Titles and abstracts

S. Albeverio (Bonn): Quantum fields, Levy processes, Stochastic (pseudo) PDEs

A. Barchielli (Milano): Quantum and classical stochastic differential equations in the quantum theory of open systems

Abstract. Stochastic differential equations for processes with values in Hilbert spaces are now largely used in the quantum theory of open systems and of measurements continuous in time. In this talk I present a class of such equations and explain how they are derived from purely quantum mechanical models, based on quantum stochastic calculus and quantum stochastic differential equations. An essential tool is an isomorphism between the bosonic Fock space and the Wiener space, which allows to connect certain quantum objects with probabilistic ones.

V. Bogachev (Moscow): Elliptic equations for measures: regularity and existence of solutions, a priori estimates, and applications to nonlinear stochastic partial differential equations

Abstract. The talk focuses on several problems related to second order elliptic equations satisfied by invariant measures of diffusion processes in finite and infinite dimensional spaces. For the diffusion on ${\rm {\bf R}}^d$ generated by the stochastic differential equation

$$d\xi_t=\sqrt{2A(\xi _t)}dW_t+B(\xi_t)dt$$

with a matrix-valued mapping A=(A^ij) and vector-valued mapping B=(Bⁱ), the elliptic equation in question is

$$L_{A,B}^{*}\mu=0 \eqno (1),$$

where

$$L_{A,B}\varphi=\sum\limits_{i,j=1}^d A^{ij} \partial_{x_i}\pa... ...varphi +\sum\limits_{i=1}^d B^i\partial_{x_i}\varphi \eqno (2)$$

and (1) is understood in the following weak sense:

$$\int L_{A,B}\varphi\, d\mu =0, \qquad \forall\, \varphi\in C_0^\infty ({\rm {\bf R}}^d).$$

Equation (1) makes sense even without assumptions about the existence of a process. In recent years, there has been a considerable progress in the study of such equations. A short survey of recent results concerning regularity and existence of solutions to (1) in the finite dimensional case will be given, and then the infinite dimensional case will be considered. In the case of an infinite dimensional state space X, the operator L_A,B is heuristically defined by (2) with an infinite sum, which makes sense on a suitable domain; however, the fuctions Bⁱ may not correspond to any mapping B on the space X. Most typical situations are connected with stochastic Burgers, Navier-Stokes and reaction-diffusion equations, and Gibbs measures. For example, in the case of a stochastic partial differential equation with the state space X=L^p[0,1], $p\ge 2$, one can deal with functions

$$B^i(x)=\int_0^1 x(s)L^{*}\eta_i(s)\, ds +\int_0^1 F\bigl(x(s)\bigr)\eta_i(s)\, ds,$$

where $\{\eta_i\}\subset L^\infty [0,1]$is an orthonormal basis in L²[0,1], L is a linear operator on L²[0,1] such that $\{\eta_i\}\subset D(L^{*})$ and $F\colon\, L^p[0,1]\to L^1[0,1]$ is a nonlinear mapping. The corresponding mapping B would be B(x)=Lx+F(x), but this mapping may not be well-defined on the support of a solution $\mu$ to (1). It will be discussed how to prove the existence of solutions to (1) on the basis of a priori estimates obtained by a suitable extension of the Lyapunov functions technique. An apropriate concept of differentiablity of solutions will be introduced, and some results on regularity of solutions will be presented. Analogous questions for manifolds will be briefly touched. Related open problems will be mentioned.

Z. Brzezniak (Hull): Maximal inequalities and exponential estimates for stochastic convolutions in Banach spaces with applications to Euler equations (Based on joint works with S. Peszat)

Abstract. Assume that X is a Banach space with the p-th power of the norm of C² class. Given a C₀-semigroup S of contractions on X consider the stochastic convolution process

$$x(t):=\int^t_0 S(t-r)\xi(r)\d W(r),\qquad t\ge 0.$$

It is shown that x satisfies the maximal inequality

$$\mathbb{E}\sup_{0\le t \le T}\vert x(t)\vert ^p \le K\, \mathbb{E}\, \Bigl( \int_0^T \Vert \xi (t)\Vert^2\d t\Bigr)^{p/2}$$

and some exponential tail estimates.

R. Buckdahn (Brest): About uniqueness of stochastic viscosity solutions of stochastic partial differential equations

Abstract. The aim of the talk is to study uniqueness of stochastic viscosity solutions of stochastic partial differential equations (SPDEs) of the type

$$\left\{ \begin{array}{l} du=\Big\{\big({1\over 2}{\rm tr}(\si... ...\\ u(0,x)=u_0(x),\hspace{6.5cm} x\in R^n. \end{array}\right.$$

After a short review of the notion of the stochastic viscosity solution a comparison theorem between a stochastic viscosity solution and the Doss transform of an $\omega$-wise viscosity solution of an associated partial differential equation will be established. This will lead us to uniqueness results. As a byproduct an extension of a fundamental lemme of Crandall-Ishii-Lions will be obtained.

