Existence of the Fomin derivative of the invariant measure of a stochastic reaction--diffusion equation

Estimate for $P_tD$ for the stochastic Burgers equation

We consider the Burgers equation on $H=L^2(0,1)$ perturbed by white noise and the corresponding transition semigroup $P_t$. We prove a new formula for $P_tDφ$ (where $φ:H\to\R$ is bounded and Borel) which depends on $φ$ but not on its derivative. Then we deduce some new consequences for the invariant measure $ν$ of $P_t$ as its Fomin differentiability and an integration by parts formula which generalises the classical one for gaussian measures.

An integral inequality for the invariant measure of some finite dimensional stochastic differential equation

We prove an integral inequality for the invariant measure $ν$ of a stochastic differential equation with additive noise in a finite dimensional space $H=\R^d$. As a consequence, we show that there exists the Fomin derivative of $ν$ in any direction $z\in H$ and that it is given by $v_z=\langle D\logρ,z\rangle$, where $ρ$ is the density of $ν$ with respect to the Lebesgue measure. Moreover, we prove that $v_z\in L^p(H,ν)$ for any $p\in[1,\infty)$. Also we study some properties of the gradient operator in $L^p(H,ν)$ and of his adjoint.

Absolutely continuous solutions for continuity equations in Hilbert spaces

We prove existence of solutions to continuity equations in a separable Hilbert space. We look for solutions which are absolutely continuous with respect to a reference measure γwhich is Fomin-differentiable with exponentially integrable partial logarithmic derivatives. We describe a class of examples to which our result applies and for which we can prove also uniqueness. Finally, we consider the case where γis the invariant measure of a reaction-diffusion equation and prove uniqueness of solutions in this case. We exploit that the gradient operator D_x is closable with respect to L^p(H,γ) and …

Solutions of fractional reaction-diffusion equations in terms of the H-function

This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2…

On the Validity of the Girsanov Transformation Method for Sensitivity Analysis of Stochastic Chemical Reaction Networks

We investigate the validity of the Girsanov Transformation (GT) method for parametric sensitivity analysis of stochastic models of chemical reaction networks. The validity depends on the likelihood ratio process being a martingale and the commutation of a certain derivative with expectation. We derive some exponential integrability conditions which imply both these requirements. We provide further conditions in terms of a reaction network that imply these exponential integrability conditions.

随机反应网络中多个同时参数扰动的通用灵敏度方法

A general-purpose sensitivity method for multiple simultaneous parameter perturbations in stochastic reaction networks

针对随机反应网络多参数扰动灵敏度估计的效率问题，提出多路径堆叠耦合（MSC）方法，经实验验证其在三类应用中均能实现最小均方根误差，性能优于现有方法。