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## Über dieses Buch

At the end of the summer 1989, an international conference on stochastic analysis and related topics was held for the first time in Lisbon (Portu­ gal). This meeting was made possible with the help of INIC and JNICT, two organizations devoted to the encouragement of scientific research in Portugal. The meeting was interdiciplinary since mathematicians and mathematical physicists from around the world were invited to present their recent works involving probability theory, analysis, geometry and physics, a wide area of cross fertilization in recent years. Portuguese scientific research is expanding fast, these days, faster, some­ times, than the relevant academic structures. The years to come will be determinant for the orientation of those young Portuguese willing to take an active part in the international scientific community. Lisbon's summer 89 meeting should initiate a new Iberic tradition, attrac­ tive both for these researchers to be and, of course, for the selected guests. Judging by the quality of contributions collected here, it is not unrealistic to believe that a tradition of "southern randomness" may well be established.

## Inhaltsverzeichnis

### Conformally Invariant Random Fields and Processes—Old and New

Abstract
Introduction. In [7], based on earlier articles of which we refer to [1, 2, 3, 6], we studied the equation
$$\bar \partial A = F,$$
where Aand F are four-component random fields of four variables, and $$\bar \partial$$ is a quaternionic version of the classical Cauchy-Riemann operator in C. Here F, the given source, is a white noise, not necessarily gaussian.
S. Albeverio, K. Iwata, T. Kolsrud

### Diffusion Processes with singular Dirichlet forms

Abstract
In this paper we construct diffusion processes associated with such Dirichlet forms which are so singular that their form domains contain no continuous functions different from the zero function. In quantum theory such singular Dirichlet forms correspond to Schrödinger operators with potentials which are singular on each neighbourhood of every point.
Sergio Albeverio, Zhi Ming Ma

### Geometric Quantization on Wiener Manifolds

Abstract
The Geometric Quantization procedure is considered in the context of Wiener manifolds.

### Heat Kernels on Lie Groups

Abstract
Using the Elworthy—Truman “elementary” formula we obtain exact and explicit formulae for the heat kernel on Lie groups and their dual symmetric spaces. We analyze also the case of nilpotent Lie groups and by means of a faithful representation we obtain for the heat kernel, associated with the Laplace—Beltrami, a recursion formula on the dimension of the representation.
Maria Teresa Arede

### Weak Sobolev Inequalities

Abstract
A weak Sobolev inequality (WSI) is a weakened form of a Sobolev inequality associated to a Dirichlet form. It turns out that it is in fact equivalent to a Sobolev inequality. If there is a spectral gap and a (WSI) holds, then we can get a tight weake Sobolev quality (TWSI). Starting with a TWSI. we get upper and lower bounds on the density of the heat semigroup associated with the Dirichlet form, when t → 0 as well as when t → ∞. In the last chapter, we give a Γ2 criterium to get a TWSI on a manifold.
Dominique Bakry

### Feynman’s Functional Calculus and Stochastic Calculus of Variations

Abstract
The mathematical structure of Quantum Mechanics is usually introduced as a calculus of non-commuting self-adjoint (unbounded) operators, the “observables,” on a Hilbert space of “states” (cf. [15]). There is no doubt that Quantum Mechanics is consistent and describes correctly many experiments, but we are supposed to renounce completely the visualization of quantum phenomena in space-time.
Ana Bela Cruzeiro, Jean Claude Zambrini

### Manifolds and Graphs with Mostly Positive Curvatures

Abstract
Bakry—Emery theory is used to define a ‘Ricci curvature’ for graphs. An upper bound for the spectral abcissa of the Laplacian of graphs with more than one end is given in terms of this quantity. This is similar to an existing result for manifolds, and the proof of that is also given.
K. D. Elworthy

### The Thom Class of Mathai and Quillen and Probability Theory

Abstract
As we will explain in this talk, the construction by Mathai and Quillen of explicit differential form representatives of the Thom class and Euler class of a vector bundle [5] gives a framework for understanding in unified way a number of ideas in stochastic differential geometry. We will also show briefly how Witten’s topological quantum field theories fit into this formalism. For the moment, the picture that he envisages is inaccessible to rigourous methods; but even in the more humdrum world of Brownian motion, the point of view presented here illuminates quite a few of the other talks that were given at this conference.
Ezra Getzler

### The General Mathematical Framework of Euclidean Quantum Mechanics, an Outline

Abstract
In Quantum Physics, the term “Euclidean” means that one, compared to the conventional formulation, works in “imaginary time.” It is a general fact that objects associated with Euclidean as opposed to Minkowski space are easier to deal with; this is seen for instance when comparing the usual Laplacian, an elliptic operator, with its Minkowski space counterpart, the hyperbolic d’Alembertian. In Quantum Mechanics, the term “Euclidean” is usually associated with approaches based on the so-called Feynman-Kac formula. The latter originates in attempts to give Feynman’s ideas [10, 11] on path integrals a solid mathematical founding.
Torbjörn Kolsrud, Jean-Claude Zambrini

