Stochastic flows and an interface SDE on metric graphs

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作者：Hatem Hajri^a ; ^{Hatem.Hajri.fn@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Olivier Raimond^b ; ^{oraimond@u-paris10.fr" class="auth_mail" title="E-mail the corresponding author}
关键词：Stochastic flows ; Reflected Brownian motion in a wedge ; Walsh&rsquo ; s Brownian motion
刊名：Stochastic Processes and their Applications
出版年：2016
出版时间：January 2016
年：2016
卷：126
期：1
页码：33-65
全文大小：371 K

文摘

This paper consists in the study of a stochastic differential equation on a metric graph, called an interface SDE class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si1.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=edfd0bac3318f937fe568c31c4863933" title="Click to view the MathML source">(ISDE)class="mathContainer hidden">class="mathCode">

croll"> (ISDE)

. To each edge of the graph is associated an independent white noise, which drives class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si1.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=edfd0bac3318f937fe568c31c4863933" title="Click to view the MathML source">(ISDE)class="mathContainer hidden">class="mathCode">

croll"> (ISDE)

on this edge. This produces an interface at each vertex of the graph. This study is first done on star graphs with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si3.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=236e5b04af09d519eedaaccabb7543e9" title="Click to view the MathML source">N≥2class="mathContainer hidden">class="mathCode">

croll"> N \geq 2

rays. The case class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si4.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=631c800483b417284c4c1dc03a4bbc10" title="Click to view the MathML source">N=2class="mathContainer hidden">class="mathCode">

croll"> N = 2

corresponds to the perturbed Tanaka’s equation recently studied by Prokaj (2013) and Le Jan and Raimond (2014) among others. It is proved that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si1.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=edfd0bac3318f937fe568c31c4863933" title="Click to view the MathML source">(ISDE)class="mathContainer hidden">class="mathCode">

croll"> (ISDE)

has a unique in law solution, which is a Walsh’s Brownian motion. This solution is strong if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si4.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=631c800483b417284c4c1dc03a4bbc10" title="Click to view the MathML source">N=2class="mathContainer hidden">class="mathCode">

croll"> N = 2

Solution flows are also considered. There is a (unique in law) coalescing stochastic flow of mappings class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si7.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=eaa7418d9b2d0c2e5dafc6b16d88d4d4" title="Click to view the MathML source">φclass="mathContainer hidden">class="mathCode"> $croll"> φ$ solving class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si1.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=edfd0bac3318f937fe568c31c4863933" title="Click to view the MathML source">(ISDE)class="mathContainer hidden">class="mathCode"> $croll"> (ISDE)$ . For class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si4.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=631c800483b417284c4c1dc03a4bbc10" title="Click to view the MathML source">N=2class="mathContainer hidden">class="mathCode"> $croll"> N = 2$ , it is the only solution flow. For class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si10.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=a457f0ff72fa0a948076a6e94bf9c219" title="Click to view the MathML source">N≥3class="mathContainer hidden">class="mathCode"> $croll"> N \geq 3$ , class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si7.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=eaa7418d9b2d0c2e5dafc6b16d88d4d4" title="Click to view the MathML source">φclass="mathContainer hidden">class="mathCode"> $croll"> φ$ is not a strong solution and by filtering class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si7.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=eaa7418d9b2d0c2e5dafc6b16d88d4d4" title="Click to view the MathML source">φclass="mathContainer hidden">class="mathCode"> $croll"> φ$ with respect to the family of white noises, we obtain a (Wiener) stochastic flow of kernels solution of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si1.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=edfd0bac3318f937fe568c31c4863933" title="Click to view the MathML source">(ISDE)class="mathContainer hidden">class="mathCode"> $croll"> (ISDE)$ . There are no other Wiener solutions. By applying our previous work Hajri and Raimond (2014), these results are extended to more general metric graphs.

The proofs involve the study of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si14.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=a847a6f9f54be40e78994c841bc0b809" title="Click to view the MathML source">(X,Y)class="mathContainer hidden">class="mathCode"> $croll"> (X, Y)$ a Brownian motion in a two dimensional quadrant obliquely reflected at the boundary, with time dependent angle of reflection. We prove in particular that, when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si15.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=202adf1f185060d09f36948460f50999" title="Click to view the MathML source">(X₀,Y₀)=(1,0)class="mathContainer hidden">class="mathCode"> $croll"> (X_{0}, Y_{0}) = (1, 0)$ and if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si16.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=be95793c0449fc478906c285c0eba98f" title="Click to view the MathML source">Sclass="mathContainer hidden">class="mathCode"> $croll"> S$ is the first time class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si17.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=d2420a5404623ec94a0972bff0157cbf" title="Click to view the MathML source">Xclass="mathContainer hidden">class="mathCode"> $croll"> X$ hits class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si18.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=7bc6ac2cdb7e0763ef7c9cbe2ca43ba2" title="Click to view the MathML source">0class="mathContainer hidden">class="mathCode"> $croll"> 0$ , then class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si19.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=21075f42625b50970b07cb0439e1323a">class="imgLazyJSB inlineImage" height="17" width="16" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0304414915001921-si19.gif">cript>cal-align:bottom" width="16" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0304414915001921-si19.gif">cript>class="mathContainer hidden">class="mathCode"> $croll"> Y_{S}^{2}$ is a beta random variable of the second kind. We also calculate class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si20.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=9658ef67036fed886a4cb1a337be9b8f" title="Click to view the MathML source">E[L_σ₀]class="mathContainer hidden">class="mathCode"> $croll"> ck">E [L_{σ_{0}}]$ , where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si21.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=2599a037e06941c91d2a9b3dc5ee6243" title="Click to view the MathML source">Lclass="mathContainer hidden">class="mathCode"> $croll"> L$ is the local time accumulated at the boundary, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si22.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=1073f91733fe1447df82b2344e29da97" title="Click to view the MathML source">σ₀class="mathContainer hidden">class="mathCode"> $croll"> σ_{0}$ is the first time class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si14.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=a847a6f9f54be40e78994c841bc0b809" title="Click to view the MathML source">(X,Y)class="mathContainer hidden">class="mathCode"> $croll"> (X, Y)$ hits class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304414915001921&_mathId=si24.gif&_user=111111111&_pii=S0304414915001921&_rdoc=1&_issn=03044149&md5=ac20fed7f3b66e48e242051f1cd670fe" title="Click to view the MathML source">(0,0)class="mathContainer hidden">class="mathCode"> $croll"> (0, 0)$ .

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