The talk is based on to two common works with Jin Ma (Purdue University, West Lafayette, U.S.A.)

S. Cerrai (Firenze): Classical solutions of Kolmogorov equations in infinite dimensions

A. Chojnowska-Michalik (Lodz): Transition semigroups for stochastic semilinear equations on Hilbert spaces

Abstract. Let

$$dX=\lbrack AX+F(X)\rbrack dt-BdW, \eqno (\ast)$$

where A generates a C₀-semigroup of operators on a Hilbert space H.

Under the assumption that the corresponding (nonsymmetric) Ornstein-Uhlenbeck process has an invariant measure $\mu$, properties of the transition semigroup for $(\ast)$ in $L^p(H,\mu)$ spaces are investigated.

P. Chow (Detroit): On some stochastic hyperbolic equations

Abstract. For stochastic parabolic PDEs, the existence and regularity questions have studied extensively. However, relatively few such results were obtained for stochastic PDEs of hyperbolic type. In this talk we shall first present some recent results on the regularity of solutions to a class of linear and semilinear stochastic wave equations. In contrast with the parabolic equations, in general, the Ito formula does not hold for stochastic hyperbolic equations due to lack of regularity. However, under suitable conditions, it is possible to derive a stochastic energy inequality, based on which some a priori estimates can be established. For the linear case, higher-order or escalated energy inequalities can also be obtained. By means of such inequalities, it is proved that there exists a unique global solution with strong regularity properties in the space variables. But the escalated energy inequalities rarely exist for the nonlinear case. As a result, the solution if exists, may explode in finite time. We will also discuss some possible generalization of the above mentioned regularity results to some stochastic hyperbolic systems in one space-dimension. In addition to the energy inequality approach, the method of characteristics will be described.

H. Crauel (Berlin): Attractors of white noise systems have empty interior

Abstract. Let $\varphi$ be the stochastic flow generated by $dx=f(x)\,dt+\sigma\,dW(t)$ with $\sigma>0$ (additive noise), $x\in{\bf R}^d$. If $\varphi$ has a random attractor we show that `typically' the (Hausdorff, fractal or box covering) dimension of the attractor is smaller than the dimension of the ambient space. This holds, in particular, for $dx=f(x)\,dt+\varepsilon\,dW(t)$ and $\varepsilon>0$arbitrary - even if the limiting system $\dot x=f(x)$has an attractor containing unstable fixed points, and therefore having full dimension.

A.B. Cruzeiro (Lisboa): Tangent processes: some applications

Abstract. The study of Geometry on the Path space of a Riemannian manifold has shown the necessity to enlarge the usual notion of tangent space of the Wiener space; the notion of tangent process was introduced. In fact these processes are also useful to derive some finite dimensional results.

N. Cutland (Hull): Existence of a global stochastic flow, perfect cocyle and stochastic attractor for 2-D Navier-Stokes equations (joint work with Marek Capinski)

Abstract. For 2-D stochastic Navier Stokes equations on the torus (i.e. with periodic boundary conditions) with a special form of multiplicative noise we construct a global stochastic flow that is a perfect cocyle, and show the existence of a global random compact attractor. The equations considered do not admit a pathwise method of solution. The underlying probability space that carries these objects is a rich space that is a quotient of a Loeb space.

R. C. Dalang (Lausanne): Time reversal in certain hyperbolic s.p.d.e.'s

Abstract. We consider the question of changes of variables in the reduced hyperbolic equation in the plane, driven by two-parameter white noise. The simplest changes of variables are reversals in one or both coordinates, and we are interested in describing the process viewed in reversed coordinates as the solution of some s.p.d.e. We show that this is possible in certain cases in which the solution is closely related to the Brownian sheet, and in these cases we provide formulas for the coefficients of the equation satisfied by the process in reversed coordinates. We also provide examples that show that even minor changes in the initial conditions can prevent the process in reversed coordinates from solving any reasonable s.p.d.e. This is joint work with J.B. Walsh.

