5 research outputs found
On a class of stochastic partial differential equations
In this paper, we study the stochastic partial differential equation with
multiplicative noise ,
where is the generator of a symmetric L\'evy process and is a Gaussian noise. For the equation in the Stratonovich sense, we show
that the solution given by a Feynman-Kac type of representation is a mild
solution, and we establish its H\"older continuity and the Feynman-Kac formula
for the moments of the solution. For the equation in the Skorohod sense, we
obtain a sufficient condition for the existence and uniqueness of the mild
solution under which we get Feymnan-Kac formula for the moments of the
solution, and we also investigate the H\"older continuity of the solution. As a
byproduct, when is a nonnegative and nonngetive-definite function,
a sufficient and necessary condition for to be exponentially integrable is
obtained.Comment: 46 page
Stochastic quantisation of Yang–Mills
We review two works arXiv:2006.04987 and arXiv:2201.03487 which study the
stochastic quantisation equations of Yang-Mills on two and three dimensional
Euclidean space with finite volume. The main result of these works is that one
can renormalise the 2D and 3D stochastic Yang-Mills heat flow so that the
dynamic becomes gauge covariant in law. Furthermore, there is a state space of
distributional -forms to which gauge equivalence
extends and such that the renormalised stochastic Yang-Mills heat flow projects
to a Markov process on the quotient space of gauge orbits .
In this review, we give unified statements of the main results of these works,
highlight differences in the methods, and point out a number of open problems.Comment: 32 pages, 1 figure. Fixed typos and updated references. To appear in
Journal of Mathematical Physics for the proceedings of ICMP 202
Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains
In this work, we consider the two and three-dimensional stochastic convective
Brinkman-Forchheimer (2D and 3D SCBF) equations driven by irregular additive
white noise for in
unbounded domains (like Poincar\'e domains)
() where is a Hilbert space valued Wiener process on
some given filtered probability space, and discuss the asymptotic behavior of
its solution. For with and with
(for with ), we first prove the existence and
uniqueness of a weak solution (in the analytic sense) satisfying the energy
equality for SCBF equations driven by an irregular additive white noise in
Poincar\'e domains by using a Faedo-Galerkin approximation technique. Since the
energy equality for SCBF equations is not immediate, we construct a sequence
which converges in Lebesgue and Sobolev spaces simultaneously and it helps us
to demonstrate the energy equality. Then, we establish the existence of random
attractors for the stochastic flow generated by the SCBF equations. One of the
technical difficulties connected with the irregular white noise is overcome
with the help of the corresponding Cameron-Martin space (or Reproducing Kernel
Hilbert space). Finally, we address the existence of a unique invariant measure
for 2D and 3D SCBF equations defined on Poincar\'e domains (bounded or
unbounded). Moreover, we provide a remark on the extension of the above
mentioned results to general unbounded domains also