# An Introduction to Stochastic Processes with Applications to Biology, Second Edition

Diekmann, eds. Michelson, K. Ito, H. Tran and J. Leith, Stochastic models for subpopulation emergence in heterogeneous tumors ,, Bulletin of Mathematical Biology , 51 , Sinko and W. Streifer, A new model for age-size structure of a population ,, Ecology , 48 , Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Entropy dissipation of Fokker-Planck equations on graphs.

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Invariant measures for a stochastic Fokker-Planck equation. Optimization of a model Fokker-Planck equation. Marco Torregrossa , Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. On a relativistic Fokker-Planck equation in kinetic theory.

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Michael Herty , Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus.

### Nonlinear stochastic Markov processes and modeling uncertainty in populations

Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics , , 3 2 : Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Giuseppe Toscani.

A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Vladimir I. Bogachev , Stanislav V. Shaposhnikov , Alexander Yu. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. John W. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers.

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Electronic Research Announcements , , Linghua Chen , Espen R. Joseph G. Simon Plazotta. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations.

Krunal B. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators.

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## An introduction to stochastic processes with applications to biology - CERN Document Server

Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. American Institute of Mathematical Sciences. We consider an alternative approach to the use of nonlinear stochastic Markov processes which have a Fokker-Planck or Forward Kolmogorov representation for density in modeling uncertainty in populations. These alternate formulations, which involve imposing probabilistic structures on a family of deterministic dynamical systems, are shown to yield pointwise equivalent population densities.

Moreover, these alternate formulations lead to fast efficient calculations in inverse problems as well as in forward simulations. Here we derive a class of stochastic formulations for which such an alternate representation is readily found. Keywords: forward Kolmogorov , uncertainty , Fokker-Planck , probabilistic structures on deterministic systems , pointwise equivalence.

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## B5.1 Stochastic Modelling of Biological Processes (2017-2018)

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## Stochastic Processes: From Applications to Theory

We assume that the reader is already familiar with the basic motivations and notions of probability theory. We commence along the lines of the founding work of Kolmogorov by regarding stochastic processes as a family of random variables defined on a probability space and thereby define a probability law on the set of trajectories of the process. More specifically, stochastic processes generalize the notion of finite-dimensional vectors of random variables to the case of any family of random variables indexed in a general set T.

The financial industry is one of the most influential driving forces behind the research into stochastic processes. This is due to the fact that it relies on stochastic models for valuation and risk management. But perhaps more surprisingly, it was also one of the main drivers that led to their initial discovery.