Szymon Peszat

Jagiellonian University | UJ · Institute of Mathematics

Let (Pt) be the transition semigroup of the Markov family (Xx(t)) defined by SDE dX=b(X)dt+dZ,X(0)=x,where Z=Z1,…,Zd∗ is a system of independent real-valued Lévy processes. Using the Malliavin calculus we establish the following gradient formula ∇Ptf(x)=EfXx(t)Y(t,x),f∈Bb(Rd),where the random field Y does not depend on f. Moreover, in the important...

We study inhomogeneous Dirichlet boundary value problems associated to a linear parabolic equation $\frac{du}{dt}=Au$ with strongly elliptic operator $A$ on bounded and unbounded domains with white noise boundary data. Our main assumption is that the heat kernel of the corresponding homogeneous problem enjoys the Gaussian type estimates taking into...

We study the existence of the stochastic flow associated to a linear stochastic evolution equation $$d X= AX\,d t +\sum_{k} B_k X\,d W_k, $$ on a Hilbert space. Our first result covers the case where $A$ is the generator of a $C_0$-semigroup, and $(B_k)$ is a sequence of bounded linear operators such that $\sum_k\|B_k\|<+\infty$. We also provide su...

We consider a consumption-investment problem in which the investor has an access to the bond market. In our approach prices of bonds with different maturities are described by the general HJM factor model. We assume that the bond market consist of entire family of rolling bonds and the investment strategy is a general signed measure distributed on...

We consider a stochastic version of a system of coupled two equations formulated by Burgers with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness of the solution as well as the irreducibility property of such system were given by Twardowska and Zabczyk. In the paper the existence of a uniqu...

Let $(P_t)$ be the transition semigroup of the Markov family $(X^x(t))$ defined by SDE $$ d X= b(X) dt + d Z, \qquad X(0)=x, $$ where $Z=\left(Z_1, \ldots, Z_d\right)^*$ is a system of independent real-valued L\'evy processes. Using the Malliavin calculus we establish the following gradient formula $$ \nabla P_tf(x)= \mathbb{E}\, f\left(X^x(t)\righ...

In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier-Stokes equation (N.S.E.) system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant measure for the system and give a sufficient condition under which it is unique and sto...

Inverse problems for stochastic linear transport equations driven by a temporal or spatial white noise are discussed. We analyse stochastic linear transport equations which depend on an unknown potential and have either additive noise or multiplicative noise. We show that one can approximate the potential with arbitrary small error when the solutio...

These are the notes for my two 90 minutes talks on some aspects of SPDEs with Lévy noise, presented during a semester on SPDEs in EPF Lausanne and then in the Institute of Applied Mathematics, Chinese Academy of Sciences. The first talk was devoted to analytical aspects of the theory: the form of the generator of a Markov semigroup in finite and in...

Let $(P_t)$ be the transition semigroup of a L\'evy process $L$ taking values in a Hilbert space $H$. Let $\nu$ be the L\'evy measure of $L$. It is shown that for any bounded and measurable function $f$, $$ \int_H\left\vert P_tf(x+y)-P_tf(x)\right\vert ^2 \nu (\dif y)\le \frac 1 t P_tf^2(x) \qquad \text{for all $t>0$, $x\in H$.} $$ As $\nu$ can be...

The dilation theorem of Nagy is applied to establish time regularity of the solutions to a class of stochastic evolutionary Volterra equations.

K. It\^{o} characterised in \cite{ito} zero-mean stationary Gauss Markov-processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of It\^{o} in the case of Hilbert spaces: Gauss-Markov families that are time-homogenous are identified as solutions to linear stochastic differential equations with singular coeffici...

In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existenc...

The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert space HH, driven by a Lévy process taking values in a Hilbert space U↩HU↩H is established. The so-called cylindrical càdlàg property is investigated as well. A special emphasis is put on infinite systems of linear equations driven by independent Lévy pr...

In this paper we prove the law of large numbers and central limit theorem for trajectories of a particle carried by a two dimensional Eulerian velocity field. The field is given by a solution of a stochastic Navier--Stokes system with a non-degenerate noise. The spectral gap property, with respect to Wasserstein metric, for such a system has been s...