### Naturality of Quasi-Invariance of Some Measures

Abstract
The purpose of this paper is to show by examples that, in a noncommutative setting, many natural measures do not enjoy the quasi-invariance à la Cameron—Martin which comes immediately to mind. Noncommutativity means the nontriviality of the bracket of the natural vector fields involved. In finite dimension brackets of the natural vector fields associated with an elliptic operator do not increase the tangent space, which is, by the ellipticity assumption, generated already by those vector fields. In infinite dimension the phenomena is quite different: starting with an elliptic operator (for instance the Laplacian on a Riemann Hilbert manifold M), the bracket phenomena will produce immediately a new tangent space, which could not necessarily contain the given tangent space of M. A well-known fact is that measures associated to the Brownian motion on M have to be realized as a Borelian measure on a bigger space than M. In the same way, a new tangent space to M has to be realized.
Paul Malliavin

### Stochastic Differential Equations with Boundary Conditions

Abstract
The stochastic calculus with anticipating integrands has been recently developed by several authors (see in particular [5,6,9] and the references therein). This new theory allows to study different types of stochastic differential equations driven by a d—dimensional Brownian motion {W(t), 0 ≤ t≤ 1}, where the solutions turn out to be non necessarily adapted to the filtration generated by W. We refer the reader to [12] for a survey of the applications of the anticipating stochastic calculus to stochastic differential equations. In particular one can consider stochastic differential equations of the form
$$\matrix{ {d{X_t} = f\left( {{X_t}} \right) + \sum\limits_{i = 1}^k {{g_i}} \left( {{X_t}} \right)odW_t^i,} & {0 \le t \le 1,} \cr }$$
and, instead of giving the value of the process at time zero, we impose a boundary condition of the form h(X 0, X 1)= h 0. In general, the solution {X t , 0 ≤ t≤ 1} will not be an adapted process, and the stochastic integral $$f_0^t{g_i}\left( {{X_s}} \right)o\,dW_s^i$$ is taken in the extended Stratonovich sense. The existence and uniqueness of a solution for an equation of this type has been investigated in some particular cases, and the Markov property of the solution has been studied. More precisely the following particular situations have been considered:
(a)
The functions f, g and h are affine. (Ocone-Pardoux [11])

(b)
k = d and the function g is a constant equal to the identity matrix. (Nualart—Pardoux [7])

(c)
k = d = 1, and h is linear. (Donati—Martin [2])

(d)
Second order stochastic differential equations in dimension one of the following type
$$\matrix{ {{{\ddot X}_t} + f\left( {{X_t},{{\dot X}_t}} \right) = {{\dot W}_t},} & {0 \le t \le 1,} \cr }$$
with Dirichlet boundary conditions X 0 = a, X 1 = b. (Nualart—Pardoux [8)).

David Nualart, Etienne Pardoux

### Sobolev Inequalities and Polynomial Decay of Convolution Powers and Random Walks

Abstract
Let F be a symmetric, finitely supported, positive function on ℤ d which charges the closest neighbors of the origin and has total mass one. Denote by F (n) the n-th convolution power of F. One has the following well known result:
$$\left\| {{F^{\left( n \right)}}} \right\|\infty \le c{n^{ - {d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2}}},\,\,n \in N*.$$
Laurent Saloff-Coste

### Microcanonical Distributions for one Dimensional Lattice Gases

Abstract
In equilibrium statistical mechanics, a central rôle is played by the Gibbs’ state of a system, and the goal of this note is to understand, on the basis of large deviation theory, why Gibbs’ states arise. In order to keep everything very simple, we will restrict our attention here to a very special class of systems known as one dimensional lattice gases with finite range interaction (cf. the Remark 2.15 below). To be precise, let Ebe a compact metric space, λ a probability measure on (E, ß), set Ω =Eℤ, and let $$u = \left[ {{U_F}:{\rm{\theta }} \ne {\rm{F}} \subset \subset } \right] \subseteq C\left( {\Omega ;} \right)$$ be a family of continuous functions with the properties that, for all non-empty, finite subsets Fof ℤ,
1)
U F (x) = U F (y) when x, y ∈ Ω with x k = y k for all kF;

2)
for all k ∈ ℤ. U k+F = U F S k , where S : Ω → Ω is the shift transformation determined by (S x k = x k+1 for all k ∈ ℤ and x ∈ Ω;

3)
there is an R ∈ ℤ+ such that U F ≡ 0 if 0 ∊ F and F ⊈ [−R, R].

Daniel W. Stroock
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