6.2 Equilibrium measures

Equilibrium measures are probability measures that minimize energy functionals related to logarithmic potentials. They play a crucial role in potential theory, with applications in complex analysis and mathematical physics. Understanding equilibrium measures helps unlock insights into capacity, energy, and logarithmic potentials.

Existence and uniqueness of equilibrium measures are fundamental questions in potential theory. Frostman's theorem characterizes their existence on compact sets with positive capacity. Uniqueness is tied to the convexity of energy functionals. The support and potential of equilibrium measures provide valuable information about their properties and behavior.

Definitions of equilibrium measures

• Equilibrium measures are probability measures that minimize a certain energy functional, which is related to the logarithmic potential of the measure
• They play a crucial role in potential theory and have applications in various areas of mathematics, such as complex analysis, approximation theory, and mathematical physics

Capacity and energy

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• Capacity is a set function that measures the size of a set in terms of its ability to support an equilibrium measure
• The energy of a measure $\mu$ is defined as the double integral of the logarithmic kernel: $I(\mu) = \iint \log \frac{1}{|x-y|} d\mu(x) d\mu(y)$
• Sets with positive capacity can support an equilibrium measure, while sets with zero capacity cannot

Logarithmic potential

• The logarithmic potential of a measure $\mu$ at a point $z$ is defined as: $U^\mu(z) = \int \log \frac{1}{|z-x|} d\mu(x)$
• It represents the potential energy generated by the measure $\mu$ at the point $z$
• The logarithmic potential is a fundamental tool in the study of equilibrium measures and their properties

Minimizing energy functionals

• Equilibrium measures are characterized as the minimizers of the energy functional $I(\mu)$ among all probability measures supported on a given set
• The existence and uniqueness of equilibrium measures can be studied using variational principles and techniques from convex analysis
• Minimizing energy functionals leads to important properties of equilibrium measures, such as their support and regularity

Existence of equilibrium measures

• The existence of equilibrium measures is a fundamental question in potential theory
• It is closely related to the concept of capacity and the properties of the underlying set

Frostman's theorem

• Frostman's theorem is a key result that characterizes the existence of equilibrium measures
• It states that a compact set $K$ in the complex plane admits an equilibrium measure if and only if it has positive capacity
• The theorem provides a link between the capacity of a set and the existence of an equilibrium measure supported on that set

Compact sets in the plane

• Equilibrium measures are typically studied on compact sets in the complex plane
• Compact sets have nice topological properties that allow for the application of various analytical techniques
• Examples of compact sets include closed intervals, circles, and more general compact subsets of the plane

Sets of finite logarithmic capacity

• A set $K$ is said to have finite logarithmic capacity if there exists a probability measure $\mu$ supported on $K$ with finite energy $I(\mu)$
• Sets of finite logarithmic capacity are important in the study of equilibrium measures, as they guarantee the existence of an equilibrium measure
• The logarithmic capacity of a set can be computed using various methods, such as the Fekete-Szegő theorem or the transfinite diameter

Uniqueness of equilibrium measures

• The uniqueness of equilibrium measures is another fundamental question in potential theory
• It is closely related to the convexity properties of the energy functional and the geometry of the underlying set

Convexity of energy functionals

• The energy functional $I(\mu)$ is a convex functional on the space of probability measures
• Convexity plays a crucial role in the study of equilibrium measures and their uniqueness
• The convexity of the energy functional allows for the application of powerful tools from convex analysis, such as the Hahn-Banach theorem and the Krein-Milman theorem

Strict convexity for compact sets

• For compact sets in the complex plane, the energy functional $I(\mu)$ is strictly convex
• Strict convexity implies that the minimizer of the energy functional, i.e., the equilibrium measure, is unique
• The strict convexity of the energy functional is a consequence of the logarithmic kernel's properties and the compactness of the set

Application of minimizing properties

• The uniqueness of equilibrium measures can be proved using the minimizing properties of the energy functional
• If two measures $\mu_1$ and $\mu_2$ both minimize the energy functional on a compact set $K$, then their convex combination $(1-t)\mu_1 + t\mu_2$ also minimizes the energy functional for $0 \leq t \leq 1$
• By the strict convexity of the energy functional, this implies that $\mu_1 = \mu_2$, proving the uniqueness of the equilibrium measure

Support of equilibrium measures

• The support of an equilibrium measure is the smallest closed set on which the measure is concentrated
• Understanding the support of equilibrium measures is important for studying their properties and applications

Relation to polynomial inequalities

• The support of an equilibrium measure is closely related to polynomial inequalities
• For a compact set $K$ in the complex plane, the support of the equilibrium measure can be characterized using the Bernstein-Walsh inequality for polynomials
• The Bernstein-Walsh inequality relates the growth of polynomials on $K$ to their values on the support of the equilibrium measure

Polynomial convexity

• A compact set $K$ is said to be polynomially convex if it is equal to its polynomial convex hull, i.e., the set of points $z$ such that $|p(z)| \leq \max_{x \in K} |p(x)|$ for all polynomials $p$
• Polynomial convexity is a crucial property in the study of the support of equilibrium measures
• For polynomially convex sets, the support of the equilibrium measure coincides with the set itself

