Nonlinear Fokker-Planck equations driven by Gaussian linear\n multiplicative noise

2017/08/29 by Viorel Barbu, Michael Röckner, Barbu, Viorel +1
Economics, Econometrics and Finance · Physics and Astronomy · #47H05 #47J05 #60H15 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1708.08768

openalex publication_date 2017/08/29 · openalex created_date 2022/09/26 · openalex updated_date 2026/07/28

Abstract

Existence and uniqueness of a strong solution in H^-1( mathbb R^d) is\nproved for the stochastic nonlinear Fokker-Planck equation dX-
rm\ndiv(DX)dt-
Delta
beta(X)dt=X
,dW
mbox in (0,T)
times
mathbb R^d,
X(0)=x,\nvia a corresponding random differential equation. Here d\≥ 1, W is a\nWiener process in H^-1( mathbb R^d), D\∈ C¹( mathbb R^d, mathbb R^d)\nand \β is a continuous monotonically increasing function. The solution\nexists for x\∈ L¹\∩ L^\∞ and preserves positivity. If \β \∈\nL¹ _{rm loc}( mathbb R), the solution is pathwise Lipschitz continuous with\nrespect to initial data in H^-1( mathbb R^d). Stochastic Fokker-Planck\nequations with nonlinear drift of the form dX- rm\ndiv(a(X))dt-\Δ\β(X)dt=X ,dW are also considered for Lipschitzian\ncontinuous functions a: mathbb R\→ mathbb R^d.\n

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