Harnack inequality and Interacting Particle Systems: Difference between pages

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The (second order) integro-differential equation
\newcommand{\dd}{\mathrm{d}}
\[ \begin{array}{rl}
\newcommand{\R}{\mathbb{R}}
\partial_t \rho &= \text{div} \left( D(\rho) \nabla \rho+\sigma(\rho) \nabla V[\rho]\right )\\
$$
V[\rho] & = J * \rho
\end{array}\]
describes at the macroscopic scale the phase segregation in  a gas whose particles are interacting at long ranges, as shown by Giacomin and Lebowitz <ref name="GL97"/>. This equation not only arises as the limit of the microscopic system but the approximation is good enough to capture both qualitative and quantitative phenomena of the microscopic system <ref name="GL97"/>. More concretely, the above equation arises as the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics. Giacomin and Lebowitz also note that unlike the standard Cahn-Hilliard equation, the above integro-differential equation has been shown rigorously to arise as the macroscopic limit of a microscopic model of interacting particles <ref name="GL97"/>.


The Harnack inequality refers to a control of the maximum of a nonnegative solution of an equation by its minimum. Unlike the [[Harnack inequality (local)|local case]], for nonlocal equations one needs to assume that the function is nonnegative in the full space.


The Harnack inequality is tightly related to [[Holder estimates]]. Whenever one holds, the other is expected to hold as well, either as a direct consequence or by applying the same proof method.
== The interacting particle system ==


The result can hold either in the parabolic or elliptic setting. The parabolic Harnack inequality trivially implies the elliptic one. The reverse implication is not automatic, and the proof in the parabolic case may have some extra difficulties compared to the elliptic case.
At the microscopic level, the system is described by a function


== Elliptic case ==
\[ \eta  : \Lambda_\gamma \to \{ 0,1\} \]


In the elliptic setting, the Harnack inequality refers to the following type of result: if a function $u: \R^n \to \R$ satisfies an elliptic equation $ L_x u (x) = f(x)$ in the unit ball $B_1$ and is nonnegative in the full space $\R^n$, then
where  $\gamma>0$ taken very small represents the spatial scale and  $\Lambda_\gamma $ denotes the finite $d$-dimensional lattice
\[ \sup_{B_{1/2}} u \leq C \left( \inf_{B_{1/2}} u + ||f|| \right). \]


The norm $||f||$ may depends on the type of equation.
\[ \Lambda_\gamma = \{ 1,2,...,[\gamma^{-1}]\}^d\]


== Parabolic case ==
in other words, a cube inside $\mathbb{Z}^d$ with sides given by $[\gamma^{-1}]$, as $\gamma \to 0$, this exhausts all of $\mathbb{Z}^d$. The set of all posible configurations $\eta$ will be denoted by $\Omega_\gamma$, this is the state space where the (microscopic scale) dynamics takes place. As $\gamma \to 0$ we expect to recover the above as a limiting dynamical system the integro-differential equation listed above, of course first we have to describe the microscopic dynamics.


In the parabolic setting, the Harnack inequality refers to the following type of result: if a function $u: [-1,0] \times \R^n \to \R$ satisfies an elliptic equation $ u_t - L_x u (x) = f(x)$ in the unit cylinder $(-1,0) \times B_1$ and is nonnegative in the full space $[-1,0] \times \R^n$, then
Given any initial condition $\eta_0 : \Lambda_\gamma \to \{0,1\}$, we consider a stochastic Poisson jump process with values in $\Lambda_\gamma$ generated by the operator
\[ \sup_{[-1/2,-1/4] \times B_{1/2}} u \leq \left(\inf_{[-1/4,0] \times B_{1/2}} u + ||f|| \right). \]


The norm $||f||$ may depend on the type of equation.
\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\]


== Concrete examples ==
where $\eta^{x,y}$ denotes the state $\eta$ where the values at $x$ and $y$ have been interchanged and the kernel $c_\gamma(x,y;\eta)$ is defined as


