Stochastic Reaction-Diffusion Systems With Hölder Continuous Multiplicative Noise
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In this article, the authors prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with a locally Lipschitz continuous reaction term of polynomial growth and Holder continuous multiplicative noise.Abstract:
We prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with a locally Lipschitz continuous reaction term of polynomial growth and Holder continuous multiplicative noise. Under additional assumptions on the coefficients, we also prove positivity of the solutions.read more
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Stochastic Integration in Banach Spaces - a Survey
TL;DR: In this article, a brief survey of the theory of stochastic integration in Banach spaces is presented, as well as some applications of the latter to vector-valued Malliavin calculus and the Stochastic maximal regularity problem.
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On a class of martingale problems on Banach spaces
TL;DR: In this paper, the local martingale problem associated to semilinear stochastic evolution equations driven by a cylindrical Wiener process was introduced and established a one-to-one correspondence between solutions of the Martingale Problem and (analytically) weak solutions of a seminear evolution equation.
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Global martingale solutions for a stochastic population cross-diffusion system
TL;DR: The existence of global nonnegative martingale solutions to a stochastic cross-diffusion system for an arbitrary but finite number of interacting population species is shown in this paper.
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A Derivative-Free Milstein Type Approximation Method for SPDEs covering the Non-Commutative Noise case.
TL;DR: A derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations that need not to fulfill a commutativity condition for the noise term and which can flexibly be combined with some approximation method for the involved iterated integrals is proposed.
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Enhancing the Order of the Milstein Scheme for Stochastic Partial Differential Equations with Commutative Noise
Claudine Leonhard,Andreas Rößler +1 more
TL;DR: In this article, a derivative-free higher-order Milstein scheme for stochastic partial differential equations with trace class noise is proposed. And the authors prove that the effective order of convergence of the proposed derivative free scheme is significantly higher than that of the original Milstein method.
References
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Impulses and Physiological States in Theoretical Models of Nerve Membrane
TL;DR: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle, which qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve.
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Stochastic Equations in Infinite Dimensions
Giuseppe Da Prato,Jerzy Zabczyk +1 more
TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
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An Active Pulse Transmission Line Simulating Nerve Axon
TL;DR: In this paper, an active pulse transmission line using tunnel diodes was made to electronically simulate an animal nerve axon, and the equation of propagation for this line is the same as that for a simplified model of nerve membrane treated elsewhere.
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Analytic Semigroups and Optimal Regularity in Parabolic Problems
TL;DR: In this article, the authors propose the generation of analytic semigroups by elliptic operators and derive the space of continuous and holder continuous functions in the intermediate spaces of continuous functions.
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Stochastic evolution equations
Nicolai V Krylov,Boris Rozovskii +1 more
TL;DR: In this paper, the theory of strong solutions of Ito equations in Banach spaces is expounded, and the results of this theory are applied to the investigation of strongly parabolic Ito partial differential equations.