Theoretical analysis demonstrates weak solutions to conservation laws using nonlinear stochastic differential equations, suggesting a rigorous probabilistic basis for sticky particle dynamics.
This work presents a construction of a solution for the nonlinear stochastic differential equation Xₜ= X₀+ ∫₀ᵗ E[u₀(X₀)|Xₛ]ds, t ≥ 0. The random variable X₀ with values in R and the function u₀ are given. We denote by Pₜ the probability distribution of Xₜ and u(x,t) = E[u₀(X₀)|Xₜ= x]. We prove that (Pₜ,u(·,t), t≥ 0) is a weak solution for a system of conservation laws arising in adhesion particle dynamics.
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Azzouz Dermoune (1999) studied this question.
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