Dislocation dynamics and Optimal stopping problem: Difference between pages

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imported>Nestor
(Created page with "Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal). If we have a finite number of parallel (horizontal) ...")
 
imported>Luis
(Created page with "This is an excerpt from the main article on the optimal stopping problem. Given a Levy process $X_t$ we consider the following problem. We want to find the optimal stopp...")
 
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Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal).  
This is an excerpt from the main article on the [[optimal stopping problem]].


If we have a finite number of parallel (horizontal) lines on a 2D crystal each given by the equation $y=y_i$ ($y_i \in \mathbb{R}$) then a simplified model for the evolution of these lines says that the positions of these lines evolve according to the system of ODEs
Given a [[Levy process]] $X_t$ we consider the following problem. We want to find the optimal stopping time $\tau$ to maximize the expected value of $\varphi(X_{\tau})$ for some given (payoff) function $\varphi$.


\[ \dot{y}_i=F-V\;'_0(y_i) - \sum \limits_{j \neq i} V\;'(y_i-y_j) \;\;\;\text{ for } i=1,...,N, \]
In this setting, the value of $\varphi(X_{\tau})$ depends on the initial point $x$ where the Levy process starts. We can call $u(x) = \mathbb E[\varphi(X_{\tau}) | X_0 = x]$. Under this choice, the function $u$ satisfies the obstacle problem with $L$ being the generator of the Levy process $X_t$.
 
One can consider the case in which $N \to +\infty$ and consider the evolution of a density of dislocation lines. If $u(x,t)$ denotes the limiting density, then the  it solves the integro-differential equation
 
\[ u_t +|u_x|\Lambda^s u = 0 \;\;\;\text{  for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+  \]
 
in the case where the interaction potential $V\;\;$ satisfies $V\;'(y)=-\frac{1}{y^s}$. For this one dimensional model (which enjoys a maximum principle) a complete theory in terms of viscosity solutions, including existence, uniqueness and regularity was recently developed <ref name="BMK" />.
 
== References ==
{{reflist|refs=
<ref name="BMK">{{Citation | last1=Biler | first1=Piotr | last2=Monneau | first2=Régis | last3=Karch | first3=Grzegorz | title=Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions | doi=10.1007/s00220-009-0855-8 | year=2009 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=294 | issue=1 | pages=145–168}}</ref>
}}

Revision as of 13:52, 28 January 2012

This is an excerpt from the main article on the optimal stopping problem.

Given a Levy process $X_t$ we consider the following problem. We want to find the optimal stopping time $\tau$ to maximize the expected value of $\varphi(X_{\tau})$ for some given (payoff) function $\varphi$.

In this setting, the value of $\varphi(X_{\tau})$ depends on the initial point $x$ where the Levy process starts. We can call $u(x) = \mathbb E[\varphi(X_{\tau}) | X_0 = x]$. Under this choice, the function $u$ satisfies the obstacle problem with $L$ being the generator of the Levy process $X_t$.