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| A Lévy process is an important type of [[stochastic process]] (namely, a family of $\mathbb{R}^d$ valued random variables each indexed by a positive number $t\geq 0$). In the context of parabolic integro-differential equations they play the same role that Brownian motion and more general diffusions play in the theory of second order parabolic equations.
| | Dislocations are microscopic defects in crystals that change over time (due for instance to shear stresses on the crystal). |
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| Informally speaking, a Lévy process is a random trajectory, generalizing the concept of Brownian motion, which may contain jump discontinuities. A prototypical example would be $X(t)=B(t)+N(t)$ where $B(t)$ is the standard [[Brownian motion]] and $N(t)$ is a [[Compound Poisson process]], the trajectory described by typical sample path of this process would look like the union of several disconnected Brownian motion paths.
| | 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 |
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| == Definition == | | \[ \dot{y}_i=F-V\;'_0(y_i) - \sum \limits_{j \neq i} V\;'(y_i-y_j) \;\;\;\text{ for } i=1,...,N, \] |
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| A stochastic process $X=\{X(t)\}_{t \geq 0}$ with values in $\mathbb{R}^d$ is said to be a Lévy process if
| | 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 |
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| 1.For any sequence $0 \leq t_1 < t_2 <...<t_n$ the random variables $X(t_0),X(t_1)-X(t_0),...,X(t_n)-X(t_{n-1})$ are independent.
| | \[ u_t +|u_x|\Lambda^s u = 0 \;\;\;\text{ for all } (x,t) \in \mathbb{R}\times\mathbb{R}_+ \] |
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| 2.For any positive times $s\leq t$ the random variables $X(t-s)$ and $X(t)-X(s)$ have the same probability law.
| | 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" />. |
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| 3.Almost surely, the trajectory of $X(t)$ is continuous from the right, with limit from the left also known as "càdlàg" for its acronym in french.
| | == References == |
| | | {{reflist|refs= |
| == Lévy-Khintchine Formula == | | <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> |
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| It follows from the first two properties above that if $X$ is a Lévy process and we further assume $X(0)=0$ a.s. then for each fixed positive $t$ the random variable $X(t)$ is infinitely divisible, that is, it can be written as the sum of $n$ independent and identically distribued random variables, for all $n\in\mathbb{N}$. Indeed, let $h=\tfrac{t}{n}$, then
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| \[X(t) = \left( X(h)-X(0)\right)+\left( X(2h)-X(h)\right)+...+\left( X(t)-X((n-1)h)\right)\]
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| and by the above definition the differences $X(kh)-X((k-1)h)$ are independent and distributed the same as $X(h)$. From the infinite divisibility of $X(t)$ it follows by a theorem of Lévy and Khintchine that for any $\xi \in \mathbb{R}^d$ we have
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| \[ \mathbb{E} \left [ e^{i\xi\cdot X_t}\right ] = e^{t\eta(\xi)}\]
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| the function $\eta(\xi)$ given by
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| \[\eta(\xi)=i y\cdot b -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{i \xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y) \]
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| where $b$ is a vector, $A$ is a positive matrix, $B_1$ is the unit ball and $\mu$ is a Lévy measure, that is, a Borel measure in $\mathbb{R}^d$ such that
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| \[ \int_{\mathbb{R}^d}\frac{|y|^2}{1+|y|^2}d\mu(y) <+\infty. \]
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| The interpretation of this measure $\mu$ is that ''jumps'' from some point $x$ to $x+y$ with $y$ in some set $A$ occur as a Poisson process with intensity $\mu(A)$.
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| == Connection to linear integro-differential operators == | |
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| Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows
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| \[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t)) \right ]\]
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| Given the initial assumption on $X(0)$ it is clear that $U_0$ is the identity, and given that $X(t)-X(s)$ is distributed as $X(t-s)$ it follows that $U_t \circ U_s = U_{t+s}$.
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| As a semigroup, $U_t$ has an infinitesimal generator which turns out to be a [[Linear integro-differential operator]]. More precisely, if we let $f(x,t):=(U_tf)(x)$, then, assuming that $f(x,t)$ has enough regularity it can be checked that
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| \[\partial_t f = Lf \;\;\;\mbox{ for all } (x,t)\in\mathbb{R}^d\times \mathbb{R}_+\]
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| where for any smooth function $\phi$, we have
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| \[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \]
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| == [[Stochastic control]] and fully non-linear integro-differential operators == | |
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| A similar connection holds via the [[Isaacs equation|Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it.
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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) 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
\[ \dot{y}_i=F-V\;'_0(y_i) - \sum \limits_{j \neq i} V\;'(y_i-y_j) \;\;\;\text{ for } i=1,...,N, \]
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 [1].
References
- ↑ Biler, Piotr; Monneau, Régis; Karch, Grzegorz (2009), "Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions", Communications in Mathematical Physics 294 (1): 145–168, doi:10.1007/s00220-009-0855-8, ISSN 0010-3616