Diffusion equations in duals of nuclear spaces

Kallianpur, G.; Mitoma, Itaru; Wolpert, Robert L.

doi:10.1080/17442509008833618

Cited by 24 publications

(17 citation statements)

References 8 publications

(7 reference statements)

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“…αϊ. [10] follows. In addition, one can also get the above mentioned result of Skorokhod [13], using our methods, under slightly stronger conditions.…”

Section: Introductionmentioning

confidence: 90%

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Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Gawarecki

Mandrekar²,

Richard

1999

Random Operators and Stochastic Equations

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Several results concerning the existence and uniqueness of solutions of Ito SDE's in a real separable Hubert space have recently been reported. In this work we first obtain the existence and uniqueness of strong solutions to (not necessarily) Ito SDE's in a Hubert space under Lipschitz-type conditions on the coefficients. We assume usual continuity and linear growth conditions. For non-Lipschitz coefficients an approximation technique of Gikhman and Skorokhod is then used to prove the existence of weak solutions taking values in a larger Hubert space H-\. This result depends on an assumption that H can be compactly embedded in H-\, such that the coefficients satisfy regularity conditions with respect to H-\. This assumption is not a limitation to our method as it is necessary even in the deterministic case. Additionally, we prove the existence of martingale solutions to infinite dimensional semilinear SDE's. In both cases, coefficients F(£, ·) and J9(£, ·) may depend on the entire past of X £ C([0, T], H) and not on the value of x(t) alone.Brought to you by | provisional account Unauthenticated Download Date | 7/2/15 1:26 AM

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“…αϊ. [10] follows. In addition, one can also get the above mentioned result of Skorokhod [13], using our methods, under slightly stronger conditions.…”

Section: Introductionmentioning

confidence: 90%

“…Following Kallianpur, Mitoma and Wolpert, [10], we assume that all H k , (k > 0) have common basis {h 1 ·}, and we denote by {hj } the associated basis in H-k· Let us assume that…”

Section: In View Of Theorem 6 In the Appendix The Measure μ Is A Lawmentioning

confidence: 99%

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Gawarecki

Mandrekar²,

Richard

1999

Random Operators and Stochastic Equations

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“…We give below the result of Kallianpur et al [8] on the existence and uniqueness of solutions of SDEs. The following condition will be needed in the proof of uniqueness.…”

Section: )'-Valued Sdesmentioning

confidence: 96%

“…We give below the main result of Kallianpur et al [8]. Next, we give a moment bound followed by a tightness result, both of which are due to Baldwin et al [1].…”

Section: )'-Valued Sdesmentioning

confidence: 96%

“…From this, it follows that the support of t/is contained in the set of solutions to the L-martingale problem. Corresponding to each solution of the Lmartingale problem, we can construct a weak solution of the McKean-Vlasov equation, by the method employed by KaUianpur et al [8]. From the previous section we know that the McKean-Vlasov equation has a unique strong solution, namely YX°.…”

Section: F(2) = ~C~_{ F(y(t)[q)]) -F(y(r)[q)]) -F: L(f Y(s) S "~mentioning

confidence: 98%

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Propagation of chaos and the McKean-Vlasov equation in duals of nuclear spaces

1991

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Abstract. An interacting system of n stochastic differential equations taking values in the dual of a countable Hilbertian nuclear space is considered. The limit (in probability) of the sequence of empirical measures determined by the above systems as n tends to oo is identified with the law of the unique solution of the McKean-Vlasov equation. An application of our result to interacting neurons is briefly discussed. The propagation of chaos result obtained in this paper is shown to contain and improve the well,known finite-dimensional results.

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Nuclear space-valued stochastic differential equations with applications

Kallianpur

1992

Probabilistic and Stochastic Methods in Analysis, With Applications

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Diffusion equations in duals of nuclear spaces

Cited by 24 publications

References 8 publications

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Propagation of chaos and the McKean-Vlasov equation in duals of nuclear spaces

Nuclear space-valued stochastic differential equations with applications

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