A. Debussche (Orsay): Global existence and blow-up for a stochastic nonlinear Schrödinger equation

M. Dozzi (Nancy I): Stochastic parabolic differential equations: local and global solutions
(Joint work with B. Maslowski)

Abstract. Consider the SPDE

$${{\partial u}\over{\partial t}}(t,\xi)=Lu(t,\xi)+F(u(t,\xi))+... ..._{t\xi},\quad (t,\xi)\in{\mathbb{R} }_+\times{\cal O}\eqno (1)$$

$$u(0,\xi)=u_0(\xi)\hspace{8cm}$$

with Dirichlet boundary conditions, where ${\cal O}\subset{\mathbb{R} }^d$ is a bounded domain with smooth boundary, L is a uniformly elliptic second order differential operator with smooth coefficients and $\stackrel{\scriptscriptstyle\circ}{W}$ is a space-dependent white noise in time. The aim is to give sufficient conditions on F and G for the existence of a global solution of (1) if F and G are only locally Lipschitz.

To give a precise meaning to (1) let

$$dX_t=AX_tdt+f(X_t)dt+g(X_t)dW_t,\qquad X_0=x\, , \eqno (2)$$

be the associated infinite dimensional equation, where the covariance operator Q of W is supposed to be selfadjoint, positive and such that $Tr\; Q<\infty$. The method of Lyapunov functions, adapted to the context of (2), yields a sufficient condition for the existence of a global solution in $H_\alpha :={\cal D}((-A)^\alpha)$ for some $\alpha>0$, whenever the initial condition x is in $H_\alpha$. $H_\alpha$ is equipped with the graph norm $\vert x\vert_\alpha =\vert(-A)^\alpha x\vert$.

An application of this result to (1) with ${\cal O}=[0,1],\; L=\partial^2/\partial\xi^2,\; u(t,\xi)=0$ for $\xi\in\{0,1\}$ leads to the following condition for the existence of a global solution in H_1/2=H¹([0,1])

$$F'(\theta)+(C^2_1+C^2_2)\, Tr\, Q\, G'(\theta)^2\leq k \eqno (3)$$

for some constant k. Here C₁ and C₂ are constants related to Q. This condition is applied to the case where F and G are polynomials. Finally, the case where W is the real valued Brownian motion is discussed.

References:

M. Dozzi, B. Maslowski, ``Lyapounov conditions for the existence of global solutions of stochastic parabolic differential equations'' In preparation.
G. Leha, B. Maslowski, G. Ritter, `` Stability of solutions to semilinear stochastic evolution equations'' Preprint (1998).

A. Eberle (La Jolla): L^p uniqueness of infinite dimensional diffusion operators

Abstract : We discuss the uniqueness of semigroups on L^p spaces generated by infinite dimensional diffusion operators of type $\Delta +\beta\cdot\nabla$. In particular, typical examples of non-unique operators are presented, and different approaches for proving uniqueness are demonstrated. The positive results are applied to some examples, including $P(\varphi )_2$ quantum fields in finite volume.

F. Fagnola (Genova): Stationary States for Quantum Markov Semigroups

Abstract. A Quantum Dynamical Semigroup (QDS) on the algebra $\mathcal{B}(h)$of bounded linear operators on a Hilbert space h is a w^*-continuous semigroup $\mathcal{T}=(\mathcal{T}_t)_{t\ge 0}$ of bounded w^*-continuous completely positive linear operators on $\mathcal{B}(h)$. A QDS is called a Quantum Markov Semigroup (QMS) if it is identity preserving i.e. $\mathcal{T}_t(1)=1$for each $t\ge 0$, (1 identity operator on h). A stationary state in a positive operator $\rho$ on h with unit trace such that $\text{trace}(\rho\mathcal{T}_t(x))=0$for $t\ge 0$.

QDS appeared in the physical literature during the seventies as a suitable mathematical framework for studying the irreversible evolution of open systems in quantum mechanics. In the extensive literature on QMS, however, there is a lack of really applicable results establishing the existence, uniqueness and properties of a stationary state for a QMS.

In this talk we provide simple but efficient criteria for answering these basic questions.

Loosely speaking, a stationary state exists, whenever one can find two self-adjoint operators X,Y where, in addition, X is positive and Y is bounded from below, with finite dimensional spectral projections associated with bounded intervals such that either

$$\int_0^\infty \mathcal{T}_s(Y)ds \le X,\eqno{(1)}$$

or the generator $\mathcal{L}$ satisfies

$$\mathcal{L}(X)\leq -Y. \eqno{(2)}$$

The first inequality clearly reads as property of the potential associated with the semigroup $(\mathcal{T}_t)_{t\geq 0}$. The inequality (2) allows us, under some natural technical hypoteses, to deduce (1). Moreover it is easier to check in the physical applications where the generator (in a form sense) usually is given instead of the semigroup.