Let mu be an invariant measure for the transition semigroup (P-i) of the Markov family defined by the Ornstein-Uhlenbcck type equation dX = A X dt + dL on a Hilbert space E, driven by a Levy process L. It is shown that for any t >= 0, P-t considered on L-2(mu) is a second quantized operator on a Poisson Fock space of e(Al). From this representation...

Let β be a standard Brownian motion, let X be an α-stable process, and let f=[^(m)]f=\widehat \mu be the Fourier transform of a discrete measure. It is shown that weakly in C([0, + ∞ )), $ \eta ^{\alpha /2} \int_0^t f(\eta X_s)\text{d}s \Rightarrow \sqrt{C_{f,\alpha}}\beta_t\qquad \text{as $ \eta ^{\alpha /2} \int_0^t f(\eta X_s)\text{d}s \Rightar...

We study nonlinear heat and wave equations on a Lie group. The noise is assumed to be a spatially homogeneous Wiener process. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise.

We consider fluctuations of the solution W ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the ini...

The paper is concerned with the properties of solutions to linear evolution equation perturbed by cylindrical L\'evy processes. It turns out that solutions, under rather weak requirements, do not have c\`adl\`ag modification. Some natural open questions are also stated.

Introduction Part I. Foundations: 1. Why equations with Levy noise? 2. Analytic preliminaries 3. Probabilistic preliminaries 4. Levy processes 5. Levy semigroups 6. Poisson random measures 7. Cylindrical processes and reproducing kernels 8. Stochastic integration Part II. Existence and Regularity: 9. General existence and uniqueness results 10. Equ...

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the ex...

We establish a law of large numbers and a central limit theorem for a class of additive functionals related to the solution of a one-dimensional stochastic differential equation perturbed by a large noise.

The trajectories of a passive tracer in a turbulent flow satisfy the ordinary differential equation x(t)=V(t,x(t)), where V(t,x) is a stationary random field, the so-called Eulerian velocity. It is a nontrivial question to define the dynamics of the tracer in the case when the realizations of the Eulerian field are only spatially Hlder regular beca...

A nonlinear wave equation on driven by a spatially homogeneous Wiener process is studied. Conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise are given for an arbitrary dimension d.

We formulate a stochastic differential equation describing the Lagrangian environment process of a passive tracer in Ornstein-Uhlenbeck velocity fields. We subsequently prove a local existence and uniqueness result when the velocity field is regular. When the Ornstein-Uhlenbeck velocity field is only spatially Hölder continuous we construct and ide...

The existence of a martingale solution to 2-dimensional stochastic Euler equations is proved. The constructed solution is a limit as the viscosity converges to zero of a sequence of solutions to modified Navier–Stokes equations.

Let B be a Brownian motion, and let C\textp \mathcal{C}_{\text{p}} be the space of all continuous periodic functions f C\textp \mathcal{C}_{\text{p}} such that the stochastic convolution Xf,B (t) = ò0t f(t - s)\textdB(s),t Î [0,1] X_{f,B} (t) = \int_0^t {f(t - s){\text{d}}B(s),t \in [0,1]} does not have a modification with bounded trajecto...

We give a necessary and sufficient condition for a Gibbs measure µ on the product space Ω = (S1)d to satisfy the spectral gap or the logarithmic Sobolev inequality with the following quadratic form: where Y is a finite set and al are integers. As a consequence we prove that the generalized Kawasaki dynamics decays exponentially to equilibrium in t...

We study nonlinear wave and heat equations on ℝ d driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ℝ d -space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of t...

Criteria for non-explosion of solutions to semilinear parabolic evolution equations on Banach spaces are given. The abstract results are applied to classical semilinear parabolic equations: heat equation with first order nonlinear perturbations, fourth order Cahn-Hilliard, FitzHugh-Nagumo system.

Let mu(Y) and mu((Y) over tilde) be the laws on C([0, T]; R(d)) Of the Gaussian processes Y(t) = integral(0)(t) K(t - s) dW(s) and (Y) over tilde(t) = integral(0)(t) (K) over tilde(t - s) dW(s), where K and (K) over tilde are entire matrix valued mappings, and W is a Wiener process. We give a necessary and sufficient condition for the mutual absolu...

