One-dimensional Wiener process with the properties of partial reflection and delay

In this paper, we construct the two-parameter semigroup of operators associated with a certain one-dimensional inhomogeneous diffusion process and study its properties. We are interested in the process on the real line which can be described as follows. At the interior points of the half-lines separated by a point, the position of which depends on the time variable, this process coincides with the Wiener process given there and its behavior on the common boundary of these half-lines is determined by a kind of the conjugation condition of Feller-Wentzell's type. The conjugation condition we consider is local and contains only the first-order derivatives of the unknown function with respect to each of its variables. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding conjugation problem for a second order linear parabolic equation to which the above problem is reduced. Its classical solvability is obtained by the boundary integral equations method under the assumption that the initial function is bounded and continuous on the whole real line, the parameters characterizing the Feller-Wentzell conjugation condition are continuous functions of the time variable, and the curve defining the common boundary of the domains is determined by the function which is continuously differentiable and its derivative satisfies the Hölder condition with exponent less than $1/2$.

Bayesian inversion of a diffusion model with application to biology

A common task in experimental sciences is to fit mathematical models to real-world measurements to improve understanding of natural phenomenon (reverse-engineering or inverse modelling). When complex dynamical systems are considered, such as partial differential equations, this task may become challenging or ill-posed. In this work, a linear parabolic equation is considered as a model for protein transcription from MRNA. The objective is to estimate jointly the differential operator coefficients, namely the rates of diffusion and self-regulation, as well as a functional source. The recent Bayesian methodology for infinite dimensional inverse problems is applied, providing a unique posterior distribution on the parameter space continuous in the data. This posterior is then summarized using a Maximum a Posteriori estimator. Finally, the theoretical solution is illustrated using a state-of-the-art MCMC algorithm adapted to this non-Gaussian setting.

Time and space meshes adaptivity for the resolution of a semi-linear heat equation

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on $ {\mathbb{R}}^N $

<p style='text-indent:20px;'>In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of <inline-formula><tex-math id="M2">\begin{document}$ L^2( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> space. We establish the Wong-Zakai approximations of solutions in <inline-formula><tex-math id="M3">\begin{document}$ L^l( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> for arbitrary <inline-formula><tex-math id="M4">\begin{document}$ l\geq q $\end{document}</tex-math></inline-formula> in the sense of upper semi-continuity of their random attractors, where <inline-formula><tex-math id="M5">\begin{document}$ q $\end{document}</tex-math></inline-formula> is the growth exponent of the nonlinearity. The <inline-formula><tex-math id="M6">\begin{document}$ L^l $\end{document}</tex-math></inline-formula>-pre-compactness of attractors is proved by using the truncation estimate in <inline-formula><tex-math id="M7">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> and the higher-order bound of solutions.</p>