One-dimensional stochastic heat equation with discontinuous conductance

Fractional stochastic heat equation with piecewise constant coefficients

“…An explicit expression of the fundamendal solution of equation (1.6) was given in [6,[22][23][24]. SPDEs with the operator L have been introduced and investigated in [25,26], with an additive noise W defined as a centered Gaussian field W = {W (t, C); t ∈ [0, T ], C ∈ B b (R)} with covariance E(W (t, C)W (s, D)) = (t ∧ s)λ(C ∩ D),…”

“…An explicit expression of the fundamendal solution of equation (1.6) was given in [6,[22][23][24]. SPDEs with the operator L have been introduced and investigated in [25,26], with an additive noise W defined as a centered Gaussian field W = {W (t, C); t ∈ [0, T ], C ∈ B b (R)} with covariance E(W (t, C)W (s, D)) = (t ∧ s)λ(C ∩ D), where λ denotes the Lebesgue measure. Comparing to those two articles, the equation (1.2) contains more complicated noise, which is multiplicative and fractional.…”

On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity

“…Equation ( 1) represents a natural extension of the stochastic heat equation driven by space-time white noise, which has been widely studied in the literature (see, e. g., Dalang et al (2009); Khoshnevisan (2014); Walsh (1986) and references therein). Equation ( 1) has been introduced in Zili and Zougar (2018) because of its interest in modeling diffusion phenomena in medium consisting of two kinds of materials, undergoing stochastic perturbations. In Zili and Zougar (2018), the authors proved the existence of the mild solution to (1), they presented explicit expressions of its covariance and variance functions, and they analyzed some regularity properties of its sample paths.…”

“…Equation ( 1) has been introduced in Zili and Zougar (2018) because of its interest in modeling diffusion phenomena in medium consisting of two kinds of materials, undergoing stochastic perturbations. In Zili and Zougar (2018), the authors proved the existence of the mild solution to (1), they presented explicit expressions of its covariance and variance functions, and they analyzed some regularity properties of its sample paths. Then, Zili and Zougar (2019a) presented an estimation method of the parameters a 1 and a 2 appearing in (3) and, after that, in Zili and Zougar (2019b), they made a deep study of the spatial quadratic variations of the mild solution process.…”

Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator

“…We note that Equation ( 7) is a particular case of (5), and it could be a good model for diffusion phenomena in a medium consisting of two kind of materials, undergoing stochastic perturbations. Equation (7) has been introduced in [30] where the authors proved the existence of the solution and they presented explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths have also been analyzed.…”