By Sidney Redner
First-passage homes underlie quite a lot of stochastic strategies, corresponding to diffusion-limited progress, neuron firing, and the triggering of inventory innovations. This ebook offers a unified presentation of first-passage procedures, which highlights its interrelations with electrostatics and the ensuing strong results. the writer starts with a contemporary presentation of primary idea together with the relationship among the profession and first-passage chances of a random stroll, and the relationship to electrostatics and present flows in resistor networks. the results of this idea are then built for easy, illustrative geometries together with the finite and semi-infinite periods, fractal networks, round geometries and the wedge. numerous functions are offered together with neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin structures, and the kinetics of diffusion-controlled reactions. Examples mentioned comprise neuron dynamics, self-organized criticality, kinetics of spin platforms, and stochastic resonance.
Read or Download A Guide to First-Passage Processes PDF
Best stochastic modeling books
Selected Topics in Integral Geometry: 220
The miracle of critical geometry is that it's always attainable to get well a functionality on a manifold simply from the information of its integrals over sure submanifolds. The founding instance is the Radon remodel, brought in the beginning of the 20 th century. due to the fact then, many different transforms have been stumbled on, and the overall concept was once constructed.
Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation
The foremost thrust of this booklet is the research of pointwise habit of Sobolev services of integer order and BV features (functions whose partial derivatives are measures with finite overall variation). the improvement of Sobolev capabilities contains an research in their continuity houses when it comes to Lebesgue issues, approximate continuity, and high quality continuity in addition to a dialogue in their larger order regularity houses by way of Lp-derivatives.
With contributions through prime mathematicians, this complaints quantity displays this system of the 8th foreign convention on $p$-adic practical research held at Blaise Pascal college (Clemont-Ferrand, France). Articles within the ebook supply a accomplished evaluation of analysis within the zone. a variety of subject matters are lined, together with easy ultrametric useful research, topological vector areas, degree and integration, Choquet idea, Banach and topological algebras, analytic features (in specific, in reference to algebraic geometry), roots of rational features and Frobenius constitution in $p$-adic differential equations, and $q$-ultrametric calculus.
Elements of Stochastic Modelling
This is often the improved moment version of a winning textbook that offers a huge creation to big parts of stochastic modelling. the unique textual content used to be constructed from lecture notes for a one-semester direction for third-year technology and actuarial scholars on the college of Melbourne. It reviewed the fundamentals of likelihood idea after which lined the next themes: Markov chains, Markov choice tactics, bounce Markov tactics, parts of queueing idea, easy renewal concept, components of time sequence and simulation.
- Random Matrices and Their Applications
- Simulated Annealing [math]
- Mathematics of Two-Dimensional Turbulence
- Convexity
Additional resources for A Guide to First-Passage Processes
Sample text
Suppose that the boundary sites in B+ are fixed at unit potential while the sites in B_ are grounded. The net current at each interior site i of the network must be zero, as there is current input and output only at the boundary. This current conservation condition is E g, (V, j — V 1 ) = 0. 1 ) where g, 1 is the conductance of the bond between sites i and j, V, is the voltage at site i, and the sum pans over the nearest neighbors j of site i. Solving for Vi gives E . 2) where the last step applies for a homogeneous network.
9) is the first-passage probability to a given point, F s = is the probability of eventually hitting this p9int. 5. 4. 1 1} exist, then F (s) in Eq. 10) contains only the Taylor series terms. 12) Thus the Laplace transform is a moment generating function, as it contains all the positive integer moments of the probability distribution F(t). This is one of the reasons why the Laplace transform is such a useful tool for first-passage processes. In summary, the small-s behavior of the Laplace transform, or, equivalently, the z 1 behavior of the generating function, are sufficient to determine the long-time behavior of the function itself.
This crossover between the intermodiate-time power law and the long-time exponential decay can be formulated more generally by an approach similar to that given in Chap. 1 for determining the asymptotics of generating functions. Consider the situation in which the first-passage probability has the generic form J() = t —ce e —Eir with r >> 1, and where j(t) is vanishingly small for t << 1. The power law represents the asymptotic behavior of an infinite system, and the exponential factor represents a finite-size cutoff.