29,401 research outputs found
Stochastic Infinity-Laplacian equation and One-Laplacian equation in image processing and mean curvature flows: finite and large time behaviours
The existence of pathwise stationary solutions of this stochastic partial differential equation (SPDE, for abbreviation) is demonstrated.
In Part II, a connection between certain kind of state constrained controlled Forward-Backward Stochastic Differential Equations (FBSDEs) and Hamilton-Jacobi-Bellman equations (HJB equations) are demonstrated. The special case provides a probabilistic representation of some geometric flows, including the mean curvature flows.
Part II includes also a probabilistic proof of the finite time existence of the mean curvature flows
Variational Principles for Stochastic Fluid Dynamics
This paper derives stochastic partial differential equations (SPDEs) for
fluid dynamics from a stochastic variational principle (SVP). The Legendre
transform of the Lagrangian formulation of these SPDEs yields their Lie-Poisson
Hamiltonian form. The paper proceeds by: taking variations in the SVP to derive
stochastic Stratonovich fluid equations; writing their It\^o representation;
and then investigating the properties of these stochastic fluid models in
comparison with each other, and with the corresponding deterministic fluid
models. The circulation properties of the stochastic Stratonovich fluid
equations are found to closely mimic those of the deterministic ideal fluid
models. As with deterministic ideal flows, motion along the stochastic
Stratonovich paths also preserves the helicity of the vortex field lines in
incompressible stochastic flows. However, these Stratonovich properties are not
apparent in the equivalent It\^o representation, because they are disguised by
the quadratic covariation drift term arising in the Stratonovich to It\^o
transformation. This term is a geometric generalisation of the quadratic
covariation drift term already found for scalar densities in Stratonovich's
famous 1966 paper. The paper also derives motion equations for two examples of
stochastic geophysical fluid dynamics (SGFD); namely, the Euler-Boussinesq and
quasigeostropic approximations.Comment: 19 pages, no figures, 2nd version. To appear in Proc Roy Soc A.
Comments to author are still welcome
A flow-based approach to rough differential equations
These are lecture notes for a Master 2 course on rough differential equations
driven by weak geometric Holder p-rough paths, for any p>2. They provide a
short, self-contained and pedagogical account of the theory, with an emphasis
on flows. The theory is illustrated by some now classical applications to
stochastic analysis, such as the basics of Freidlin-Wentzel theory of large
deviations for diffusions, or Stroock and Varadhan support theorem.Comment: 63 page
On post-Lie algebras, Lie--Butcher series and moving frames
Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on
differential manifolds. They have been studied extensively in recent years,
both from algebraic operadic points of view and through numerous applications
in numerical analysis, control theory, stochastic differential equations and
renormalization. Butcher series are formal power series founded on pre-Lie
algebras, used in numerical analysis to study geometric properties of flows on
euclidean spaces. Motivated by the analysis of flows on manifolds and
homogeneous spaces, we investigate algebras arising from flat connections with
constant torsion, leading to the definition of post-Lie algebras, a
generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately
associated with euclidean geometry, post-Lie algebras occur naturally in the
differential geometry of homogeneous spaces, and are also closely related to
Cartan's method of moving frames. Lie--Butcher series combine Butcher series
with Lie series and are used to analyze flows on manifolds. In this paper we
show that Lie--Butcher series are founded on post-Lie algebras. The functorial
relations between post-Lie algebras and their enveloping algebras, called
D-algebras, are explored. Furthermore, we develop new formulas for computations
in free post-Lie algebras and D-algebras, based on recursions in a magma, and
we show that Lie--Butcher series are related to invariants of curves described
by moving frames.Comment: added discussion of post-Lie algebroid
A qualitative approach to the existence of random periodic solutions
In this thesis, we study the existence of random periodic solutions of random dynamical systems (RDS) by geometric and topological approach. We employed an extension of ergodic theory to random setting to prove that a random invariant set with some kind of dissipative structure, can be expressed as union of random periodic curves. We extensively characterize the dissipative structure by random invariant measures and Lyapunov exponents. For stochastic flows induced by stochastic differential equations (SDEs), we studied the dissipative structure by two point motion of the SDE and prove the existence exponential stable random periodic solutions
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