Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Stochastic differential equations and diffusion processes

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

Measure-valued Markov processes

A comparison theorem is derived for a class of function valued stochastic partial differential equations (SPDE's) with Lipschitz coefficients driven by cylindrical and regular Hilbert space valued Brownian motions. Moreover, we obtain necessary and sufficient conditions for the positivity of the mild solutions of the SPDE's where the sufficiency follows from the comparison theorem. Thereby it is, e.g., possible to identify a class of SPDE's, which can serve as stochastic space-time models for the density of particles. As a consequence we can construct unique mild solutions of SPDE's on the cone of positive functions with non-Lipschitz drift parts including the case of arbitrary polynomialsR(x) withR(O)≧O and leading negative coefficient.

Comparison methods for a class of function valued stochastic partial differential equations

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita.

/pdf/macroscopic-limits-for-stochastic-partial-differential-5cp8jw1bb2.pdf

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

A system ofN particles inR

 d
 with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by “extending” the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL
2(R

 d
 ,dr), wheredr is the Lebesgue measure, the solution will have densities inL
2(R

 d
 ,dr) and strong uniqueness (in the Ito sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

The solutionX of a nonlinear reaction-diffusion equation on then-dimensional unit cubeS is approximated by a space-time jump Markov processX v,N (law of large numbers (LLN)).X v,N is constructed on a gridS N onS ofN cells, wherev is proportional to the initial number of particles in each cell. The deviation ofX v,N fromX is computed by a central limit theorem (CLT). The assumptions on the parametersv, N are for the LLN: υ → ∞, asN → ∞, and for the CLT:\(\frac{N}{\upsilon } \to 0\), asN → ∞. The limitY =Y X in the CLT, which is a generalized Ornstein-Uhlenbeck process, is represented as the mild solution of a linear stochastic partial differential equation (SPDE) and its best possible state spaces are described. The problem of stationary solutions ofY X in dependence ofX is also investigated.

Peter Kotelenez

Papers

Comparison methods for a class of function valued stochastic partial differential equations

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

High density limit theorems for nonlinear chemical reactions with diffusion

Fractional step method for stochastic evolution equations