The Kolmogorov Operator Associated to a Burgers SPDE in Spaces of Continuous Functions

Abstract

We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.

Kolmogorov operator and Fokker–Planck equation associated to a stochastic Burgers equation driven by Lévy noise

“…The aim of the present paper is to extend the results of [18] for stochastic Burgers equation driven by Wiener noise to the equation driven by Lévy noise. That is, we will investigate the transition semigroup of the solution in the space of continuous functions weighted by a proper potential and we show that the infinitesimal generator is the closure of the Kolmogorov operator associated to the equation in a suitable topology.…”

“…In recent years, many people have been interested in studying Burgers turbulence in the presence of random forces. Most works are with white noise, for example, [1], [2], [3], [4], [5], [6], [7], [16], [12], [13], [14], [17], [18], [19], [21] and references therein. As far as we know, there are only a few articles dealing with the case of non-white noise, see [11], [22], [23] for the Lévy noise.…”

“…Moreover, the semigroup e tA can be extended to L p (0, 1) (p > 1), (1.1) e tA x L p ≤ e γpt |x| L p , x∈ L p (0, 1), where γ p = 2p −1 (p−1)π 2 . Note that the usual Sobolev space W k,p (0, 1) can be extended to W s,p (0, 1), for s ∈ R, then by [18], e tA (t ≥ 0) have the smoothing properties: for any s 1 , s 2 ∈ R with s 1 ≤ s 2 , r ≥ 1, e tA : W s1,r (0, 1) → W s2,r (0, 1) and there exists a constant C 1 depending on s 1 , s 2 , r such that (1.2) e tA z W s 2 ,r (0,1) ≤ C 1 1 + t (s1−s2)/2 |z| W s 1 ,r (0,1) , z ∈ W s1,r (0, 1).…”

Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation

“…The introduction of randomness in Burgers equation produced a number of very interesting new directions; directions connected with dynamical systems aspects of the equation, e.g. existence and properties of invariant measures (see for instance the important contributions by E et al [19] or Goldys and Maslowski [23]), directions related to various questions on the well-posedeness of the equation in various functional settings using techniques from infinite dimensional stochastic analysis (see for instance the important contributions of Da Prato, Deboussche, Nualart and others [15,16,25,24,33,40]), interesting connections with geometry (see for instance the contributions by Cruzeiro and Malliavin [13] or Davies, Truman, and Zhao [18]), connections with the theory of superprocesses in [4] etc. This theoretical work was inspired by issues related to turbulence (see e.g.…”

Forward - backward stochastic differential equations with random coefficients and applications to finance

“…The introduction of randomness in Burgers equation produced a number of very interesting new directions; directions connected with dynamical systems aspects of the equation, e.g. existence and properties of invariant measures (see for instance the important contributions by E et al [100] or Goldys and Maslowski [48]), directions related to various questions on the well-posedeness of the equation in various functional settings using techniques from infinite dimensional stochastic analysis (see for instance the important contributions of Da Prato, Deboussche, Nualart and others [33,34,50,49,60,73]), interesting connections with geometry (see for instance the contributions by Cruzeiro and Malliavin [30] or Davies, Truman, and Zhao [36]), connections with the theory of superprocesses in [13] etc. This theoretical work was inspired by issues related to turbulence (see e.g.…”