7 research outputs found
Boundary regularity of stochastic PDEs
The boundary behaviour of solutions of stochastic PDEs with Dirichlet
boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as
shown by Krylov, for any one can find a simple -dimensional
constant coefficient linear equation whose solution at the boundary is not
-H\"older continuous.
We obtain a positive counterpart of this: under some mild regularity
assumptions on the coefficients, solutions of semilinear SPDEs on domains
are proved to be -H\"older continuous up to the boundary with some
.Comment: 29 page
Sharp regularity near an absorbing boundary for solutions to second order SPDEs in a half-line with constant coefficients
We prove that the weak version of the SPDE problem \begin{align*} dV_{t}(x) &
= [-\mu V_{t}'(x) + \frac{1}{2} (\sigma_{M}^{2} + \sigma_{I}^{2})V_{t}"(x)]dt -
\sigma_{M} V_{t}'(x)dW^{M}_{t}, \quad x > 0, \\ V_{t}(0) &= 0 \end{align*} with
a specified bounded initial density, , and a standard Brownian
motion, has a unique solution in the class of finite-measure valued processes.
The solution has a smooth density process which has a probabilistic
representation and shows degeneracy near the absorbing boundary. In the
language of weighted Sobolev spaces, we describe the precise order of
integrability of the density and its derivatives near the origin, and we relate
this behaviour to a two-dimensional Brownian motion in a wedge whose angle is a
function of the ratio . Our results are sharp: we
demonstrate that better regularity is unattainable
Finite time extinction for the 1D stochastic porous medium equation with transport noise
We establish finite time extinction with probability one for weak solutions
of the Cauchy-Dirichlet problem for the 1D stochastic porous medium equation
with Stratonovich transport noise and compactly supported smooth initial datum.
Heuristically, this is expected to hold because Brownian motion has average
spread rate whereas the support of solutions to the
deterministic PME grows only with rate . The rigorous
proof relies on a contraction principle up to time-dependent shift for
Wong-Zakai type approximations, the transformation to a deterministic PME with
two copies of a Brownian path as the lateral boundary, and techniques from the
theory of viscosity solutions.Comment: 38 page
Finite time extinction for the 1D stochastic porous medium equation with transport noise
We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate O(t12) whereas the support of solutions to the deterministic PME grows only with rate O(t1m+1). The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions
Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence
In this paper we develop a new approach to nonlinear stochastic partial
differential equations with Gaussian noise. Our aim is to provide an abstract
framework which is applicable to a large class of SPDEs and includes many
important cases of nonlinear parabolic problems which are of quasi- or
semilinear type. This first part is on local existence and well-posedness. A
second part in preparation is on blow-up criteria and regularization. Our
theory is formulated in an -setting, and because of this we can deal with
nonlinearities in a very efficient way. Applications to several concrete
problems and their quasilinear variants are given. This includes Burger's
equation, the Allen-Cahn equation, the Cahn-Hilliard equation,
reaction-diffusion equations, and the porous media equation. The interplay of
the nonlinearities and the critical spaces of initial data leads to new results
and insights for these SPDEs. The proofs are based on recent developments in
maximal regularity theory for the linearized problem for deterministic and
stochastic evolution equations. In particular, our theory can be seen as a
stochastic version of the theory of critical spaces due to
Pr\"uss-Simonett-Wilke (2018). Sharp weighted time-regularity allow us to deal
with rough initial values and obtain instantaneous regularization results. The
abstract well-posedness results are obtained by a combination of several
sophisticated splitting and truncation arguments.Comment: Accepted for publication in Nonlinearit