt;(u)over cap&gt;(y) := lim inf(z) -&gt; y u(z) &lt; u(y), is finite. 2. The consequent use of (only) the strong Markov property. 3. The proof of the equality integral u d mu = inf{integral w d mu: w is an element of epsilon(X), w &gt;= u} not only for measures mu satisfying integral w d mu &lt; infinity for some excessive majorant w of u, but also for all finite measures. At the end, the measurability assumption on u is weakened considerably.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Theoretical Probability

ISSN

0894-9840

e-ISSN

—

Svazek periodika

33

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

BE - Belgické království

Počet stran výsledku

17

Strana od-do

1613-1629

Kód UT WoS článku

000550905100014

EID výsledku v databázi Scopus

2-s2.0-85068096128