60,261 research outputs found

    On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

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    This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form ∂tu(x,t)=∑j=1mXj2u(x,t)+X0u(x,t)(x,t)∈RN× ]−∞,T[, \partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty ,T[, proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.Comment: The results of the present version recover most of the ones in the previous version, but, on top of it, this new version contains some further new and interesting result

    One-Dimensional Fokker-Planck Equations and Functional Inequalities for Heavy Tailed Densities

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    We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker-Planck equations modeling socio-economic problems, and one-dimensional functional inequalities of the type of Poincare, Wirtinger and logarithmic Sobolev, with weight, for probability densities with polynomial tails. As main examples, we consider inequalities satisfied by inverse Gamma densities, taking values on R+, and Cauchy-type densities, taking values on R

    Degenerate backward SPDEs in domains: non-local boundary conditions and applications to finance

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    Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied and the equation can be degenerate. Some generalized solutions based on the representation theorem are suggested. In addition to problems with a standard Cauchy condition at the terminal time, problems with special non-local boundary conditions are considered. These non-local conditions connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability and regularity results are obtained. Some applications to portfolio selection problem are considered.Comment: arXiv admin note: substantial text overlap with arXiv:1211.1460, arXiv:1208.553

    On Cauchy problem for first order nonlinear functional differential equations of non-Volterra’s type

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    summary:On the segment I=[a,b]I=[a,b] consider the problem u′(t)=f(u)(t),u(a)=c, u^{\prime }(t)=f(u)(t) , \quad u(a)=c, where f C(I,R)→L(I,R)f\:C(I,\mathbb{R})\rightarrow L(I,\mathbb{R}) is a continuous, in general nonlinear operator satisfying Carathéodory condition, and c∈Rc\in \mathbb{R}. The effective sufficient conditions guaranteeing the solvability and unique solvability of the considered problem are established. Examples verifying the optimality of obtained results are given, as well

    Stability of Steady Multi-Wave Configurations for the Full Euler Equations of Compressible Fluid Flow

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    We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.Comment: 9 figures
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