论文标题
从属原理和Feynman-KAC公式
Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations
论文作者
论文摘要
我们考虑了$$ u（t）= u_0+\ int_0^tk（t，s）lu（s）lu（s）ds $$的一般时间 - 分数进化方程，带有相当通用的内存元素$ k $和操作员$ l $是强烈连续的连续半元组的发电机。尤其是，$ l $可能是某些状态空间$ q $的Markov Process $ L_0 $的发电机$ L_0 $，或$ l：= l_0+b \ nabla+v $，用于合适的潜在$ v $和漂移$ v $和drift $ b $，或$ l $ l $生成的下属半群或schrödinger类型组。这类进化方程在特定的时间和空间分数中以及schrödinger型方程。我们表明，下属原理适用于此类演化方程，并通过使用不同随机过程（例如下属马尔可夫过程和随机缩放的高斯过程）来获取这些方程解决方案的Feynman-kac公式。特别是，我们获得了一些具有广义的灰色布朗运动和其他相关的自相似过程的Feynman-kac公式，并具有固定的增量。
We consider generalized time-fractional evolution equations of the form $$u(t)=u_0+\int_0^tk(t,s)Lu(s)ds$$ with a fairly general memory kernel $k$ and an operator $L$ being the generator of a strongly continuous semigroup. In particular, $L$ may be the generator $L_0$ of a Markov process $ξ$ on some state space $Q$, or $L:=L_0+b\nabla+V$ for a suitable potential $V$ and drift $b$, or $L$ generating subordinate semigroups or Schrödinger type groups. This class of evolution equations includes in particular time- and space- fractional heat and Schrödinger type equations. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.