We discuss then uniqueness and convergence of

$$\frac{1}{t}\int_0^t \text{trace}(\sigma\mathcal{T}_s(x)) ds,\qquad \text{and}\qquad \text{trace}(\sigma\mathcal{T}_t(x))$$

as $t\to\infty$.

Finally we outline applications to laser models and quantum spin systems.

F. Flandoli (Pisa): Probabilistic models in fluid dynamics

Abstract. After a short review of the activity of a research group in Pisa on stochastic equations of Navier-Stokes and Euler type (problems of existence and uniqueness of solutions, singularities, invariant measures and stationary solutions, ergodicity, random attractors, Kolmogorov equation) and probabilistic representation of solutions to Navier-Stokes equations, the talk will concentrate on the recent results on small scale structure in 3D fluids. We present some probabilistic model of vortex filaments based on Brownian motion and other continuous processes. We try to define continuous analogs of the lattice vortex structures introduced in the last years by A. Chorin. Vortex filaments turn out to have infinite self-energy, so we analyse the interaction energy between different filaments and we introduce vortex structures with a fractal cross section having finite energy. Gibbs measures, renormalization, vortex dynamics are discussed.

P. Florchinger (Metz): Stability and stabilization of nonlinear stochastic partial differential equations

Abstract. The aim of this talk is to provide Lyapunov criterions for the stability of nonautonomous nonlinear stochastic evolution equations. These criterions are applied to state sufficient conditions for the stabilizability of control stochastic partial differential equations.

M. Fuhrman (Milano): Some results on transition probabilities and invariant measures for stochastic evolution equations

Abstract. We investigate regularity properties for transition probabilities and invariant measure of Markov processes in Hilbert spaces, arising as solutions of stochastic evolution equations driven by Wiener process. We are concerned with proving existence of logarithmic derivatives, existence of densities (with respect to properly chosen reference measures) and their summability and regularity properties.

B. Goldys (Sidney): Invariant measures and ergodic properties for some nonsymmetric dissipative systems

Abstract. In this work we study ergodic properties of semilinear stochastic equations with dissipative drift. We assume existence of invariant measure but we do not assume that it has a density with respect to some reference measure and we do not assume that the transition semigroup is symmetric. Under some conditions we prove the spectral gap property and the Logarithmic Sobolev inequality for the invariant measure. In the case when the invariant measure has a density with respect to a Gaussian measure we give conditions for regularity of the density. As a special case we study stochastic reaction-diffusion equation with additive noise. For drifts without the strong dissipativity property we show that the transition semigroup is compact in the space L² (invariant measure) and in particular, exponential convergence to equilibrium holds in this space.

F. Gozzi (Roma I): Hamilton-Jacobi-Bellman equations for the optimal control of the Duncan-Mortensen-Zakai equation (joint work with Andrzej Swiech)

Abstract. We study a class of Hamilton-Jacobi-Bellman (HJB) equations associated to stochastic optimal control of the Duncan-Mortensen-Zakai equation. The equations are investigated in weighted L²spaces. We introduce an appropriate notion of weak (viscosity) solution of such equations and prove that the value function is the unique solution of the HJB equation. We apply the results to stochastic optimal control problems with partial observation and correlated noise.

G. Jona-Lasinio (Roma I): Gibbs measures invariant under Schroedinger evolution and quantum statistical mechanics

Abstract. If one considers the evolution of quantum mean values of a system of oscillators the classical Gibbs measure appears naturally as the equilibrium measure. It will be shown that the Gibbs measure associated to the Schroedinger equation viewed as an infinite dimensional dynamical system induces such classical equilibrium ensemble on mean values. This type of analysis can be extended to systems with superquadratic Hamiltonian and at low temperature the so called effective potential replaces the classical potential. The quantum states typical for the ensembles considered are coherent-like states.

Y. Kondratiev (Bonn): Non-equilibrium stochastic dynamics and diffusion hierarchy for continuous systems

P.M. Kotelenez (Cleveland):Stochastic Partial Differential Equations in the Derivation of Macroscopic Equations from Microscopic Equations

Abstract. Two types of particles are considered. N ``big" particles and M ``small" particles. M is much bigger then N. The time evolution of the joint system is described by a system of ordinary differential equations (the microscopic equations). Through space-time scaling and random sampling the influence of the ``small" particles on the displacement of the ``big" particles is replaced by correlated Brownian motions, where the correlation depends on the distance between the ``big" particles and the potentials governing the ordinary differential equations. The empirical distribution of the ``big" particles is the sum of the point measures supported by the positions of the ``big" particles and multiplied by their respective masses. This empirical distribution turns out to be the solution of a quasilinear stochastic partial differential equation (SPDE), which can be extended (letting N go to infinity) to an SPDE on the space of smooth measures. This SPDE is called a mezoscopic equation. As the correlation length, determined by the potentials, tends to 0, the solutions of the SPDE's tend to the solution of a macroscopic (i.e., deterministic) quasilinear partial differential equation (macroscopic limit).

