Stochastic coalescence multi-fragmentation processes

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• 作者：Eduardo Cepeda ^{eduardo.cepeda@m4x.org" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60K35 ; 60J25
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：126
• 期：2
• 页码：360-391
• 全文大小：372 K

文摘

We study infinite systems of particles which undergo coalescence and fragmentation, in a manner determined solely by their masses. A pair of particles having masses mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si1.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=8e259d90973132ce5b90f18fe3a17044" title="Click to view the MathML source">xmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mi>xmi>math> and mmlsi2" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si2.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=32e99f8038de0b51276c4d20adbe268f" title="Click to view the MathML source">ymathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><mi>ymi>math> coalesces at a given rate mmlsi3" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si3.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=165b991dd5a69e3960d575d2282b918d" title="Click to view the MathML source">K(x,y)mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><mi>Kmi><mrow><mo>(mo><mi>xmi><mo>,mo><mi>ymi><mo>)mo>mrow>math>. A particle of mass mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si1.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=8e259d90973132ce5b90f18fe3a17044" title="Click to view the MathML source">xmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mi>xmi>math> fragments into a collection of particles of masses mmlsi5" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si5.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=c85844105a9b2acb6eada80e731aaa56" title="Click to view the MathML source">θ₁x,θ₂x,…mathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><msub><mrow><mi>θmi>mrow><mrow><mn>1mn>mrow>msub><mi>xmi><mo>,mo><msub><mrow><mi>θmi>mrow><mrow><mn>2mn>mrow>msub><mi>xmi><mo>,mo><mo>…mo>math> at rate mmlsi6" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si6.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=df6a738977a68e7fcd15ebb5b2a6511c" title="Click to view the MathML source">F(x)β(dθ)mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><mi>Fmi><mrow><mo>(mo><mi>xmi><mo>)mo>mrow><mi>βmi><mrow><mo>(mo><mi>dmi><mi>θmi><mo>)mo>mrow>math>. We assume that the kernels mmlsi7" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si7.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=3ee26289a2ba3838bb6dce4ae3262ed2" title="Click to view the MathML source">KmathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mi>Kmi>math> and mmlsi8" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si8.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=6e87fed9569be98cb3cd936c462c001e" title="Click to view the MathML source">FmathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll"><mi>Fmi>math> satisfy H&ouml;lder regularity conditions with indices mmlsi9" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si9.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=99b082c7ddf8248277684c5d83063040" title="Click to view the MathML source">λ∈(0,1]mathContainer hidden">mathCode"><math altimg="si9.gif" overflow="scroll"><mi>λmi><mo>∈mo><mrow><mo>(mo><mn>0mn><mo>,mo><mn>1mn><mo>]mo>mrow>math> and mmlsi10" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si10.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=76b5c08da41bd76ec8ced42c27592729" title="Click to view the MathML source">α∈[0,∞)mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll"><mi>αmi><mo>∈mo><mrow><mo>[mo><mn>0mn><mo>,mo><mi>∞mi><mo>)mo>mrow>math> respectively. We show existence of such infinite particle systems as strong Markov processes taking values in mmlsi11" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si11.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=da93cdd5583217de0c9815ce2bfbb224" title="Click to view the MathML source">ℓ_λmathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll"><msub><mrow><mi>ℓmi>mrow><mrow><mi>λmi>mrow>msub>math>, the set of ordered sequences mmlsi12" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si12.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=26a6f701a5ba0dcf9fae0131ad6277e4" title="Click to view the MathML source">(m_i)_i≥1mathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll"><msub><mrow><mrow><mo>(mo><msub><mrow><mi>mmi>mrow><mrow><mi>imi>mrow>msub><mo>)mo>mrow>mrow><mrow><mi>imi><mo>≥mo><mn>1mn>mrow>msub>math> such that mmlsi13" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915002227&_mathId=si13.gif&_user=111111111&_pii=S0304414915002227&_rdoc=1&_issn=03044149&md5=ab8f5e13e03327a6d0a7c8d100536293">mg class="imgLazyJSB inlineImage" height="18" width="83" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0304414915002227-si13.gif">mathContainer hidden">mathCode"><math altimg="si13.gif" overflow="scroll"><msub><mrow><mo>&sum;mo>mrow><mrow><mi>imi><mo>≥mo><mn>1mn>mrow>msub><msubsup><mrow><mi>mmi>mrow><mrow><mi>imi>mrow><mrow><mi>λmi>mrow>msubsup><mo><mo><mi>∞mi>math>. We show that these processes possess the Feller property. This work relies on the use of a Wasserstein-type distance, which has proved to be particularly well-adapted to coalescence phenomena.