Gaussian random fields,infinite dimensional Ornstein-Uhlenbeck processes,and symmetric Markov processes

A general treatment of infinite dimensional Ornstein-Uhlenbeck processes (OUPs) is presented. Emphasis is put on their connection with ordinary Gaussian random fields, and OUPs as symmetric Markov processes. We also discuss the relation to second quantisation and Gaussian Markov random fields.Supported in part by the Swedish Natural Science Research Council, NFR.     

On the continuity of Ornstein-Uhlenbeck processes in infinite dimensions

Summary Existence and continuity of Ornstein-Uhlenbeck processes in Banach and Hilbert spaces are investigated under various assumptions.This work was partly written when W. Smoleski visited the Mathematics Department in Angers     

LARGE DEVIATIONS FOR INFINITE DIMENSIONAL AND REVERSIBLE REACTION－DIFFUSION PROCESSES

ＬＡＲＧＥＤＥＶＩＡＴＩＯＮＳＦＯＲＩＮＦＩＮＩＴＥＤＩＭＥＮＳＩＯＮＡＬＡＮＤＲＥＶＥＲＳＩＢＬＥＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＣＥＳＳＥＳＣＨＥＮＪＩＮＷＥＮ（陈金文）（ＤｅｐｅｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，Ｔ...     

Eigenfunction expansions for the Schrödinger operator

We obtain an eigenfunction expansion for the operator ?°+V under assump-tions (1.2)–(1.5) given below.     

Local continuity modulus for wiener and infinite dimensional OU processes

In this paper, we gave a proof for the local continuity modulus theorem of the Wiener process, i.e., $$\mathop {\lim }\limits_{t \to 0} \mathop {\sup }\limits_{0 \leqslant s \leqslant t} |W(s)|/(2s\log \log (1/s))^{1/2} = 1$$ a.s. This result was given by Csörgö and Révész (1981), but the proof gets them nowhere. We also gave a similar local continuity modulus result for the infinite dimensional OU processes.     

Free Ornstein-Uhlenbeck processes

Free Ornstein-Uhlenbeck processes are studied in finite von Neumann algebras. It is shown that a free self-decomposable probability measure on R can be realized as the distribution of a stationary free Ornstein-Uhlenbeck process driven by a free Levy process. A characterization of a probability measure on R to be the stationary distribution of a periodic free Ornstein-Uhlenbeck process driven by a free Levy process is given in terms of the Levy measure of the measure. Finally, the notion of a free fractional Brownian motion is introduced. It is proved that the free stochastic differential equation driven by a fractional free Brownian motion has a unique solution. We call the solution a fractional free Ornstein-Uhlenbeck process.     

Critical Ornstein-Uhlenbeck processes

The Ornstein-Uhlenbeck position process with the invariant measure is shown to satisfy a variational principle quite analogous to Hamilton's least action principle of classical mechanics. To prove this, a stochastic calculus of variations is developed for processes with differentiable sample paths, and which form a diffusion together with their derivative. The key tool in the derivation of stochastic Euler-Lagrange-type equations is a symmetric variant of Nelson's integration by parts formula for semimartingales simultaneously adapted to an increasing and a decreasing family of-algebras. An energy conservation theorem is also proved.     

The Ornstein-Uhlenbeck semigroup in an infinite dimensional L2 setting

We study the number operator, N, of quantum field theory as a partial differential operator in infinitely many variables. Informally Nu(x) = ?Δu(x) + x · grad u(x). A large core for N is constructed which is invariant under e^?tN and on which this informal expression may be given a precise and natural meaning.     

Gradient estimate for Ornstein-Uhlenbeck jump processes

By using absolutely continuous lower bounds of the Lévy measure, explicit gradient estimates are derived for the semigroup of the corresponding Lévy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time t under the condition that the process jumps before t. Finally, by using bounded perturbations of the Lévy measure, the resulting gradient estimates are extended to linear SDEs driven by Lévy-type processes.     

Densities for Ornstein-Uhlenbeck processes with jumps

We consider an Ornstein–Uhlenbeck process with valuesin ⁿ driven by a Lévy process (Z_t) taking values in ^dwith d possibly smaller than n. The Lévy noise can havea degenerate or even vanishing Gaussian component. Under a controllabilityrank condition and a mild assumption on the Lévy measureof (Z_t), we prove that the law of the Ornstein–Uhlenbeckprocess at any time t > 0 has a density on ⁿ. Moreover, whenthe Lévy process is of -stable type, (0, 2), we showthat such density is a C-function.     

Potential theory of infinite dimensional Lévy processes

We study the potential theory of a large class of infinite dimensional Lévy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e., excessive functions with compact level sets. Then many techniques from classical potential theory carry over to this infinite dimensional setting. Thus a number of potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as, e.g., formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron-Martin space is polar, in the Lévy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data.