On reflecting diffusion processes and Skorokhod decompositions

Summary LetG be ad-dimensional bounded Euclidean domain, H¹ (G) the set off in L²(G) such that f (defined in the distribution sense) is in L²(G). Reflecting diffusion processes associated with the Dirichlet spaces (H¹(G), ) on L²(G, dx) are considered in this paper, where A=(a^ij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thata ^ij H¹(G) for eachi,j, and H¹(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H¹(G), ) having starting points inG under a mild condition which is satisfied when G has finite (d–1)-dimensional lower Minkowski content.     

Two-parameter diffusion processes and martingales

We introduce a class of two-parameter processes which are diffusions on each coordinate and satisfy a particular Markov property related to the partial ordering in R²₊. These processes can be expressed as solutions of some stochastic integral equations driven by a two-parameter Wiener process and two families of ordinary Brownian motions. This result is based on a characterization of two-parameter martingales with orthogonal increments.     

Approximations of non-smooth integral type functionals of one dimensional diffusion processes

In this article, we obtain the weak and strong rates of convergence of time integrals of non-smooth functions of a one dimensional diffusion process. We propose the use of the exact simulation scheme to simulate the process at discretization points. In particular, we also present the rates of convergence for the weak and strong errors of approximation for the local time of a one dimensional diffusion process as an application of our method.     

Dirichlet forms and diffusion processes for fermion random point fields

We construct the Dirichlet forms and the associated diffusion processes on the configuration space of particles moving on the Euclidean space , for which certain fermion random point fields are invariant.     

Construction of infinite dimensional interacting diffusion processes through Dirichlet forms

Summary. By the theory of quasi-regular Dirichletforms and the associated special standard processes, the existence of symmetric diffusion processes taking values in the space of non-negative integer valued Radon measures on and having Gibbs invariant measures associated with some given pair potentials is considered. The existence of such diffusions can be shown for a wide class of potentials involving some singular ones. Also, as a consequence of an application of stochastic calculus, a representation for the diffusion by means of a stochastic differential equation is derived. Received: 5 September 1995 / In revised form: 14 March 1996     

Boundary blow-up solutions with a spike layer

We prove that for large λ>0, the boundary blow-up problem

     

Aging for interacting diffusion processes

We study the aging phenomenon for a class of interacting diffusion processes {Xt(i),i∈Zd}. In this framework we see the effect of the lattice dimension d on aging, as well as that of the class of test functions f(Xt) considered. We further note the sensitivity of aging to specific details, when degenerate diffusions (such as super random walk, or parabolic Anderson model), are considered. We complement our study of systems on the infinite lattice, with that of their restriction to finite boxes. In the latter setting we consider different regimes in terms of box size scaling with time, as well as the effect that the choice of boundary conditions has on aging. The key tool for our analysis is the random walk representation for such diffusions.     

Small time asymptotics of diffusion processes

We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity.     

Well-posedness of the martingale problem for some degenerate diffusion processes occurring in dynamics of populations

In this paper we establish the well-posedness in C([0,∞);[0,1]^d), for each starting point x∈[0,1]d, of the martingale problem associated with a class of degenerate elliptic operators which arise from the dynamics of populations as a generalization of the Fleming-Viot operator. In particular, we prove that such degenerate elliptic operators are closable in the space of continuous functions on [0,1]^d and their closure is the generator of a strongly continuous semigroup of contractions.     

Boundary layer and spike layer solutions for a bistable elliptic problem with generalized boundary conditions

We show that for large λ>0, the “generalized” boundary value problem

     

On order-preservation and positive correlations for multidimensional diffusion processes

Summary As a continuation of the study by Herbst and Pitt (1991), this note presents two criteria. The first one is on the order-preservation for two (may be different) multidimensional diffusion processes. The second one is on the preservation of positive correlations for a diffusion process.Research supported in part by the Ying-Tung Fok Educational Foundation and the National Natural Science Foundation of China     

Determinantal martingales and noncolliding diffusion processes

Two aspects of noncolliding diffusion processes have been extensively studied. One of them is the fact that they are realized as harmonic Doob transforms of absorbing particle systems in the Weyl chambers. Another aspect is integrability in the sense that any spatio-temporal correlation function can be expressed by a determinant. The purpose of the present paper is to clarify the connection between these two aspects. We introduce a notion of determinantal martingale and prove that, if the system has determinantal-martingale representation, then it is determinantal. In order to demonstrate the direct connection between the two aspects, we study three processes.     

Weak Dirichlet processes with a stochastic control perspective

The motivation of this paper is to prove verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term, in the case where the value function is assumed to be continuous in time and once differentiable in the space variable (C^0,1$C^{0,1}$) instead of once differentiable in time and twice in space (C^1,2$C^{1,2}$), like in the classical results. For this purpose, the replacement tool of the Itô formula will be the Fukushima–Dirichlet decomposition for weak Dirichlet processes. Given a fixed filtration, a weak Dirichlet process is the sum of a local martingale M$M$ plus an adapted process A$A$ which is orthogonal, in the sense of covariation, to any continuous local martingale. The decomposition mentioned states that a C^0,1$C^{0,1}$ function of a weak Dirichlet process with finite quadratic variation is again a weak Dirichlet process. That result is established in this paper and it is applied to the strong solution of a Cauchy problem with final condition.     

Probabilistic approach to homogenization of a non-divergence form semilinear PDE with non-periodic coefficients

We consider a semilinear partial differential equation (PDE) of non-divergence form perturbed by a small parameter. We then study the asymptotic behavior of Sobolev solutions in the case where the coefficients admit limits in C?esaro sense. Neither periodicity nor ergodicity will be needed for the coefficients. In our situation, the limit (or averaged or effective) coefficients may have discontinuity. Our approach combines both probabilistic and PDEs arguments. The probabilistic one uses the weak convergence of solutions of backward stochastic differential equations (BSDE) in the Jakubowski S-topology, while the PDEs argument consists to built a solution, in a suitable Sobolev space, for the PDE limit. We finally show the existence and uniqueness for the associated averaged BSDE, then we deduce the uniqueness of the limit PDE from the uniqueness of the averaged BSDE.     

Conditional law of risk processes given that ruin occurs

A risk process that can be Markovised is conditioned on ruin. We prove that the process remains a Markov process. If the risk process is a PDMP, it is shown that the conditioned process remains a PDMP. For many examples the asymptotics of the parameters in both the light-tailed case and the heavy-tailed case are discussed.     

Nonlinear Free Boundary Problems with Singular Source Terms

We prove the existence of solutions to nonlinear free boundary problem with singularities at given points.