The macroscopic limit is joint work with T. Kurtz.

N. Krylov (Minneapolis): L_q(L_p)-theory of SPDEs in bounded domains

Abstract. Existence and uniqueness theorems are presented for evolution stochastic partial differential equations of second order in L_p-spaces with weights allowing derivatives to blow up near the boundary. It is allowed for the powers of summability with respect to space and time variables to be different.

H. Kunita (Fukuoka): Densities of conditional distributions of a nonlinear filter

Abstract. Suppose that the signal process (system process) is governed by a canonical SDE with jumps

$$d\xi_t=V_0(\xi_t)dt+\sum_{j=1}^mV_j(\xi_t)\diamond Z^j_t,$$

where V₀,...,V_m are smooth vector fields and Z_t=(Z¹_t,...,Z^m_t) is a Lévy process. The observation is employed by

$$Y_t=\int^t_0h(\xi_s)ds+W_t,$$

where W_t is a Brownian motion independent of the signal process. Let $\pi$ be the conditional distribution of $\xi_t$ based on the observation data $\sigma(Y_s; s\leq t)$: $\pi(A)=P(\xi_t\in A\vert\sigma(Y_s;s\leq t))$. The existence of the smooth density of $\pi(dx)$ has been studied by Michel, Bismut-Michel, Kusuoka-Stroock, Kunita and others, in connection with the Malliavin calculus.

In this talk, we discuss the case where the signal process is a jump process. Suppose that the driving process Z_t is a nondegenerate Lévy process not including the Gaussian part such that its Lévy measure satifies an order condition $\mu(B_\epsilon)\sim \epsilon^{(\alpha-2)}$ for some $0<\alpha<2$, where $B_\epsilon=\{z\in {\bf R}^m;\vert z\vert<\epsilon\}$. Suppose further that the function h together with its derivatives is bounded. Then, if the family of the vector fields $\{V_0,V_1,...,V_m\}$satisfies a uniform Hörmander condition, then the conditional distribution $\pi(dx)$ has a smooth density.

The existence of the smooth density of the conditional distribution of the nonlinear filter was studied in the case where the signal process is a diffusion process.

P. Malliavin (Paris): Radonification on the diffeomorphism group of the circle

R. Manthey (Jena): Existence of invariant measures for stochastic reaction-diffusion equations

Abstract. We give sufficient conditions guaranteeing existence of invariant measures for a stochastic reaction-diffusion equation with multiplicative noise.

B. Maslowski (Praha): Strong exponential and geometric convergences of Markov semigroups associated to SPDE's

Abstract. Strong convergence of probability laws of solutions to the stochastic evolution equation of the abstract form

$$dX_t= (AX_t + f(t,X_t))dt +Q^{1/2}dW_t,\quad t\in R_+, \eqno(1)$$

are studied, where A is an unbounded linear operator on a separable Hilbert space H, $Q^{1/2}\in {\cal L}(H)$, W_tdenotes a cylindrical Wiener process on H, and $f:R_+\times E \to H$, E being a Banach space continuously embedded into H. If f(t,x)=f(x) does not depend on $t\in R_+$ and the equation (1) induces a homogeneous Markov process it is shown to be geometrically (or exponentially) ergodic under respective natural conditions. As a by-product, a formula for density of the Markov transition measure with respect to a Gaussian measure is derived, which can be of independent interest. Part of the results were obtained jointly with Isabel Simao.

J.L. Menaldi (Detroit): Impulse control of stochastic 2-D Navier-Stokes equations Joint work with S.S. Sritharan (US Navy, San Diego, USA)

Abstract. We are interested in the impulse control of a stochastic Navier-Stokes equation in a two-dimensional domain. To be able to use the semigroup technique, we first consider a 2-D Navier-Stokes equation with random (Gaussian) forcing field. Under suitable condition and based on a local monotone property, we show existence and uniqueness of strong solutions and therefore a Markov-Feller process is constructed. At this point, the quasi-variational inequality technique can be used to deal with the impulse control in a weak sense.

