276 research outputs found
Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces
Let be a compact Riemannian homogeneous space (e.g. a Euclidean sphere).
We prove existence of a global weak solution of the stochastic wave equation
\mathbf D_t\partial_tu=\sum_{k=1}^d\mathbf
D_{x_k}\partial_{x_k}u+f_u(Du)+g_u(Du)\,\dot Wd\ge 1fgW\mathbb R^d$ with finite spectral measure. A
nonstandard method of constructing weak solutions of SPDEs, that does not rely
on martingale representation theorem, is employed
Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noise
The paper is concerned with spatial and time regularity of solutions to
linear stochastic evolution equation perturbed by L\'evy white noise "obtained
by subordination of a Gaussian white noise". Sufficient conditions for spatial
continuity are derived. It is also shown that solutions do not have in general
\cadlag modifications. General results are applied to equations with fractional
Laplacian. Applications to Burgers stochastic equations are considered as well.Comment: This is an updated version of the same paper. In fact, it has already
been publishe
Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces
Let be a Banach space such that, for some , the
function is of class and its first and second
Fr\'{e}chet derivatives are bounded by some constant multiples of -th
power of the norm and -th power of the norm and let be a
-semigroup of contraction type on . We consider the
following stochastic convolution process \begin{align*}
u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\;
t\geq 0, \end{align*} where is a compensated Poisson random measure
on a measurable space and is an -predictable function. We
prove that there exists a c\`{a}dl\`{a}g modification a of the
process which satisfies the following maximal inequality \begin{align*}
\mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E}
\left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d}
z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all and
with .Comment: This version is only very slightly updated as compared to the one
from September 201
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