We report on recent progress in the study of evolution processes involving degenerate parabolic equations which may exhibit free boundaries. The equations we have selected follow two recent trends in diffusion theory: considering anomalous diffusion with long-range effects, which leads to fractional operators or other operators involving kernels with large tails; and the combination of diffusio...

in the present article, we focus on the numerical approximation of stochastic partial differential equations of itˆo type with space-time white noise process, in particular, parabolic equations. for each case of additive andmultiplicative noise, the numerical solution of stochastic diffusion equations is approximated using two stochastic finite difference schemes and the stability and consisten...

We present a second order, finite volume scheme for the constant-coefficient diffusion equation on curved, parametric surfaces. While our scheme is applicable to general quadrilateral surface meshes based on smooth or piecewise smooth coordinate transformations, our primary motivation for developing the present scheme is to solve diffusion problems on a particular set of circular and spherical ...

In this paper, we consider the following type of non-local (pseudo-differential) operators L on Rd: Lu(x) = 1 2 d ∑ i,j=1 ∂ ∂xi ( aij(x) ∂ ∂xj ) + lim ε↓0 ∫ {y∈Rd: |y−x|>ε} (u(y) − u(x))J(x, y)dy, where A(x) = (aij(x))1≤i,j≤d is a measurable d × d matrix-valued function on Rd that is uniformly elliptic and bounded and J is a symmetric measurable non-trivial nonnegative kernel on Rd × Rd satisfy...

In this paper we explicit the derivative of the flows of one dimensional reflected diffusion processes. We then get stochastic representations for derivatives of viscosity solutions of one dimensional semilinear parabolic partial differential equations and parabolic variational inequalities with Neumann boundary conditions. Key-words: Reflected backward stochastic differential equations, semili...

We prove existence and uniqueness of the solution of a parabolic SPDE in one space dimension driven by space-time white noise, in the case of a measurable drift and a constant diffusion coefficient, as well as a comparison theorem.

Consider classical solutions u ∈ C(R×(0,∞))∩C(R× [0,∞)) to the parabolic reaction diffusion equation ut = Lu+ f(x, u), (x, t) ∈ R n × (0,∞); u(x, 0) = g(x) ≥ 0, x ∈ R; u ≥ 0,

We study the interior regularity properties of the solutions to the degenerate parabolic equation, ∆xu+ b∂yu− ∂tu = f, (x, y, t) ∈ R × R× R, which arises in mathematical finance and in the theory of diffusion processes.

We study the limit, when k → ∞, of the solutions u = uk of (E) ∂tu−∆u+ h(t)uq = 0 in RN × (0,∞), uk(., 0) = kδ0, with q > 1, h(t) > 0. If h(t) = e−ω(t)/t where ω > 0 satisfies to R 1 0 p ω(t)t−1dt < ∞, the limit function u∞ is a solution of (E) with a single singularity at (0, 0), while if ω(t) ≡ 1, u∞ is the maximal solution of (E). We examine similar questions for equations such as ∂tu−∆u + h...