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OF GENERALIZED STOCHASTIC PROCESSES广义随机过程

OF GENERALIZED STOCHASTIC PROCESSES广义随机过程

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时间：2019-08-08

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1、LINEARANDNON-LINEARTHEORYOFGENERALIZEDFUNCTIONSANDITSAPPLICATIONSBANACHCENTERPUBLICATIONS,VOLUME88INSTITUTEOFMATHEMATICSPOLISHACADEMYOFSCIENCESWARSZAWA2010EQUATIONSINDIFFERENTIALSINTHEALGEBRAOFGENERALIZEDSTOCHASTICPROCESSESNADZEYAV.BEDZIUKBelarusianStateUniversityNezavisimosti4,220030Minsk,Belar

2、usE-mail:nbedyuk@gmail.comALEHL.YABLONSKIBelarusianStateUniversityNezavisimosti4,220030Minsk,BelarusE-mail:yablonski@bsu.by,yablonski_oleg@rambler.ruAbstract.Weconsideranordinaryorstochasticnonlinearequationwithgeneralizedcoeffi-cientsasanequationindifferentialsinthealgebraofnewgeneralizedfunctions

3、inthesenseof[8].Consequently,thesolutionofsuchanequationisanewgeneralizedfunction.Weformu-lateconditionsunderwhichthesolutionofagivenequationinthealgebraofnewgeneralizedfunctionsisassociatedwithanordinaryfunctionorprocess.Moreovertheclassofallpossibleassociatedfunctionsandprocessesisdescribed.1.

4、Introduction.Thetheoryofgeneralizedfunctionsisoneofthemostpowerfultoolsforinvestigatinglineardifferentialequations.However,thedistributiontheoryfromtheverybeginninghasanessentialdisadvantage:itisinapplicabletosolutionsofnonlinearproblems.Therefore,variousinterpretationsofsolutionsofnonlineardiffer

5、entialequa-tionswereproposedbysomemathematicians.Unfortunately,differentinterpretationsofthesameequationlead,ingeneral,todifferentsolutions;seee.g.[2,4,6,11,14,18].Usually,differentialequationsareusedtodescribethedynamicsofrealsystemsorphenomena.Inordertochooseanadequateinterpretationofsuchequa-tio

6、nsonehastoconsiderreasonsthatareusedformodellingthedynamicsoftherealsystems.2000MathematicsSubjectClassification:Primary34A37,60H20;Secondary46F30,60G20.Keywordsandphrases:algebraofgeneralizedfunctionsandstochasticprocesses,associatedsolutions,differentialequationswithgeneralizedcoefficients.Thepape

7、risinfinalformandnoversionofitwillbepublishedelsewhere.DOI:10.4064/bc88-0-3[31]cInstytutMatematycznyPAN,201032N.V.BEDZIUKANDA.L.YABLONSKIInthispaperwewillconsiderthefollowingnonlinearequationwithgeneralizedcoef-ficientsX_(t)=f

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