Characterization of support

• The support of the equilibrium measure can be characterized using various analytical and geometric properties
• One important characterization is the Frostman's maximum principle, which states that the logarithmic potential of the equilibrium measure is constant on its support
• Other characterizations involve the fine topology, the notion of regular points, and the balayage of measures

Equilibrium potential

• The equilibrium potential is the logarithmic potential generated by the equilibrium measure
• It plays a central role in the study of equilibrium measures and their properties

Definition and basic properties

• For a compact set $K$ with equilibrium measure $\mu_K$, the equilibrium potential is defined as: $U^{\mu_K}(z) = \int \log \frac{1}{|z-x|} d\mu_K(x)$
• The equilibrium potential is harmonic outside the support of the equilibrium measure and superharmonic on the whole complex plane
• It satisfies the Frostman's maximum principle, which states that $U^{\mu_K}(z) \leq I(\mu_K)$ for all $z \in \mathbb{C}$, with equality on the support of $\mu_K$

Frostman's maximum principle

• Frostman's maximum principle is a fundamental result in potential theory
• It characterizes the equilibrium potential as the unique function that satisfies certain maximality and minimality properties
• The principle states that the equilibrium potential is constant on the support of the equilibrium measure and smaller than this constant value outside the support

Continuity properties

• The equilibrium potential is a continuous function on the complex plane
• It is harmonic outside the support of the equilibrium measure and superharmonic on the whole plane
• The continuity of the equilibrium potential is a consequence of the regularity properties of the logarithmic kernel and the equilibrium measure

Examples of equilibrium measures

• Studying examples of equilibrium measures helps to understand their properties and behavior
• Some classical examples include intervals on the real line, circles, and more general compact sets

Intervals on the real line

• For an interval $[a, b]$ on the real line, the equilibrium measure is given by the arcsine distribution: $d\mu_{[a,b]}(x) = \frac{1}{\pi \sqrt{(x-a)(b-x)}} dx$
• The support of the equilibrium measure is the entire interval $[a, b]$
• The equilibrium potential is constant on the interval and behaves like a square root at the endpoints

Circles and circular arcs

• For a circle $\{z : |z| = r\}$, the equilibrium measure is the uniform distribution on the circle: $d\mu(z) = \frac{1}{2\pi r} |dz|$
• The support of the equilibrium measure is the entire circle
• For a circular arc, the equilibrium measure can be obtained by restricting the uniform measure on the circle to the arc

Cantor sets and self-similar measures

• Cantor sets are examples of compact sets with intricate geometric structure
• The equilibrium measure on a Cantor set is often a self-similar measure, which satisfies certain invariance properties under the set's contraction mappings
• The support of the equilibrium measure on a Cantor set is typically the entire set, but the measure can have a highly non-uniform distribution

Equilibrium measure vs harmonic measure

• Equilibrium measures and harmonic measures are two important concepts in potential theory
• While they share some similarities, they also have distinct properties and applications

Definitions and basic properties

• The equilibrium measure minimizes the energy functional among all probability measures supported on a given set
• The harmonic measure, on the other hand, is defined using the solution to the Dirichlet problem on the complement of the set
• Harmonic measures are related to the concept of Brownian motion and have probabilistic interpretations

Coincidence for regular sets

• For certain classes of sets, called regular sets, the equilibrium measure and the harmonic measure coincide
• Regular sets include sets with smooth boundaries or those satisfying certain geometric conditions, such as the cone condition
• The coincidence of equilibrium and harmonic measures for regular sets is a consequence of the regularity properties of the Dirichlet problem

Counterexamples for irregular sets

• For irregular sets, the equilibrium measure and the harmonic measure can be different
• Examples of irregular sets include sets with cusps, corners, or highly non-smooth boundaries
• The difference between equilibrium and harmonic measures for irregular sets highlights the importance of regularity in potential theory

Computation of equilibrium measures

• Computing equilibrium measures is an important task in potential theory and its applications
• Various methods and techniques have been developed to approximate and compute equilibrium measures

Discretization methods

• Discretization methods approximate the equilibrium measure by a discrete measure supported on a finite set of points
• These methods often involve solving a finite-dimensional optimization problem, such as minimizing the discrete energy functional
• Examples of discretization methods include the Fekete points method and the Gauss-Jacobi quadrature

Orthogonal polynomials

• Orthogonal polynomials play a crucial role in the computation of equilibrium measures
• The zeros of orthogonal polynomials with respect to the equilibrium measure tend to distribute according to the equilibrium measure as the degree of the polynomials increases
• This property allows for the approximation of the equilibrium measure using the zeros of orthogonal polynomials

Numerical approximation techniques

• Numerical approximation techniques are used to compute equilibrium measures in cases where analytical solutions are not available
• These techniques often involve solving a discretized version of the energy minimization problem using optimization algorithms
• Examples of numerical approximation techniques include the finite difference method, the finite element method, and the boundary integral method