2019/04/09 by Marco Rehmeier, Rehmeier, Marco
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #35Q84 #60J60 #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1904.04756
openalex publication_date 2019/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let the coefficients aij and bi, i,j \≤ d, of the linear\nFokker-Planck-Kolmogorov equation (FPK-eq.)\n
partialt
mut =
partiali
partialj(aij
mut)-
partiali(bi
mut)\nbe Borel measurable, bounded and continuous in space. Assume that for every s\n\∈ [0,T] and every Borel probability measure \ν on \ℝd there is\nat least one solution \μ = (\μt)t \∈ [s,T] to the FPK-eq. such that\n\μs = \ν and t \↦ \μt is continuous w.r.t. the topology of weak\nconvergence of measures. We prove that in this situation, one can always select\none solution \μs,\ν for each pair (s,\ν) such that this family of\nsolutions fulfills\n
mus,
nut =
mu^r,
mus,
nurt
text for all 0
leq s
leq r\n
leq t
leq T,which one interprets as a flow property of this solution\nfamily. Moreover, we prove that such a flow of solutions is unqiue if and only\nif the FPK-eq. is well-posed.\n