Andrzej Korzeniowski

An example in the theory of hypercontractive semigroups

Let L = x(d2/dx2) + (1 x)(d/dx) on C.((O, oo)) be the Laguerre operator. It is shown that for t > 0, and 1 0) a conservative Markov semigroup on B(E) for which m is a reversible measure (i.e. for each t > 0, Pt is symmetric on L2(m)). Then, as an easy application of Jensens inequality, IIPtIlLP(m)-LP(m) 0 and p E [1, oc]. In particular, each Pt admits a unique extension Pt as a bounded operator on L2(m) and { Pt: t > 0) is a semigroup of selfadjoint contractions. A well-studied example of this situation is the Ornstein-Uhlenbeck semigroup {rt(d): t > 0) on B(Rd): E = Rd, m(dx) = y(d)(dx) g(d)(1, x) dx, and Pt = rt(d) is given by vt(d)f(x) = Jg(d)(I e-2t, y e-tx)f(y) dy where g(d) (T,) = (2.TT)-d/2exp(-ItI2/2T), (T, {) E (0, oc) X Rd. In connection with his work on constructive field theory, E. Nelson [2] discovered that { rt(d): t > 0) enjoys a hypercontractivity property. Namely, he showed that for given 1 (p 1)/(q 1), then I t(d) LP () -Lq() = 00. Since Nelsons initial discovery, many other examples of hypercontractive semigroups have been found (cf. F. Weissler [7, 8], F. Weissler and C. Mueller [9], and 0. Rothaus [3-5]). In most cases the difficult part of the analysis lies in the attempt to obtain the optimal result (i.e. the smallest T(p, q) > 0 such that IIPtIILP(m)>Lq(m) T( p, q)). The work of L. Gross [1] shows that this question is closely related to that of finding the smallest a > 0 for which the logarithmic Sobolev inequality (1) JIf | logIf 2 dm a a(ff, ) + |If || 2(m) logllf 12 L2(m) Received by the editors March 27, 1984 and, in revised form, June 6, 1984. 1980 Mathematics Subject Classification. Primary 47D05; Secondary 46E30.

On logarithmic Sobolev constant for diffusion semigroups

Abstract This paper is concerned with estimates for the constant in the logarithmic Sobolev inequality. As an application, we show that for the Laguerre semigroup with the generator xD 2 − ( x − λ ) D , the logarithmic Sobolev constant α λ satisfies log (3 + 6 λ )/ log 3 ⩽ α λ ⩽ 16 log (3 + 6 λ ) for 0 1 2 and α λ = 4 for 1 2 ⩽ λ .

Stability of Stochastic Differential Equations Under Discretization

Abstract Long time behavior of stochastic differential equations (SDE) involves two instances of exponential mean square stability (EMS-stability). First deals with stability of the original continuous time system while the second is concerned with stability after the time step discretization. By considering a linear operator S associated with SDE, we show that the discrete system is EMS-stable if and only if S is a positive contraction on the set of symmetric positive definite matrices.

On computer simulation of Feynman-Kac path-integrals

Abstract Consider a path-integral E x exp {∞ t o V ( X ( s )) ds } f ( X ( t )) which is the solution to a diffusion version of the generalized Schrodingers equation ∂u ∂t = Hu , u(0,x) = ƒ(x) . Here H = A + V , where A is an infinitesimal generator of a strong continuous Markov semigroup corresponding to the diffusion process { X ( s ), 0⩽ s ⩽ t , X (0) = x }. To see a connection to quantum mechanics, take A = 1 2 Δ and replace V by − V . Then one obtains H = −H = − 1 2 Δ + V , which is a quantum mechanical Hamiltonian corresponding to a particle of mass 1 (in atomic units) subject to interaction with potential V . Path-integrals play a role in obtaining physical quantities such as ground state energies. This paper will be concerned with explanations of two approaches in the actual computer evaluations of path-integrals through simulations of the diffusion processes. The results will be presented by comparing, in concrete examples, the computational advantages or disadvantages depending on whether the diffusion process X ( t ) is ergodic or not.

Microscopic turbulence in water

The purpose of this paper is to demonstrate, by means of supercomputer simulations, that in the absence of random perturbations from the initial equilibrium positions one can observe a transition to turbulence by increasing the kinetic energy of the molecules.

On Simulating Wiener Integrals and Their Expectations

An approximation scheme for evaluating Wiener integrals by simulating Brownian motion is studied. The rate of convergence and numerical results are given, including an application to the heat equation by using the Feynman-Kac formula.

On diffusions that cannot escape from a convex set

Let Xt be a diffusion on D with a generator , where f is a probability density such that f> 0 on an open convex set D and vanishes outside. Then [integral operator]D 1/f = [infinity] implies that Xt never leaves D. As an application we extend the asymptotics of Wiener integrals of Donsker--Varadhan from 1 to D.

Quantum mechanical simulation of the hydrogen molecule

Abstract Using the probabilistic path integral representation of the time dependent solution to the Schrodinger equation, we devise a numerical algorithm to obtain the binding energy, diameter and vibrational frequency of the hydrogen molecule. The results agree to a high degree of accuracy with experimental data.

On principal eigenvalues for random evolutions

Using Large Deviation Principle of Donsker–Varadhan a variational formula is established for the principal eigenvalue of the higher order elliptic operator driven by the Brownian Motion

Dynamic systems driven by Markov processes

Consider a differential equation Y=V(X(t))Y(t), where X(t) is a random function. Sufficient conditions for asymptotic stability of the solution in terms of a generator of the stochastic process X(t) are given. The results are illustrated by several examples.