Journal of Evolution Equations ( IF 1.1 ) Pub Date : 2020-03-25 , DOI: 10.1007/s00028-020-00569-y Marco Rehmeier
Let the coefficients \(a_{ij}\) and \(b_i\), \(i,j \le d\), of the linear Fokker–Planck–Kolmogorov equation (FPK-eq.)
$$\begin{aligned} \partial _t\mu _t = \partial _i\partial _j(a_{ij}\mu _t)-\partial _i(b_i\mu _t) \end{aligned}$$be Borel measurable, bounded and continuous in space. Assume that for every \(s \in [0,T]\) and every Borel probability measure \(\nu \) on \(\mathbb {R}^d\) there is at least one solution \(\mu = (\mu _t)_{t \in [s,T]}\) to the FPK-eq. such that \(\mu _s = \nu \) and \(t \mapsto \mu _t\) is continuous w.r.t. the topology of weak convergence of measures. We prove that in this situation, one can always select one solution \(\mu ^{s,\nu }\) for each pair \((s,\nu )\) such that this family of solutions fulfills
$$\begin{aligned} \mu ^{s,\nu }_t = \mu ^{r,\mu ^{s,\nu }_r}_t \text { for all }0 \le s \le r \le t \le T, \end{aligned}$$which one interprets as a flow property of this solution family. Moreover, we prove that such a flow of solutions is unique if and only if the FPK-eq. is well-posed.
中文翻译：
线性Fokker-Planck-Kolmogorov方程流的存在及其与适定性的关系
令线性Fokker-Planck-Kolmogorov方程（FPK-eq。）的系数\（a_ {ij} \）和\（b_i \），\（i，j \ le d \）
$$ \ begin {aligned} \ partial _t \ mu _t = \ partial _i \ partial _j（a_ {ij} \ mu _t）-\ partial _i（b_i \ mu _t）\ end {aligned} $$Borel在空间上可测量，有界且连续。假设对于每\（S \在[0，T] \）和每一个的Borel概率测度\（\ NU \）上\（\ mathbb {R} ^ d \）有至少一个溶液\（\亩= （\ mu _t）_ {t \ in [s，T]} \）到FPK-eq。使得\（\ mu _s = \ nu \）和\（t \ mapsto \ mu _t \）在度量弱收敛的拓扑结构上是连续的。我们证明，在这种情况下，总是可以为每对\ {{s，\ nu} \}选择一个解决方案\ {\ mu ^ {s，\ nu} \）使得该系列解决方案能够满足
$$ \ begin {aligned} \ mu ^ {s，\ nu} _t = \ mu ^ {r，\ mu ^ {s，\ nu} _r} _t \ text {对于所有} 0 \ le s \ le r \ le t \ le T，\ end {aligned} $$它被解释为该解决方案系列的流量属性。而且，我们证明，只有当FPK-eq时，这种解决方案流才是唯一的。状况良好。
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