Theory of Stochastic Processes
A.A. 2024/25

Motivation

• Brownian Motion: Einstein and Langevin description.
• Birth-Death processes: Master equation.
• Shot Noise: Stochastic Differential Equation.

Stochastic Processes

• Definitions.
• Bernoulli Trials.
• Markov Processes.
• The Chapman-Kolmogorov equation.
• Continuous Markov processes; examples: Brownian motion and Cauchy process.
• Differential Chapman-Kolmogorov equation.
• The master equation: jump processes.
• The Liouville equation: deterministic processes.
• The Fokker-Planck equation: diffusive processes.

Fokker-Planck Equation

• Backward differential Chapman-Kolmogorov equation.
• Forward Kramers-Moyal equation.
• Stationary and homogeneous Markov Processes.
• The Wiener Process.
• The Ornstein-Uhlembeck proceess.
• The method of characteristics for quasilinear partial differential equations.
• Probability current and boundary conditions for the Fokker-Planck equation.
• Stationary solution for the one-dimensional homegeneous Fokker-Planck equation: Potential solutions.
• Stationary solution in a gravitational field.
• Stationary solution of the Ornstein-Uhlembeck process.
• Stationary solution for the Brownian motion.
• The eigenvalue method: Wiener process with absorbing and reflectin barriers.
• First Passage Time for homogeneous processes.
• Escape over a potential barrier, the Arrhenius formula.

Stochastic Differential Equations

• The Langevin equation and the Wiener process.
• The stochastic integration: the Ito Integral.
• Nonanticipating functions.
• dW(t)2 and dW(t)2+N.
• The Stratonovich Integral.
• The Ito differentiation rule.
• The Dirac delta function and the Heaviside theta function.
• Numerical integration of SDE: Stochastic second order Runge-Kutta Algorithm.
• The Ito formula.
• Ito SDE and Fokker-Planck equation.
• Ito and Stratonovich SDE.
• Stratonovich SDE and Fokker-Planck equation.
• Example: SDE with linear multiplicative noise.
• Example: Linear oscillator with random frequency.
• Formalismo di Martin-Siggia-Rose

Path Integral Methods for Stochastic Differential Equations

• Path Integral formulation for additive white noise Langevin equations.
• The Martin-Siggia-Rose-Janssen-de Domicis (MSRJD) Action.
• The Onsager-Machlup-Graham-Bausch-Wegner (OMGBW) Action.
• Derivation of the Fokker-Planck equation from the path integral.
• Response functions and response fields.
• Quadratic MSRJD action.
• Example: The one dimensional Ornstein-Uhlembeck process.
• Path Integral formulation for multiplicative white noise Langevin equations.
• From Path Integral to Fokker-Planck equation for multiplicative white noise processes: state dependent diffusion.
• Thermodynamic equilibrium conditions for multiplicative white noise processes.
• Fluctuation-Dissipation Theorem for additive white noise processes.
• Detailed Balance condition.
• Fluctuation-Dissipation Theorem for multiplicative white noise processes.

Dynamical Field Theory

• The Novikov and Wick theorems.
• Perturbation theory in the nonlinearity (interaction).
• Perturbative evaluation of (equilibrium) two points functions.
• The Dyson equation.
• The Dyson equation for the zero-dimensional λϕ4 model: first order calculation.
• Perturbation theory in the non linearity with non zero external field.
• Dynamical Effective potential Γ[φ,φ̂].
• Dynamical Effective potential for the Ornstein-Uhlembeck process.
• Loop expansion of the Dynamical Effective potential.
• Dynamical Effective potential for the zero-dimensional λϕ4 model: zero and one loop calculation.
• Conditional Probability Function: the φ̂ ≠ 0 solution.
• Dynamical Effective Potential and Conditional Probability Function.
• Arrhenius Law.
• Quantum averages over the Close Time Path (CTP) contour.
• Bosonic Coherent States.
• CTP calculation of Z[0] for the bosonic system Ĥ = ω₀ b̂⁺ b̂.
• CTP Green function for the bosonic system Ĥ = ω₀ b̂⁺ b̂.
• The Keldysh formalism.
• Quantum Harmonic Oscillator.

Dynamical Field Theory for disordered systems.

• Quenched disorder average in statics and dynamics.
• Random non-symmetric two-body interaction neural network model: averaged dynamical generating function.
• The spherical p-spin and 2+p-spin spin-glass models: averaged dynamical generating function and effective dynamics.
• Equilibrium dynamics of the spherical 2+p-spin spin-glass models.
• Ergodic dynamics of the 2+3-spin spin-glass model: critical slowing down and continuous transition. ^*
• Ergodic dynamics of the 2+3-spin spin-glass model: dynamical arrest and discontinuous transition. ^*
• Non-ergodic dynamcis of the 2+3-spin spin-glass model: two-steps relaxation, dynamical marginal condition and 1RSB phase. ^*
• Path-Integral approach to a fully connected neural network model with random independent asymmetric couplings. ^*