🔢Potential Theory Unit 10 – Brownian Motion in Potential Theory

Introduction to Brownian Motion

• Brownian motion models the random motion of particles suspended in a fluid (liquid or gas) resulting from collisions with molecules
• Named after botanist Robert Brown who first observed this phenomenon in 1827 while studying pollen grains in water
• Mathematically formalized by Albert Einstein in 1905 and Norbert Wiener in 1923
  • Einstein's work explained the motion using kinetic theory and provided a way to estimate Avogadro's number
  • Wiener's work laid the foundation for the rigorous mathematical treatment of Brownian motion as a stochastic process
• Key properties of Brownian motion include continuous sample paths, independent increments, and Gaussian distribution of displacements
• Serves as a fundamental model in various fields such as physics, chemistry, biology, and finance (stock price fluctuations)
• Generalizations of Brownian motion include fractional Brownian motion and multifractional Brownian motion
• Plays a crucial role in the development of stochastic calculus and the study of stochastic differential equations

Mathematical Foundations

• Brownian motion is a continuous-time stochastic process $\{B_t, t \geq 0\}$ with the following properties:
  • $B_0 = 0$ almost surely
  • Independent increments: for any $0 \leq t_1 < t_2 < \cdots < t_n$, the increments $B_{t_2} - B_{t_1}, B_{t_3} - B_{t_2}, \ldots, B_{t_n} - B_{t_{n-1}}$ are independent random variables
  • Stationary increments: the distribution of $B_{t+h} - B_t$ depends only on $h$ and not on $t$
  • Gaussian increments: for any $t > 0$ and $h > 0$, $B_{t+h} - B_t$ is normally distributed with mean 0 and variance $h$
  • Continuous sample paths: $B_t$ is continuous in $t$ almost surely
• The covariance function of Brownian motion is given by $\mathbb{E}[B_s B_t] = \min(s, t)$ for $s, t \geq 0$
• Brownian motion is a martingale with respect to its natural filtration
• The quadratic variation of Brownian motion over the interval $[0, t]$ is equal to $t$ almost surely
• Brownian motion satisfies the strong Markov property: for any stopping time $\tau$, the process $\{B_{\tau+t} - B_\tau, t \geq 0\}$ is a Brownian motion independent of the filtration $\mathcal{F}_\tau$
• The reflection principle relates the distribution of the maximum of Brownian motion to its distribution at a fixed time

Brownian Motion in Potential Theory

• Potential theory studies the behavior of harmonic functions and related objects, such as Green's functions and Poisson kernels
• Brownian motion is intimately connected to potential theory through the Dirichlet problem and the concept of harmonic measure
• The Dirichlet problem seeks to find a harmonic function in a domain $D$ with prescribed boundary values on $\partial D$
  • Probabilistic solution: the solution at a point $x \in D$ is the expected value of the boundary function at the first hitting point of Brownian motion started at $x$
• Harmonic measure $\omega_x$ at a point $x \in D$ is a probability measure on $\partial D$ that describes the distribution of the first hitting point of Brownian motion started at $x$
  • Harmonic measure solves the Dirichlet problem with boundary values given by the indicator function of a subset of $\partial D$
• The Poisson kernel $P_D(x, y)$ is the density of the harmonic measure with respect to the surface measure on $\partial D$
• Green's function $G_D(x, y)$ is the expected time spent by Brownian motion at $y$ before exiting $D$ when started at $x$
  • Related to the fundamental solution of the Laplace equation with Dirichlet boundary conditions
• Brownian motion can be used to prove the existence and uniqueness of solutions to the Dirichlet problem under mild regularity assumptions on the domain and boundary function

