Existence and regularity results for stochastic fractional pseudo-parabolic equations driven by white noise

<p style='text-indent:20px;'>Solutions of a direct problem for a stochastic pseudo-parabolic equation with fractional Caputo derivative are investigated, in which the non-linear space-time-noise is assumed to satisfy distinct Lipshitz conditions including globally and locally assumptions. The main aim of this work is to establish some existence, uniqueness, regularity, and continuity results for mild solutions.</p>

Refined existence and regularity results for a class of semilinear dissipative SPDEs

We prove the existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the [Formula: see text]th moment of solutions depends on the integrability of the initial datum, in the whole range [Formula: see text]. Lipschitz continuity of the solution map in [Formula: see text]th moment is established, under a Lipschitz continuity assumption on the diffusion coefficient, in the even larger range [Formula: see text]. A key role is played by an Itô formula for the square of the norm in the variational setting for processes satisfying minimal integrability conditions, which yields pathwise continuity of solutions. Moreover, we show how the regularity of the initial datum and of the diffusion coefficient improves the regularity of the solution and, if applicable, of the invariant measures.

Dehn functions and Hölder extensions in asymptotic cones

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.