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Stefan problem in a 2D case

Piotr Bogusław Mucha (2006)

Colloquium Mathematicae

The aim of this paper is to analyze the well posedness of the one-phase quasi-stationary Stefan problem with the Gibbs-Thomson correction in a two-dimensional domain which is a perturbation of the half plane. We show the existence of a unique regular solution for an arbitrary time interval, under suitable smallness assumptions on initial data. The existence is shown in the Besov-Slobodetskiĭ class with sharp regularity in the L₂-framework.

Stochastic calculus and degenerate boundary value problems

Patrick Cattiaux (1992)

Annales de l'institut Fourier

Consider the boundary value problem (L.P): ( h - A ) u = f in D , ( v - Γ ) u = g on D where A is written as A = 1 / 2 i = 1 m Y i 2 + Y 0 , and Γ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields Y i ( 0 i m ), for regular open sets D with a non-characteristic boundary.Our study lies on the stochastic representation of u and uses the stochastic calculus of variations for the ( A , Γ ) -diffusion process...

Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions

Mireille Bossy, Mamadou Cissé, Denis Talay (2011)

Annales de l'I.H.P. Probabilités et statistiques

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.

Stochastic Taylor expansions and heat kernel asymptotics

Fabrice Baudoin (2012)

ESAIM: Probability and Statistics

These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern–Gauss–Bonnet theorem.

Strangely sweeping one-dimensional diffusion

Ryszard Rudnicki (1993)

Annales Polonici Mathematici

Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t))dW(t) + b(X(t))dt. We analyse the asymptotic behaviour of p(t) = ProbX(t) ≥ 0 as t → ∞ and construct an equation such that l i m s u p t t - 1 0 t p ( s ) d s = 1 and l i m i n f t t - 1 0 t p ( s ) d s = 0 .

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