Key Concepts and Definitions

• Filtration: an increasing sequence of $\sigma$-algebras $\{\mathcal{F}_t, t \geq 0\}$ representing the information available up to time $t$
• Stopping time: a random variable $\tau$ such that the event $\{\tau \leq t\}$ is in $\mathcal{F}_t$ for all $t \geq 0$
• Martingale: a stochastic process $\{M_t, t \geq 0\}$ adapted to a filtration $\{\mathcal{F}_t, t \geq 0\}$ such that $\mathbb{E}[|M_t|] < \infty$ and $\mathbb{E}[M_t | \mathcal{F}_s] = M_s$ for all $s \leq t$
• Quadratic variation: for a stochastic process $\{X_t, t \geq 0\}$, the quadratic variation over the interval $[0, t]$ is defined as the limit in probability of $\sum_{i=1}^n (X_{t_i} - X_{t_{i-1}})^2$ for any sequence of partitions $0 = t_0 < t_1 < \cdots < t_n = t$ with mesh size tending to zero
• Harmonic function: a twice continuously differentiable function $u$ satisfying the Laplace equation $\Delta u = 0$
• Green's function: a fundamental solution of the Laplace equation with Dirichlet boundary conditions
  • For a domain $D$, the Green's function $G_D(x, y)$ satisfies $\Delta_y G_D(x, y) = -\delta_x(y)$ for $y \in D$ and $G_D(x, y) = 0$ for $y \in \partial D$
• Poisson kernel: the density of the harmonic measure with respect to the surface measure on the boundary of a domain

Applications in Physics and Finance

• Diffusion processes: Brownian motion is used to model the diffusion of particles in a medium, such as the motion of molecules in a gas or the spread of heat in a solid
  • The diffusion equation $\frac{\partial u}{\partial t} = D \Delta u$ describes the evolution of the concentration $u(x, t)$ of particles, with $D$ being the diffusion coefficient
  • The fundamental solution of the diffusion equation is the probability density function of Brownian motion with variance $2Dt$
• Fluctuation-dissipation theorem: relates the response of a system to an external perturbation to the fluctuations of the system at equilibrium
  • For example, the mobility of a Brownian particle (response to an external force) is proportional to its diffusion coefficient (fluctuations at equilibrium)
• Financial mathematics: Brownian motion is the building block for many models of asset price dynamics, such as the Black-Scholes model for option pricing
  • The geometric Brownian motion $S_t = S_0 \exp(\mu t + \sigma B_t)$ is used to model the evolution of stock prices, with $\mu$ being the drift (average return) and $\sigma$ the volatility
  • The Black-Scholes formula for the price of a European call option is derived using the properties of geometric Brownian motion and the principle of risk-neutral valuation
• Random walks: Brownian motion can be approximated by random walks on a lattice, where a particle moves to a neighboring site with equal probability at each time step
  • The central limit theorem ensures that the scaling limit of a random walk converges to Brownian motion under appropriate conditions
  • Random walks are used in various applications, such as polymer physics (self-avoiding walks) and statistical physics (spin systems)

Analytical Techniques

• Stochastic calculus: extends the concepts of calculus to stochastic processes, such as Brownian motion
  • The Itô integral $\int_0^t f(s) dB_s$ is defined as the limit in probability of Riemann sums $\sum_{i=1}^n f(t_{i-1}) (B_{t_i} - B_{t_{i-1}})$ for a suitable class of integrands $f$
  • Itô's lemma provides a change of variables formula for functions of stochastic processes: for a twice continuously differentiable function $f(t, x)$, the process $Y_t = f(t, B_t)$ satisfies $dY_t = \frac{\partial f}{\partial t}(t, B_t) dt + \frac{\partial f}{\partial x}(t, B_t) dB_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2}(t, B_t) dt$
• Stochastic differential equations (SDEs): differential equations driven by Brownian motion or more general stochastic processes
  • An SDE of the form $dX_t = b(t, X_t) dt + \sigma(t, X_t) dB_t$ describes the evolution of a stochastic process $X_t$, with $b$ being the drift coefficient and $\sigma$ the diffusion coefficient
  • The solution of an SDE is a stochastic process that satisfies the integral equation $X_t = X_0 + \int_0^t b(s, X_s) ds + \int_0^t \sigma(s, X_s) dB_s$
  • Existence and uniqueness of solutions to SDEs can be proven under Lipschitz and linear growth conditions on the coefficients
• Feynman-Kac formula: establishes a connection between SDEs and partial differential equations (PDEs)
  • For a suitable function $f(x)$ and a stochastic process $X_t$ satisfying the SDE $dX_t = b(X_t) dt + \sigma(X_t) dB_t$, the function $u(t, x) = \mathbb{E}[f(X_T) | X_t = x]$ solves the PDE $\frac{\partial u}{\partial t} + b(x) \frac{\partial u}{\partial x} + \frac{1}{2} \sigma^2(x) \frac{\partial^2 u}{\partial x^2} = 0$ with terminal condition $u(T, x) = f(x)$
  • The Feynman-Kac formula provides a probabilistic representation for the solution of certain parabolic PDEs and is used in option pricing and other applications