S. Mitter (MIT Cambridge): Stochastic control information theory and statistical mechanics

J.M. Noble: Euler Lagrange equations and the problems of global minimisation

Abstract. In the first part of the talk, construction of a process which has the quantum probability densities will be outlined for time independent potentials and it will be shown that the $\hbar \rightarrow 0$limit yields the classical mechanics. The technique is based upon eigenvalue problems and therefore may not be extended to time dependent potentials. Furthermore, the result may not be true in the time dependent setting. The `dominated convergence theorem' is discussed, showing that it is not possible to conclude that the `global minimiser' of an action functional, taken as the limit of a minimising sequence, necessarily minimises the action functional. It means that the global minimiser does not necessarily satisfy the Euler Lagrange equations. An example of an smooth, bounded, periodic potential where it does not is given; namely, the potential V(t,x) = cos (x + sin t). This casts doubt over the `intuition' suggesting that the $\hbar \rightarrow 0$ limit of a quantum dynamics should yield a classical dynamics. It also exposes a flaw in the key step of the argument in recent work by Weinan E, Khanin, Mazel and Sinai on stochastic Burgers' equation, where the method of proof is wrong and the results are also wrong.

S. Peszat (Poland): The Cauchy problem for a nonlinear stochastic wave equation in any dimension

Abstract. Conditions for the existence of a function valued solution to a nonlinear stochastic wave equation are given. The noise is a spatially homogeneous Wiener process. All dimensions are considered.

J. Potthoff (Mannheim): White noise analysis approach to SPDE's

M. Roeckner (Bielefeld): Uniqueness of invariant measures and maximal dissipativity of diffusion operators on L¹

Abstract. It is proved that there exists at most one probability measure $\mu$on $\bf R^d$, so that $L^*\mu = 0$, where $L = a^{ij} \partial_i\partial_j + b^i\partial_i$, provided $(L,C^\infty_0(\bf R))$ is maximally dissipative on $L^1(\bf R^d,\nu)$ for at least one $\nu$, so that $L^*\nu=0$. Here it is assumed that (a^ij is non-degenerate, $a^{ij} \in H^{p,1}_{loc}$, and $b^i \in L^p_{loc}$. We also present a whole class of examples (even for $a^{ij} = \delta^{ij}$), where $L^*\mu = 0$ has more than one solution. Furthermore, recent related results are reviewed.

B.L. Rozovskii (Los Angeles): Stochastic fluid flows and Wiener chaos

Abstract. In this talk we are concerned with fluid dynamics described by stochastic flows of diffeomorphisms. Stochastic Euler and Navier-Stokes equations will be derived from the conservation laws of mass and momentum. Well-posedness of these equations shall be discussed. A Wiener chaos expansion of the velocity field will be presented and formulas for its statistical moments will be derived.
The talk is based on a joint work with R. Mikulevicius.

C. Rovira (Paris 13): Varadhan estimates for a stochastic delay equation (joint work with M. Ferrante and M. Sanz-Sole)

We consider the stochastic delay equation dX(t) = H(t,X)dt+g(t,X(t-r))dW(t), t >0 and X(t) = f(t) if -r < t <=0, where r is a positive time delay and H depends on the whole path X. Using partial Malliavan calculus we obtain Varadhan estimates for the family of densities obtained from small perturbations of our initial equation.

F. Russo (Paris 13): On some stochastic differential equations with distributional drift and Lyons-Zheng processes
Joint works with F. Flandoli (Pisa) and J. Wolf (Jena).

Abstract. We study a martingale problem related to a parabolic PDE operator L with continuous (non-degenerate) diffusion term and with drift being the derivative of a continuous function. In several cases, this martingale problem turns out to be a true stochastic differential equation even though the solution X is generally not a semimartingale. However X is a special case of Dirichlet process, that is to say a Lyons-Zheng (LZ) process. For LZ processes, a framework of stochastic calculus has been developed. In case the solution is a semimartingale, we also establish an Itô formula under very weak conditions for f(X). For the moment, this study is only restricted to dimension 1.

M. Sanz-Solé (Barcelona): Hölder properties of some classes of spde's

Abstract. Let $M=\{M_{t}(A),\,t\in {\bf R}_{+},\,A\in {\cal B}_{b}({\bf R}^{d})\}$be a Gaussian martingale measure with covariance function $E(M_t(A) M_t(B))= t\int_{{\bf R}^d} \Big(\,1_A(\cdot)\ast\,\,\tilde 1_B(\cdot)\Big)(x) \Gamma(dx)$where $\Gamma$ is a non-negative tempered measure. Consider a space-time distribution S such that the function $t\rightarrow S(t)$ takes its values in the space of distributions with rapid decrease. Moreover, if $\mu$is the spectral measure of $\Gamma$ and ${\cal F}$ denotes the Fourier transform, suppose that for any T>0,

$$\int_{0}^{T}ds\int_{{\bf R}^{d}}\mu (d\xi ) \vert{\cal F} S(s,\cdot )(\xi )\vert^{2}\,<+\infty \eqno (1)$$

and

$$\lim_{h\downarrow 0}\int_{0}^{T}ds\int_{{\bf R} ^{d}}\mu (d\x... ...dot )(\xi )-{\cal F} S(r,\cdot )(\xi )\Big\vert^{2}=0.\eqno (2)$$

