A Stochastic Parabolic Equation with Nonlinear Flux on the Boundary Driven by a Gaussian Noise

Absolute continuity of the law for solutions of stochastic differential equations with boundary noise

“…We consider in this section the non homogeneous diffusion equation (1). In the first part of the section we impose condition (g1) on the function g: g is Lipschitz continuous and belongs to the class C 1 (R) with |∂ u g(u)| ≤ L, g(u) ≤ c(1 + |u|) for some finite constant L > 0. and prove that it is a well-defined mapping from the space L p W ([0, T ] ; L p (∂D)) into itself.…”

“…Here, we aim to investigate further the properties of the solution for the problem introduced in [1]. We consider the case when the boundary condition are given by the sum of a nonlinear function of the solution and a stochastic perturbation:…”

“…We also prove that its law admits a density with respect to the Lebesgue measure. In Section 5 we recall the results concerning the existence and regularity of the solution of (3) proved in [1] and we study the Malliavin derivative of the solution. Finally in Sections 6 we study the regularity of the solution in the Malliavin sense.…”

A Stochastic Heat Equation with Nonlinear Dissipation on the Boundary

“…(On these lines see [12,13].) Also, one might treat the more general case where is replaced by a nonlinear elliptic operator of the form y → div a(∇ y), where a : R d → R d is monotone, coercive, and with polynomial growth.…”

Existence and regularity of the density for solutions of stochastic differential equations with boundary noise

“…1 One of the main issues is the regularity of the solution inside the domain as well as near the boundary. In a recent paper, 2 it was studied a class of nonlinear heat diffusions with stochastic perturbation of Brownian and fractional Brownian motion type. In this note, we proceed with the analysis of such equations, by considering in particular the problem of regularity in the Malliavin sense of the solution of such problem.…”