Stackelberg Solutions of Feedback Type for Differential Games with Random Initial Data

2012 ◽  
Vol 3 (3) ◽  
pp. 341-358 ◽  
Author(s):  
Alberto Bressan ◽  
Deling Wei
Keyword(s):  
Initial Data ◽  
Differential Games ◽  
Random Initial Data ◽  
Feedback Type ◽  
Stackelberg Solutions

scholarly journals SCALING LIMITS FOR TIME-FRACTIONAL DIFFUSION-WAVE SYSTEMS WITH RANDOM INITIAL DATA

2010 ◽  
Vol 10 (01) ◽  
pp. 1-35 ◽  
Author(s):  
GI-REN LIU ◽  
NARN-RUEIH SHIEH
Keyword(s):  
Initial Data ◽  
Initial Conditions ◽  
Scaling Limit ◽  
Reaction Diffusion ◽  
Equation System ◽  
Random Initial Data ◽  
Diffusion Type ◽  
Diffusion Wave ◽  
Decoupling Method ◽  
Time Space

Let w (x, t) := (u, v)(x, t), x ∈ ℝ3, t > 0, be the ℝ2-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), ut(x,0), and v(x,0), vt(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

scholarly journals Numerical Stackelberg Solutions in a Class of Positional Differential Games

2018 ◽  
Vol 51 (32) ◽  
pp. 326-330
Author(s):  
Dmitry R. Kuvshinov ◽  
Sergei I. Osipov
Keyword(s):  
Differential Games ◽  
Stackelberg Solutions

scholarly journals Solving the 4NLS with white noise initial data

2020 ◽  
Vol 8 ◽  
Author(s):  
Tadahiro Oh ◽  
Nikolay Tzvetkov ◽  
Yuzhao Wang
Keyword(s):  
Invariant Measure ◽  
Initial Data ◽  
White Noise ◽  
Fourth Order ◽  
Random Initial Data ◽  
Fourier Restriction ◽  
Singular Component ◽  
Gauge Transform ◽  
Nonlinear Decomposition ◽  
Noise Measure

Abstract We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

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