In [Dalang, EJP, 1999] it is proved that, assuming (1) and (2) the indefinite stochastic integral of S with respect to Mcan be defined as a real-valued martingale with respect to the filtration generated by M. Moreover, if S(t) is positive the integrator M can be extended to martingale measures obtained by stochastic integration of previsible processes with respect to M.

These results are the first step for proving the existence and uniqueness of real valued solutions for some classes of spde's with spatial dimension $d\geq 1$driven by the noise M. In particular, one can consider the nonlinear stochastic wave equation with $1\leq d\leq 3$and the stochastic heat equation for any d (see also Peszat-Zabczyk (1998) and Karczewska-Zabczyk (1997, 1999) for related work).

We are interested in the Hölder continuity in time and in space of the solutions of these equations. We study two particular examples: (a) the semilinear stochastic wave equation ${{\partial^2u}\over{\partial t^2}}-\Delta u= \dot M+b(u)$, with suitable initial conditions, $1\leq d\leq 3$, (b) the nonlinear stochastic heat equation ${{\partial u}\over{\partial t}} + {1\over 2} \Delta u = \sigma(u)\dot M + b(u)$with a fixed initial condition and any spatial dimension d.

Assume that there exists $\eta \in (0,1)$ such that

$$\int_{{\bf R}^d}(1+\vert\xi\vert^2)^{-\eta}\mu(d\xi)<\infty\eqno (3),$$

then the corresponding solution processes are $\gamma$-Hölder continuous in time with $\gamma\in (0,1-\eta]$ in case (a) and $\gamma\in (0,1-1/2(1-\eta)]$ in case (b). In both situations condition (3) ensures $\gamma$-Hölder continuity in space with $\gamma\in (0,1-\eta]$.

Using well-known results on the Bessel kernel, condition (3) can be expressed in terms of integrability conditions on $\Gamma$ depending on the dimension d.

I. Simão (Lisbon): Essential Self-adjointness of perturbed Ornstein-Uhlenbeck operators on Hilbert spaces (Joint work with H. Long of University of Alberta, Edmonton, Canada)

Abstract. We show the essential-selfadjointness of Ornstein-Uhlenbeck operators under certain drift and potential perturbations, on Hilbert spaces, by first establishing the existence and uniqueness of classical solutions to the associated Kolmogorov equations and then giving a gradient estimate for the classical solutions.

O.G. Smolyanov (Moscow): Stochastic Schrödinger equations for Hamiltonian systems with constraints

Abstract. A stochastic version of a system of partial differential equations obtained by a quantization of Hamiltonian systems with constraints is discussed and a Feynman path integral representation for solutions of these equations is given. Some of these results will be published in a paper written together with Prof.A.Truman.

R. Sowers (Urbana): Stochastic averaging and stratified spaces: a Markov process on a lollipop

Abstract. We consider a random perturbation of a 2-dimensional Hamiltonian ODE. Under an appropriate change of time, we identify a reduced model, which in some aspects is similar to a stochastically-averaged model. The novelty of our problem is that the set of critical points of the Hamiltonian has an interior. Thus we can stochastically average outside this set of critical points, but inside we can make no model reduction. The result is a Markov process on a stratified space which looks like a tetherball (i.e, a 2-dimensional sphere with a line attached). At the junction of the ball and the tether, glueing conditions identify the behavior of the Markov process. We discuss both the existence of the limit and some other interesting features.

W. Stannat (New York): On the Cauchy-problem for singular diffusion operators

Abstract. Let $L=a^{ij}\partial_{ij} + b^{i}\partial_i$ be a locally strictly elliptic differential operator defined on $C_0^\infty (U)$, $U\subset R^d$ open, and $\mu$ be a nonnegative measure satisfying the equation $L^*\mu = 0$. We study existence, uniqueness and quasi-regularity of maximal extensions of L in $L^1 (\mu )$ under minimal regularity assumptions on the coefficients. We also consider infinite dimensional analogues of our finite dimensional results and give applications to Nelson diffusions on possibly infinite dimensional state spaces as well as two-dimensional stochastic Navier-Stokes equations on bounded domains with periodic boundary conditions.

