Abstract
We consider the new class of the Markov measure‐valued stochastic processes with constant mass. We give the construction of such processes with the family of the probabilities which describe the motion of single particles. We also consider examples related to stochastic flows with the interactions and the local times for such processes.
Highlights
This paper is devoted to one type of the Markov measurevalued processes with constant mass
The behavior of the trajectory which starts from a certain point depends on the whole mass distribution on the space. These processes are different from the well-known Fleming-Viot processes with constant mass
Our measure-valued process will be constructed with the help of the following stochastic differential equation: dx(u, t) = a x(u, t), μt dt + b x(u, t), μt dw(t), x(u, 0) = u, u ∈ Rd, μt = μ0 · x(·, t)−1
Summary
This paper is devoted to one type of the Markov measurevalued processes with constant mass. The situation changes if the initial mass distribution is not the discrete measure In this example, the trajectory of the single particle is the Itô process, which satisfies certain stochastic differential equation. The Markov process {μt; t ≥ 0} with respect to {Ᏺt; t ≥ 0} in the space (M, γ) is called evolutionary if for every initial state μ0 ∈ M, there exists the measurable function f : Ω × [0, +∞) × X → X (2.3). Consider φ which is an X-valued random process on X correspondent to the family {R} It follows from (2.7) that φ has a measurable modification. (3) there exists C > 0 for all u1, . . . , um ∈ X and all μ1, μ2 ∈ M such that γm Q μ, u1, . . . , um , Q μ, u1, . . . , um ≤ Cγ μ1, μ2
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