1,721,068 research outputs found
Fujita type results for a class of degenerate parabolic operators
No full textWe study the interior regularity properties of the solutions to a
degenerate parabolic equation which arises in mathematical finance and in the theory of diffusion processes
A priori estimates for quasilinear degenarate parabolic equations
Full text linkWe prove some maximum and gradient estimates for classical solutions to a wide class of quasilinear degenerate parabolic equations, including first order ones. The proof is elementary and exploits the smallness of the domain in the time direction
Superparabolic Functions Related to Second Order Hypoelliptic Operators
No full textIn this paper, we consider a wide class of second order hypoelliptic partial differential operators with nonnegative characteristic form. We prove a monotone smoothing theorem and a representation formula for superparabolic functions
On stochastic Langevin and Fokker-Planck equations: The two-dimensional case
Full text linkWe prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution of the stochastic Langevin equation that is a degenerate SPDE satisfying the weak Hörmander condition. This problem naturally appears in stochastic filtering theory. We use a Wentzell's transform to reduce the SPDE to a PDE with random coefficients. After introducing an original notion of intrinsic solution, we apply a new method based on the parametrix technique to construct it. This approach avoids the use of the Duhamel's principle for the SPDE and the related measurability issues that appear in the stochastic framework. Our results are new even for the deterministic equation in that we prove existence and gradient estimates for the fundamental solution of equations whose coefficients are merely measurable with respect to the time variable. We also propose a different, possibly simpler proof for the Gaussian lower bound
Backward and forward filtering under the weak Hörmander condition
Full text linkWe derive the forward and backward filtering equations for a class of degenerate partially observable diffusions, satisfying the weak Hörmander condition. Our approach is based on the Hölder theory for degenerate SPDEs that allows to pursue the direct approaches proposed by Krylov and Zatezalo, and Veretennikov, avoiding the use of general results from filtering theory. As a by-product we also provide existence, regularity and estimates for the filtering density
The parametrix method for parabolic SPDEs
Full text linkWe consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Itô–Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients
On the viscosity solutions of a stochastic differential utility problem
No full textWe prove existence, uniqueness and gradient estimates of stochastic differential utility as a solution of the Cauchy problem for the following equation in R3: ∂xxu + u∂yu - ∂tu = f (·,u), where f is Lipschitz continuous. We also characterize the solution in the vanishing viscosity sense. © 2002 Elsevier Science (USA). All rights reserved
Hölder Regularity for a Kolmogorov Equation
No full textWe study the interior regularity properties of the solutions to the degenerate parabolic equation, Δ_x u + b∂_y u - ∂_t u = f, (x, y, t) ∈ R^N × R × R, which arises in mathematical finance and in the theory of diffusion processes
On the Harnack inequality for a class of hypoelliptic evolution equations
No full textWe give a direct proof of the Harnack inequality for a class ofdegenerate evolution operators which contains the linearized prototypes of the Kolmogorov and Fokker-Planck operators. We also improve the known results in that we find the optimal constant of the inequality
The Moser's iterative method for a class of ultraparabolic equations
No full textWe adapt the iterative scheme by Moser, to prove that the weak solutions to an ultraparabolic equation, with measurable coefficients, are locally bounded functions. Due to the strong degeneracy of the equation, our method differs from the classical one in that it is based on some ad hoc Sobolev type inequalities for solution
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