Peter M. Kotelenez

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

SummaryA comparison theorem is derived for a class of function valued stochastic partial differential equations (SPDEs) 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 SPDEs where the sufficiency follows from the comparison theorem. Thereby it is, e.g., possible to identify a class of SPDEs, which can serve as stochastic space-time models for the density of particles. As a consequence we can construct unique mild solutions of SPDEs 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.

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

SummaryA system ofN particles inRd 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 inL2(Rd,dr), wheredr is the Lebesgue measure, the solution will have densities inL2(Rd,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDEs.

High density limit theorems for nonlinear chemical reactions with diffusion

SummaryThe solutionX of a nonlinear reaction-diffusion equation on then-dimensional unit cubeS is approximated by a space-time jump Markov processXv,N (law of large numbers (LLN)).Xv,N is constructed on a gridSN onS ofN cells, wherev is proportional to the initial number of particles in each cell. The deviation ofXv,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:

Conservation of Total Vorticity for a 2D Stochastic Navier Stokes Equation

\frac{N}{\upsilon } \to 0

Itô and Stratonovich stochastic partial differential equations: Transition from microscopic to macroscopic equations

, asN → ∞. The limitY =YX 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 ofYX in dependence ofX is also investigated.

A Comparison Theorem for a Class of Stochastic Partial Differential Equations

The derivation of quasilinear stochastic partial differential equations (SPDE’s) for mass distributions and their generalizations is reviewed in Section 2. Special emphasis is given to the vorticity distribution and its macroscopic limit in a 2D-fluid. In Section 4 and 5 bilinear SPDE’s on weighted Hilbert spaces are derived from the underlying particle system. Moreover, it is shown that spatially homogeneous initial conditions imply that the solution is also spatially homogeneous. I.e., (non-Gaussian!!) homogeneous random fields are derived from an underlying particle system using SPDE methods.

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

We consider 𝑁 point vortices whose positions satisfy a stochastic ordinary differential equation on ℝ2𝑁 perturbed by spatially correlated Brownian noise. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) with a state-dependent stochastic term. As the number of vortices tends to infinity, we obtain a smooth solution to the SNSE, and we prove the conservation of total vorticity in this continuum limit.

Fractional step method for stochastic evolution equations

Solutions of quasi-linear stochastic Fokker–Planck equations for the number density of a system of solute particles in suspension are derived. The initial values and the solutions take values in a class of σ-finite Borel measures over Rd where d ≥ 1. The stochastic driving noise is defined by Ito differentials. For the special case of semi-linear stochastic Fokker–Planck equations, the solutions can be represented as solutions of first-order stochastic transport equations driven by Stratonovich differentials.