10.2 Harmonic measure

Harmonic measure is a key concept in potential theory, assigning probabilities to boundary subsets of a domain. It's closely linked to harmonic functions and Brownian motion, providing insights into boundary behavior and probabilistic interpretations.

This topic explores harmonic measure's properties, computation methods, and applications. From conformal invariance to numerical approximations, it connects various mathematical fields and offers powerful tools for analyzing complex domains and function behaviors.

Definition of harmonic measure

• Harmonic measure is a fundamental concept in potential theory that assigns a probability measure to subsets of the boundary of a domain in Euclidean space or more general settings
• It quantifies the likelihood that a Brownian motion starting from a point inside the domain will first hit the boundary within a given subset
• Harmonic measure is closely related to the study of harmonic functions and plays a crucial role in understanding the boundary behavior of these functions

Relationship to harmonic functions

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• Harmonic measure is intimately connected to the theory of harmonic functions, which are functions that satisfy Laplace's equation $\Delta u = 0$ in a domain
• The harmonic measure of a boundary subset can be interpreted as the value of the harmonic function with boundary values equal to the characteristic function of that subset
• Conversely, given a harmonic function in a domain, its boundary values determine a unique harmonic measure on the boundary

Connection with Brownian motion

• Harmonic measure has a probabilistic interpretation in terms of Brownian motion, a continuous-time stochastic process that models the random motion of particles
• The harmonic measure of a boundary subset is equal to the probability that a Brownian motion starting from a point inside the domain will first hit the boundary within that subset
• This connection allows for the use of probabilistic techniques in the study of harmonic measure and provides insights into the behavior of Brownian motion near the boundary

Harmonic measure vs surface measure

• Harmonic measure is distinct from the usual surface measure on the boundary of a domain, although they coincide in certain special cases (unit disk)
• Surface measure is determined solely by the geometry of the boundary, while harmonic measure takes into account the geometry of the domain and the behavior of harmonic functions
• In general, harmonic measure can be singular with respect to surface measure, meaning that there can be boundary subsets with positive surface measure but zero harmonic measure

Properties of harmonic measure

• Harmonic measure possesses several important properties that reflect its connection to harmonic functions and its role in potential theory
• These properties provide insights into the behavior of harmonic measure under various transformations and its relationship to the geometry of the underlying domain

Conformal invariance

• Harmonic measure is invariant under conformal mappings, which are angle-preserving transformations between domains
• If $f: \Omega_1 \to \Omega_2$ is a conformal mapping between two domains, then the harmonic measure of a boundary subset in $\Omega_1$ is equal to the harmonic measure of its image under $f$ in $\Omega_2$
• This property allows for the study of harmonic measure in complex domains by mapping them to simpler domains (unit disk) where the harmonic measure is well-understood

Doubling property

• Harmonic measure satisfies the doubling property, which means that the harmonic measure of a ball on the boundary is comparable to the harmonic measure of a concentric ball with twice the radius
• More precisely, there exists a constant $C > 0$ such that for any ball $B(x, r)$ centered at a boundary point $x$ with radius $r$, we have $\omega(B(x, 2r)) \leq C \omega(B(x, r))$, where $\omega$ denotes the harmonic measure
• The doubling property reflects the regularity of harmonic measure and has important consequences for the behavior of harmonic functions near the boundary

Support and regularity

• The support of harmonic measure, denoted by $\text{supp}(\omega)$, is the smallest closed subset of the boundary that has full harmonic measure
• In other words, it is the set of boundary points where the harmonic measure "concentrates"
• Harmonic measure is often absolutely continuous with respect to surface measure on its support, meaning that it can be represented by a density function
• The regularity of this density function is related to the regularity of the boundary and has implications for the boundary behavior of harmonic functions

Uniqueness and existence

• Harmonic measure is uniquely determined by the domain and the starting point of the Brownian motion
• In other words, for a given domain $\Omega$ and a point $x \in \Omega$, there exists a unique harmonic measure $\omega_x$ on the boundary $\partial \Omega$
• The existence of harmonic measure can be established using the Perron-Wiener-Brelot method, which constructs harmonic functions with prescribed boundary values
• The uniqueness of harmonic measure follows from the maximum principle for harmonic functions

Computation of harmonic measure

• Computing harmonic measure is a fundamental problem in potential theory, and various techniques have been developed to calculate or approximate it in different settings
• These methods often exploit the connection between harmonic measure and harmonic functions or the probabilistic interpretation in terms of Brownian motion

Poisson kernel and integral representation

• In certain domains with nice geometry, such as the unit disk or half-space, harmonic measure can be explicitly computed using the Poisson kernel
• The Poisson kernel is a function $P(x, y)$ that relates the values of a harmonic function at an interior point $x$ to its boundary values at a point $y$
• Harmonic measure can be represented as an integral of the boundary values against the Poisson kernel: $\omega_x(E) = \int_E P(x, y) d\sigma(y)$, where $E$ is a boundary subset and $\sigma$ is the surface measure

