Narrow Results

In this paper we prove Kunita-type stochastic differential equation (SDE) , t > s, driven by a spatial continuous semimartingale F (x, t) =(F¹ (x, t), …, F^m (x, t)), x ∈ ℜ^m, with local characteristic (a, b), in the sense of Kunita (Chap. 3 of Ref. 10), can produce a stochastic flow of homeomorphisms of ℜ^m into itself almost surely under the (a, b) satisfies non-Lipschitz conditions. This result bases on recent works¹² by the author.

In this article we will prove new representation for the Levi–Cività connection in terms of the stochastic flow corresponding to Brownian motion on manifold.

Recently, Cranston, Scheutzow and Steinsaltz showed that the image at time T of a bounded set under the action of a stochastic flow on R^d with Lipschitz and bounded characteristics is almost surely contained in a ball of radius KT around the origin for all sufficiently large T. Here K is a constant which depends on the dimension d and the characteristics of the flow (but not on the set ). We improve these results in three different ways: our proof is shorter and more transparent, the bound K which we obtain is sharper and we are able to improve the bounds on the tails of the distance of the most distant point in the image at time T to be genuinely sub-Gaussian.

Let ϕ_t be a Lévy process in a semisimple Lie group G of noncompact type regarded as a stochastic flow on a homogeneous space of G, called a G-flow. We will determine the Lyapunov exponents and the stable manifolds of ϕ_t, and the stationary points of an associated vector field. As examples, SL(d,R)-flows and SO(1,d)-flows on SO(d) and S^{d - 1} are discussed in details.

Random dynamical systems can be generated by stochastic differential equations (sde) on the one hand, and by random differential equations (rde), i.e. randomly parametrized ordinary differential equations on the other hand. Due to conflicting concepts in stochastic calculus and ergodic theory, asymptotic problems for systems associated with sde are harder to treat. We show that both objects are basically identical, modulo a stationary coordinate change (cohomology) on the state space. This observation opens completely new opportunities for the treatment of asymptotic problems for systems related to sde: just study them for the conjugate rde, which is often possible by simple path-by-path classical arguments. This is exemplified for the problem of local linearization of random dynamical systems, the classical analogue of which leads to the Hartman–Grobman theorem.

In many physical applications it is important to predict particle trajectories in a turbulent flow using a numerical model of the flow. One of the major reasons for a poor prediction is the limited resolution in space. We discuss the problem of predictability limits in the framework of a stochastic flow modeling inertial particles in turbulence.

Two stochastic flows called the "model" and "true" are considered. The "model" flow is the "true" flow smoothed in space. The principal measure of discrepancy of the flows is the distance between the centers of mass of two initially identical particle clusters driven by the "true" and "model" flows respectively. The predictability limit is defined as a moment when this distance exceeds the space correlation scale of the true flow. The main result is a simple formula relating the predictability limit with the smoothing scale. This relation is well supported by stochastic simulations and could be a helpful tool in planning experiments in numerical models of turbulent flows. Also the difference between particle scattering in the "true" and "model" flows is studied.

In this paper we study how σ-finite measures on ℝ^d evolve under a class of "stochastic flows" associated to stochastic differential equations with (resp. without) jumps in ℝ^d. First we show the related measure evolution processes are càdlàg (resp. continuous), strongly Markovian and weakly Fellerian. Then we extend the existing results on incompressibility in Harris [8] and Kunita [14], and prove strong Markov property of the process describing how compact subsets evolve under incompressible "stochastic flows" under a certain condition.

We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.

A natural "flow" over super-Brownian motions, which is not in Le Jan–Raimond framework on stochastic flows of measurable maps and kernels, is discovered in this paper.

This paper presents a probabilistic interpretation for the weak Sobolev solution of the obstacle problem for semilinear parabolic partial integro-differential equations (PIDEs). The results of Léandre [32] concerning the homeomorphic property for the solution of SDEs with jumps are used to construct random test functions for the variational equation for such PIDEs. This results in the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDEs), namely Feynman–Kac's formula for the solution of the PIDEs. Moreover, it gives an application to the pricing and hedging of contingent claims with constraints in the wealth or portfolio processes in financial markets including jumps.

Existence of random dynamical systems for a class of coalescing stochastic flows on $ℝ$ is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered.

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs. The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.