The Harnack inequality as above is known to hold in the following situations.
\[c_\gamma(x,y;\eta) = \left \{ \begin{array}{rl}
\Phi \left ( \beta\left [ H(\eta^{x,y}-H(\eta) \right ] \right) & \text{ if }\; |x-y|=1\\
0 & \text{ otherwise }
  \end{array}\right.\]


* '''Generalizad elliptic [[Krylov-Safonov]]'''. If $L_x u(x)$ is a symmetric integro-differential operator of the form
\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)) K(x,y) \dd y \]
with $K$ symmetric ($K(x,y)=K(x,-y)$) and uniformly elliptic of order $s$: $(2-s)\lambda |y|^{-n-s} \leq K(x,y) \leq (2-s) \Lambda |y|^{-n-s}$.


In this case the elliptic Harnack inequality is known to hold with a constant $C$ which '''does not blow up as $s\to 2$''', and $||f||$ refers to $||f||_{L^\infty(B_1)}$ <ref name="CS"/>. It is a generalization of [[Krylov-Safonov]] theorem. The corresponding parabolic Harnack inequality with a uniform constant $C$ is not known.
== References ==
 
{{reflist|refs=
* '''Elliptic equations with variable order (but strictly less than 2)'''. If $L_x u(x)$ is an integro-differential operator of the form
\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)- y \cdot \nabla u(x) \chi_{B_1}(y)) K(x,y) \dd y \]
with uniformly elliptic of variable order: $\lambda |y|^{-n-s_1} \leq K(x,y) \leq \Lambda |y|^{-n-s_2}$ and $0<s_1 < s_2 < 2$ and $s_2 - s_1 < 1$, then
the elliptic Harnack inequality holds if $f \equiv 0$<ref name="BK"/>. The constants in this result blow up as $s_2 \to 2$, so it does not generalize [[Krylov-Safonov]] theorem. The proof uses probability and was based on a previous result with fixed order <ref name="BL"/>.


It is conceivable that a purely analytic proof could be done using the method of the corresponding [[Holder estimate]] <ref name="S"/>, but such proof has never been done.
<ref name="GL97"> {{Citation | last1=Lebowitz | first1=Joel | last2=Giacomin | first2=Giambattista | title=Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits | doi=10.1007/BF02181479 | year=1997 | journal=Journal of Statistical Physics | issn=0022-4715 | volume=87 | issue=1 | pages=37–61}} </ref>


* '''Gradient flows of symmetric Dirichlet forms with variable order'''. If $u_t - L_x u(x)=0$ is the gradient flow of a [[Dirichlet form]]:
\[ \iint_{\R^n \times \R^n} (u(y)-u(x))^2 K(x,y)\, \dd x \dd y. \]
for kernels $K$ such that $K(x,y)=K(y,x)$ and $\lambda |x-y|^{-n-s_1} \leq K(x,y) \leq \Lambda |x-y|^{-n-s_2}$ for some $0<s_1<s_2<2$ and $|x-y|$ sufficiently small. Then the parabolic Harnack inequality holds if $f \equiv 0$ for some constant $C$ which a priori '''blows up as $s_2 \to 2$''' <ref name="BBCK"/>.
It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.
== References ==
{{reflist|refs=
<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder estimates for solutions of integro-differential equations like the fractional Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 | doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 | pages=1155–1174}}</ref>
<ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
<ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann | first2=Moritz | title=Harnack inequalities for non-local operators of variable order | url=http://dx.doi.org/10.1090/S0002-9947-04-03549-4 | doi=10.1090/S0002-9947-04-03549-4 | year=2005 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=357 | issue=2 | pages=837–850}}</ref>
<ref name="BL">{{Citation | last1=Bass | first1=Richard F. | last2=Levin | first2=David A. | title=Harnack inequalities for jump processes | url=http://dx.doi.org/10.1023/A:1016378210944 | doi=10.1023/A:1016378210944 | year=2002 | journal=Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis | issn=0926-2601 | volume=17 | issue=4 | pages=375–388}}</ref>
<ref name="BBCK">{{Citation | last1=Barlow | first1=Martin T. | last2=Bass | first2=Richard F. | last3=Chen | first3=Zhen-Qing | last4=Kassmann | first4=Moritz | title=Non-local Dirichlet forms and symmetric jump processes | url=http://dx.doi.org/10.1090/S0002-9947-08-04544-3 | doi=10.1090/S0002-9947-08-04544-3 | year=2009 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=361 | issue=4 | pages=1963–1999}}</ref>
<ref name="K">{{Citation | last1=Kassmann | first1=Moritz | title=A priori estimates for integro-differential operators with measurable kernels | url=http://dx.doi.org/10.1007/s00526-008-0173-6 | doi=10.1007/s00526-008-0173-6 | year=2009 | journal=Calculus of Variations and Partial Differential Equations | issn=0944-2669 | volume=34 | issue=1 | pages=1–21}}</ref>
}}
}}