Computational Methods

• Monte Carlo simulation: a method for estimating the expectation of a function of a stochastic process by generating multiple independent realizations of the process
  • For Brownian motion, sample paths can be generated using the properties of independent and stationary Gaussian increments
  • The expectation of a function $f(B_T)$ can be approximated by the sample average $\frac{1}{N} \sum_{i=1}^N f(B_T^{(i)})$, where $B_T^{(i)}$ are independent realizations of Brownian motion at time $T$
  • Variance reduction techniques, such as antithetic variates and control variates, can be used to improve the efficiency of Monte Carlo simulations
• Discrete approximations: Brownian motion can be approximated by discrete-time processes, such as the random walk or the binomial tree model
  • The Euler-Maruyama scheme approximates the solution of an SDE $dX_t = b(t, X_t) dt + \sigma(t, X_t) dB_t$ by the discrete process $X_{t_{i+1}} = X_{t_i} + b(t_i, X_{t_i}) \Delta t + \sigma(t_i, X_{t_i}) \sqrt{\Delta t} Z_i$, where $Z_i$ are independent standard normal random variables and $\Delta t$ is the time step
  • The convergence of discrete approximations to Brownian motion or the solution of an SDE can be analyzed using the concept of weak or strong convergence of stochastic processes
• Numerical methods for PDEs: the Feynman-Kac formula allows for the numerical solution of certain parabolic PDEs using probabilistic methods
  • The value of the solution $u(t, x)$ at a point $(t, x)$ can be estimated by simulating the stochastic process $X_s$ starting from $X_t = x$ and computing the expectation of $f(X_T)$
  • This approach can be combined with traditional numerical methods for PDEs, such as finite differences or finite elements, to solve problems in higher dimensions or with more complex boundary conditions

Advanced Topics and Current Research

• Stochastic partial differential equations (SPDEs): PDEs driven by space-time white noise or other random fields
  • The stochastic heat equation $\frac{\partial u}{\partial t} = \Delta u + \dot{W}$, where $\dot{W}$ is space-time white noise, models the evolution of a random field subject to random fluctuations
  • SPDEs arise in various applications, such as turbulence, image processing, and mathematical finance
  • The theory of SPDEs requires the development of new analytical and numerical tools, such as the concept of mild solutions and the use of Malliavin calculus
• Rough paths: a framework for extending the concepts of stochastic calculus to processes with lower regularity than Brownian motion
  • A rough path is a pair $(X, \mathbb{X})$ consisting of a path $X$ with values in a Banach space and a second-order process $\mathbb{X}$ that controls the quadratic variation of $X$
  • The theory of rough paths allows for the construction of integrals and solutions to differential equations driven by processes such as fractional Brownian motion or Gaussian rough paths
  • Rough paths have applications in stochastic analysis, mathematical finance, and machine learning (signature methods)
• Malliavin calculus: a variational calculus for functionals of stochastic processes, such as Brownian motion
  • The Malliavin derivative $D_t F$ of a functional $F$ measures the sensitivity of $F$ to perturbations of the underlying stochastic process at time $t$
  • The Malliavin derivative satisfies a chain rule and an integration by parts formula, which allows for the computation of densities and the proof of regularity properties of solutions to SDEs
  • Malliavin calculus has applications in mathematical finance (sensitivity analysis and Greeks computation), stochastic control, and the analysis of SPDEs
• Stochastic homogenization: the study of the effective behavior of differential equations with rapidly oscillating random coefficients
  • For example, the solution of the elliptic PDE $-\nabla \cdot (a(\frac{x}{\varepsilon}, \omega) \nabla u^\varepsilon) = f$ with random coefficient matrix $a(y, \omega)$ converges as $\varepsilon \to 0$ to the solution of a homogenized PDE $-\nabla \cdot (a^* \nabla u) = f$ with constant effective coefficient matrix $a^*$
  • Stochastic homogenization can be analyzed using probabilistic methods, such as the invariance principle for random walks in random environments or the ergodic theorem for stationary random fields
  • Applications include the study of heat and mass transfer in heterogeneous media, the modeling of composite materials, and the analysis of porous media flow