Conformal mapping techniques

• Conformal mappings provide a powerful tool for computing harmonic measure in simply connected domains
• By mapping a domain conformally to the unit disk, where the harmonic measure is given by the normalized arc length measure, one can obtain the harmonic measure in the original domain
• The conformal map and its inverse can be computed using various techniques, such as the Schwarz-Christoffel formula for polygonal domains or numerical methods for more general domains

Numerical methods and approximations

• For domains with complex geometry or in higher dimensions, explicit formulas for harmonic measure are often not available
• In such cases, numerical methods can be employed to approximate harmonic measure
• Finite element methods discretize the domain into smaller elements and solve the Laplace equation numerically to obtain approximations of harmonic functions and harmonic measure
• Probabilistic methods, such as Monte Carlo simulations, can be used to estimate harmonic measure by simulating Brownian motion and recording the hitting distribution on the boundary

Applications of harmonic measure

• Harmonic measure finds applications in various branches of mathematics, including complex analysis, potential theory, and geometric measure theory
• It provides a powerful tool for studying the boundary behavior of harmonic functions and related objects

Boundary behavior of harmonic functions

• Harmonic measure can be used to characterize the boundary behavior of harmonic functions
• The Fatou theorem states that a bounded harmonic function has non-tangential limits almost everywhere with respect to harmonic measure
• The non-tangential limit of a harmonic function at a boundary point can be interpreted as the average value of the function with respect to harmonic measure in a cone-like region touching the boundary point

Capacity and Hausdorff dimension

• Harmonic measure is closely related to the concepts of capacity and Hausdorff dimension, which quantify the size and regularity of sets
• The capacity of a boundary subset can be defined using harmonic measure, and sets with zero capacity are precisely those with zero harmonic measure
• The Hausdorff dimension of a boundary subset can be estimated using the behavior of harmonic measure on balls centered at the subset, providing a connection between harmonic measure and fractal geometry

Extremal length and quasiconformal mappings

• Harmonic measure is connected to the theory of quasiconformal mappings, which are generalizations of conformal mappings that allow for bounded distortion of angles
• The concept of extremal length, which measures the conformal invariant of a family of curves, can be expressed in terms of harmonic measure
• Quasiconformal mappings preserve the doubling property of harmonic measure and are useful in the study of deformations of domains and their impact on harmonic measure

Potential theory in complex analysis

• Harmonic measure plays a fundamental role in potential theory, particularly in the context of complex analysis
• In the complex plane, harmonic functions are closely related to analytic functions, and harmonic measure provides a bridge between the boundary behavior of these functions
• The study of harmonic measure in the complex setting leads to important results, such as the Riesz decomposition theorem and the characterization of removable singularities for bounded analytic functions

Advanced topics in harmonic measure

• The theory of harmonic measure can be extended and generalized in various directions, leading to active areas of research in mathematics
• These advanced topics involve the study of harmonic measure in non-smooth domains, its relationship to geometric properties of the boundary, and its connections to other branches of analysis

Harmonic measure for non-smooth domains

• While the classical theory of harmonic measure is well-developed for domains with smooth boundaries, the study of harmonic measure in non-smooth domains presents new challenges and opportunities
• Domains with fractal boundaries, such as the von Koch snowflake or the Julia sets of complex dynamical systems, exhibit intricate behavior of harmonic measure
• The study of harmonic measure in these settings requires the development of new tools and techniques, such as the theory of Dirichlet forms and the use of multifractal analysis

Harmonic measure and rectifiability

• The relationship between harmonic measure and the geometric notion of rectifiability, which measures the regularity of sets in terms of their approximability by smooth surfaces, is an active area of research
• The F. and M. Riesz theorem states that for simply connected planar domains, harmonic measure is absolutely continuous with respect to arc length measure if and only if the boundary is rectifiable
• Generalizations of this result to higher dimensions and more general settings have led to the development of new techniques in geometric measure theory and harmonic analysis

Harmonic measure and the Dirichlet problem

• The Dirichlet problem, which asks for the existence and uniqueness of harmonic functions with prescribed boundary values, is closely related to the study of harmonic measure
• The solvability of the Dirichlet problem in a domain is often characterized by the regularity properties of its boundary, which can be expressed in terms of harmonic measure
• The study of harmonic measure has led to important results in the theory of partial differential equations, such as the Wiener criterion for the regularity of boundary points and the Perron-Wiener-Brelot method for solving the Dirichlet problem

Generalizations and related concepts

• The concept of harmonic measure can be generalized and extended in various ways, leading to the study of related notions in potential theory and analysis
• The theory of $p$-harmonic measure, which is based on the $p$-Laplace equation, provides a generalization of harmonic measure that is useful in the study of nonlinear potential theory
• The notion of elliptic measure, which is associated with more general elliptic partial differential equations, extends the theory of harmonic measure to a wider class of operators and has applications in the study of diffusion processes and stochastic differential equations