Revision as of 00:35, 1 February 2012

The (second order) integro-differential equation \[ \begin{array}{rl} \partial_t \rho &= \text{div} \left( D(\rho) \nabla \rho+\sigma(\rho) \nabla V[\rho]\right )\\ V[\rho] & = J * \rho \end{array}\] describes at the macroscopic scale the phase segregation in a gas whose particles are interacting at long ranges, as shown by Giacomin and Lebowitz [1]. This equation not only arises as the limit of the microscopic system but the approximation is good enough to capture both qualitative and quantitative phenomena of the microscopic system [1]. More concretely, the above equation arises as the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics. Giacomin and Lebowitz also note that unlike the standard Cahn-Hilliard equation, the above integro-differential equation has been shown rigorously to arise as the macroscopic limit of a microscopic model of interacting particles [1].


The interacting particle system

At the microscopic level, the system is described by a function

\[ \eta : \Lambda_\gamma \to \{ 0,1\} \]

where $\gamma>0$ taken very small represents the spatial scale and $\Lambda_\gamma $ denotes the finite $d$-dimensional lattice

\[ \Lambda_\gamma = \{ 1,2,...,[\gamma^{-1}]\}^d\]

in other words, a cube inside $\mathbb{Z}^d$ with sides given by $[\gamma^{-1}]$, as $\gamma \to 0$, this exhausts all of $\mathbb{Z}^d$. The set of all posible configurations $\eta$ will be denoted by $\Omega_\gamma$, this is the state space where the (microscopic scale) dynamics takes place. As $\gamma \to 0$ we expect to recover the above as a limiting dynamical system the integro-differential equation listed above, of course first we have to describe the microscopic dynamics.

Given any initial condition $\eta_0 : \Lambda_\gamma \to \{0,1\}$, we consider a stochastic Poisson jump process with values in $\Lambda_\gamma$ generated by the operator

\[ L_\gamma f(\eta) = \sum \limits_{x,y\in \Lambda_\gamma} c_\gamma(x,y;\eta) \left (f(\eta^{x,y})-f(\eta) \right )\]

where $\eta^{x,y}$ denotes the state $\eta$ where the values at $x$ and $y$ have been interchanged and the kernel $c_\gamma(x,y;\eta)$ is defined as

\[c_\gamma(x,y;\eta) = \left \{ \begin{array}{rl} \Phi \left ( \beta\left [ H(\eta^{x,y}-H(\eta) \right ] \right) & \text{ if }\; |x-y|=1\\ 0 & \text{ otherwise } \end{array}\right.\]


References

  1. 1.0 1.1 1.2 Lebowitz, Joel; Giacomin, Giambattista (1997), "Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits", Journal of Statistical Physics 87 (1): 37–61, doi:10.1007/BF02181479, ISSN 0022-4715