D.W. Stroock (MIT Cambridge): An application of ultracontractive estimates to a Hodge theory for non-compact manifolds

Abstract. The analytic essence of Hodge theory on compact manifolds is provided by the existence of a "good" parametrix for the Holdge Laplacian. That is, the parametrix is "good" in the sense that it inverts the Hodge Laplacian up to a compact operator. On non-compact manifolds, even though the parametrix for the classical Hodge operator may exist, it will not be "good." In this lecture, I will discuss a modified Hodge operator which, under appropriate conditions, admits a "good" parametrix and leads to a Hodge theory for the DeRham coholmogy of tempered forms.

G. Tessitore (Genova): Semigroup Approach to the Wong-Zakai approximations for SPDEs (Joint work with J. Zabczyk)

Abstract. We apply the semigroup technique to show that the noise in an infinite dimensional stochastic partial differential equation can be approximated by polynomial approximations. As in the basic paper by Strook and Varadhan [Proc. Berkeley Symp. 72] the proof consists into two parts. The first shows using factorization lemma that the measures induced by the solutions of the approximated equations form a tight family. The second (the identification part) shows that the limiting measure is exactly the measure induced by the solution of the original equation. The purpose is to use the approximation result to obtain sufficient conditions for the invariance of sets with respect to stochastic evolutions of interest in mathematical finance.

R. Tribe (Warwick): Duality formulae for some Stochastic PDEs

Abstract. A duality formula relates the expectation of a functional of one stochastic process in terms of an expectation for a second stochastic process. This talk describes some duality formulae for Stochastic PDE's driven by space-time white noise, including some equations derived from interactive branching processes.

A. Truman (Swansea): Stochastic heat and Burgers equations and their singularities

Abstract. The effect of white noise on the shock waves and wavefronts for the inviscid limit of Burgers equation and the corresponding heat equation is discussed. In particular, we show how in general the shape of the wavefronts and shock waves can vary with the introduction of small noise.

P. Vuillermot (Nancy I): Some recent results concerning semilinear parabolic Ito equations

A.N.K. Yip (Purdue): Noise and non-uniqueness of motion by mean curvature

Abstract. We will discuss the issue of non-uniqueness in the motion by mean curvature of a surface. Such geometric motion is a simplified model for solidification processes. We will show that (white) noise has the potential of eliminating the non-uniqueness phenomena of the evolution and thus giving a selection principle for the solutions.

J. Zabczyk (Warsaw): Excessive measures and optimal stopping

Abstract. Excessive measures for several classes of finite and infinite dimensional Markov processes are constructed. Special attention is paid to the infinite dimensional Wiener process and to the Ornstein-Uhlenbeck processes. Then the measures are treated as weights in the spaces of square summable spaces in which Bellman's equations corresponding to stopping problems are studied. Applications to pricing American options both in finite and infinite markets are presented as well.

References:
D.Gatarek and M.Musiela, ``Pricing of American interest rate derivatives as optimal stopping of Ornstein-Uhlenbeck processes", manuscript, 1996.
J.Zabczyk, ``Stopping problems on Polish spaces", Annales Univ. MCS, LI(1997), 181-199.
J.Zabczyk, ``Bellman's inclusions and excessive measures", Preprints No.8, Scuola Normale Superiore di Pisa, 1998.

O. Zeitouni (Haifa): Quenched Large Deviations for one dimensional Nonlinear Filtering (Joint work with E. Pardoux)

Abstract. Consider the standard, one dimensional, nonlinear filtering problem for diffusion processes observed in small additive white noise: $d\Xi_t=\beta(\Xi_t)dt+ \sigma(\Xi_t)dB_t\,, dY_t^\epsilon=\gamma(\Xi_t)dt+\epsilon dV_t\,,$ where $B_\cdot, V_\cdot$are standard independent Brownian motions. Denote by $q^\epsilon_1(\cdot)$ the density of the law of $\Xi_1$ conditioned on $\sigma(Y_t^\epsilon: 0\leq t\leq 1)$. We provide ``quenched" large deviation estimates for the random family of measures $q^\epsilon_1(x)dx$: there exists a continuous, explicit mapping $\bar {\cal J}~: \mathbb{R} ^2\to\mathbb{R}$ such that for almost all $B_\cdot, V_\cdot$, $\bar {\cal J}(\cdot,\Xi_1)$ is a good rate function and for any measurable $G\subset \mathbb{R}$,

$$\begin{eqnarray*}\lefteqn{ -\inf_{x\in G^o} \bar {\cal J}(x,\Xi_1) \leq \liminf_... ...epsilon(x) dx \leq -\inf_{x\in \bar G} \bar {\cal J}(x,\Xi_1)\,. \end{eqnarray*}$$