Stochastic partial differential equations on ℝd are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted Lq-space. Then we obtain the existence of a space continuous solution by means of the Da P...

Space-time regularity of stochastic convolution integrals J=∫ 0 · S(·-r)Z(r)dW(r) driven by a cylindrical Wiener process W in an L 2 -space on a bounded domain is investigated. The semigroup S is supposed to be given by the Green function of a 2mth order parabolic boundary value problem, and Z is a multiplication operator. Under fairly general assu...

. We consider stochastic Navier--Stokes equations on a possibly unbounded domain O ` R d , where d is equal to 2 or 3. First we prove the existence of a martingale solution for the initial value being a probability measure on the space of square integrable R d -valued functions. Then we show the existence of a spatially homogeneous solution to the...

A semilinear parabolic equation on d with a non-additive random perturbation is studied. The noise is supposed to be a spatially homogeneous Wiener process. Conditions for the existence and uniqueness of the solution in terms of the spectral measure of the noise are given. Applications to population and geophysical models are indicated. The Freidli...

For stochastic Navier-Stokes equations in a 3-dimensional bounded domain we first show that if the initial value is sufficiently regular, then martingale solutions are strong on a random time interval and we estimate its length. Then we prove the uniqueness of the strong solution in the class of all martingale solutions.

The present paper is concerned with the existence and uniqueness of the solution for the stochastic equation where A is the generator of a C 0-semigroup on a Hilbert space H, F and G are defined on some subspace of H, and W stands for a cylindrical Wiener process on H. As a special case we consider a stochastic reaction diffusion equation

It is shown that the transition semigroup $(P_t)_{t\geq0}$ corresponding to a nonlinear stochastic evolution equation is strong Feller and irreducible, provided the nonlinearities are Lipschitz continuous and the diffusion term is nondegenerate. This result ensures the uniqueness of the invariant measure for $(P_t)_{t\geq0}$.

The large deviation principle obtained by Freidlin and Wentzell for measures associated with finite-dimensional diffusions is extended to measures given by stochastic evolution equations with non-additive random perturbations. The proof of the main result is adopted from the Priouret paper concerning finite-dimensional diffusions. Exponential tail...

Let (H,〈·,·〉) be a real separable Hilbert space and let μ be a Gaussian measure on H. The Sobolev space W 1,2 (H,dμ) is constructed. It is proved that W 1,2 (H,dμ) is compactly embedded in L 2 (H,dμ).

The behaviour of the probabilities ℙ(sup|X(t)| E ≥δ), where X is a stochastic convolution, is studied. It is proved that under some reasonable conditions these probabilities are dominated by the terms Cexp(-δ 2 /κ 2 η). Some factorization method and exponential estimates for stochastic integrals are used.

Sufficient conditions for equivalence of distributions in LZ(O, T, H) of two Ornstein-Uhlenbeck processes taking values in a Hilbert space H are given. The Girsanov theorem and some facts in the theory of perturbations of semigroup generators are used.

Sufficient and necessary conditions for the equivalence of the distributions of the solutions of some linear stochastic equations in Hilbert spaces are given. Some facts in the theory of perturbations of semigroup generators and Zabczyk’s results on law equivalence are used.

The paper is devoted to the stochastic partial differ-ential equation for the forward curve of the bond market, in the Musiela parameterizations and the Heath-Jarrow-Morton frame-work. Special attention is paid to the existence, positivity and long-time behavior of the solutions.

Having a Borel measure on a Banach space B one may define Sobolev spaces of functions on B. In this paper, the problem whether these spaces are compactly imbedded into the space of square integrable functions is considered. Some part of the paper is devoted to the case of Gaussian measures on B. In fact in this case the problem of compactness of th...

In the first part of this paper we study existence and uniqueness of strong solutions to stochastic Navier-Stokes equations in space dimension ≤3. Our main results obtained by applying a general framework developed elsewhere by the first named author. In the second part we show existence and uniqueness of global solutions to certain modified stocha...