Introduction to Stochastic Calculus with Applications

“…85,86 Roughly speaking, a martingale is a stochastic process M (t) whose expectation value is constant in time: ∂ t E[M (t)] = 0. For this reason we call consequences of Eq.…”

Section: Martingale Conditions On Partition Functionsmentioning

confidence: 99%

“…Now we set t = 0 in (78). It is then straightforward to use the stochastic equations (61) and (70) and Ito's formula 85,86 to transform this equation into a Fokker-Planck equation:…”

Section: Martingale Conditions On Partition Functionsmentioning

confidence: 99%

Quantum Hall transitions: An exact theory based on conformal restriction

2012

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We revisit the problem of the plateau transition in the integer quantum Hall effect. Here we develop an analytical approach for this transition, and for other two-dimensional disordered systems, based on the theory of "conformal restriction". This is a mathematical theory that was recently developed within the context of the Schramm-Loewner evolution which describes the "stochastic geometry" of fractal curves and other stochastic geometrical fractal objects in two-dimensional space. Observables elucidating the connection with the plateau transition include the so-called point-contact conductances (PCCs) between points on the boundary of the sample, described within the language of the Chalker-Coddington network model for the transition. We show that the disorder-averaged PCCs are characterized by a classical probability distribution for certain geometric objects in the plane (which we call pictures), occurring with positive statistical weights, that satisfy the crucial so-called restriction property with respect to changes in the shape of the sample with absorbing boundaries; physically, these are boundaries connected to ideal leads. At the transition point, these geometrical objects (pictures) become fractals. Upon combining this restriction property with the expected conformal invariance at the transition point, we employ the mathematical theory of "conformal restriction measures" to relate the disorder-averaged PCCs to correlation functions of (Virasoro) primary operators in a conformal field theory (of central charge c = 0). We show how this can be used to calculate these functions in a number of geometries with various boundary conditions. Since our results employ only the conformal restriction property, they are equally applicable to a number of other critical disordered electronic systems in two spatial dimensions, including for example the spin quantum Hall effect, the thermal metal phase in symmetry class D, and classical diffusion in two dimensions in a perpendicular magnetic field. For most of these systems, we also predict exact values of critical exponents related to the spatial behavior of various disorder-averaged PCCs.

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“…where the drift [12]). A well-known method for solving the boundary crossing problem for diffusion processes is to express them as functions of a Brownian motion, and the boundary crossing probability for diffusion processes is equivalent to a boundary crossing probability for the Brownian motion with transformed time interval and boundaries.…”

Section: Dx(t) = B(t X(t)) Dt + σ (T X(t)) Dw (T)mentioning

confidence: 99%

Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

2010

J. Appl. Probab.

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We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

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Introduction to Stochastic Calculus with Applications

Cited by 422 publications

References 0 publications

Approximate Solution of the Volterra Random Integral Equations

Approximate Solution of the Volterra Random Integral Equations

Quantum Hall transitions: An exact theory based on conformal restriction

Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

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