Vector-valued stochastic delay equations - a weak solution and its Markovian representation

2013/01/22 by Mariusz Górajski, Górajski, Mariusz
Biochemistry, Genetics and Molecular Biology · Computer Science · Economics, Econometrics and Finance · Mathematics · #34K50 #47D06 #60H15 #60H30 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Gene Regulatory Network Analysis #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories #math.FA #math.PR #msc:34K50 #msc:47D06 #msc:60H15 #msc:60H30

paper · pdf · doi:10.48550/arxiv.1301.5300

arxiv created 2013/01/22 · openalex publication_date 2013/01/22 · arxiv updated 2013/01/23 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

A class of stochastic delay equations in Banach space E driven by cylindrical Wiener process is studied. We investigate two concepts of solutions: weak and generalised strong, and give conditions under which they are equivalent. We present an evolution equation approach in a Banach space \Epp:=E× L^p(-1,0;E) proving that the solutions can be reformulated as \Epp-valued Markov processes. Based on the Markovian representation we prove the existence and continuity of the solutions. The results are applied to stochastic delay partial differential equations with an application to neutral networks and population dynamics.

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