On a McKean-Vlasov Stochastic Integro-differential Evolution Equation of Sobolev-Type

• Published in: Stochastic analysis and applications Vol. 21; no. 5; pp. 1115 - 1139
• Main Authors: Keck, David N., McKibben, Mark A.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 09.01.2003; Taylor & Francis
• Subjects: Almost sure exponential stability; Exact sciences and technology; Mathematical analysis; Mathematics; McKean-Vlasov; Ordinary differential equations; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Sobolev-type equation; Stochastic analysis; Stochastic evolution equation; Solution uniqueness; Sobolev type equation; Stochastic analysis; McKean Vlasov equation; Initial condition; Evolution equation; Global existence; Hilbert space; Almost sure exponential stability; Integrodifferential equation; Stochastic equation
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1081/SAP-120024706

We establish the global existence and uniqueness of mild solutions for a class of first-order abstract stochastic Sobolev-type integro-differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so-called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory.