Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces

Chojnowska-Michalik, Anna; Gołdys, Beniamin

doi:10.1007/bf01192465

Cited by 86 publications

(69 citation statements)

References 14 publications

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“…For the reader's convenience we will amend this section with three propositions which are minor modifications of earlier results [20], [18] and [4], in which Hypothesis 1.5 (strong Feller property and irreducibility) is verified. Proposition 2.7.…”

Section: Uniform Exponential Ergodicity and Some Auxiliary Resultsmentioning

confidence: 99%

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Uniform Exponential Ergodicity of Stochastic Dissipative Systems

Gołdys

Maslowski

2001

Czechoslovak Mathematical Journal

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Abstract. We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in R d with d ≤ 3. IntroductionIn this paper we deal with a semilinear stochastic equationin a separable Banach space (E, · ) continuously embedded into a separable Hilbert space H with the inner product ·, · and the norm |·|. We assume that F : E → E is a nonlinear mapping, (W t ) is a standard cylindrical Wiener process in H defined on a probability space (Ω, F, (F t ) , P) and Q = Q * ∈ L(H) is nonnegative. Under the assumptions stated below equation (1.1) has a unique solution which defines a Markov E-valued process with the transition semigroupand moreover, it has a unique invariant measure µ. In this paper we provide conditions under which the convergence to an invariant measure is uniformly ergodic in the following sense: There exist positive constants C and γ such thatfor any Borel probability measure ν on E, where · var denotes the norm of total variation of measures and P * t is the adjoint Markov semigroup (in some papers P * t ν is also denoted by νP t ). This result is known as the uniform exponential ergodicity of the Markov process associated with the transition semigroup (P t ). Note that the convergence in (1.2) is uniform with respect 1991 Mathematics Subject Classification. 60H15, 60J99, 37A30, 47A35.

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Section: Uniform Exponential Ergodicity and Some Auxiliary Resultsmentioning

confidence: 99%

“…The proof is a simple combination of arguments from [18] and [4] so it is only sketched. Let c > 0 be the norm of the embedding j : E → H. For m ≥ 1, and x ∈ E set…”

Section: )mentioning

confidence: 99%

Uniform Exponential Ergodicity of Stochastic Dissipative Systems

Gołdys

Maslowski

2001

Czechoslovak Mathematical Journal

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“…is satisfied for t > 0, by the Cameron-Martin formula (Lemma 3 in [7]) the integral operators [7], the operators T λ are also compact. Thus A result on the approximation of functions in Dom(L) by functions in Dom(L 0 ) is given.…”

Section: S(t) ∈ L(h)mentioning

confidence: 99%

Adaptive Control for Semilinear Stochastic Systems

Duncan¹,

Maslowski²,

Pasik-Duncan³

2000

SIAM J. Control Optim.

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Abstract. An adaptive, ergodic cost stochastic control problem for a partially known, semilinear, stochastic system in an infinite dimensional space is formulated and solved. The solutions of the Hamilton-Jacobi-Bellman equations for the discounted cost and the ergodic cost stochastic control problems require some special interpretations because they do not typically exist in the usual sense. The solutions of the parameter dependent ergodic Hamilton-Jacobi-Bellman equations are obtained from some corresponding discounted cost control problems as the discount rate tends to zero. The solutions of the ergodic Hamilton-Jacobi-Bellman equations are shown to depend continuously on the parameter. A certainty equivalence adaptive control is given that is based on the optimal controls from the solutions of the ergodic Hamilton-Jacobi-Bellman equations and a strongly consistent family of estimates of the unknown parameter. This adaptive control is shown to achieve the optimal ergodic cost for the known system. Key words. stochastic adaptive control, ergodic control, stochastic semilinear systems, stochastic optimal control, distributed parameter systems AMS subject classifications. 93C40, 93C20, 60J27, 60H15 some properties of the solution of the Hamilton-JacobiBellman (HJB) equation of optimal control are used to verify self-optimality of an adaptive control using a strongly consistent family of estimates of the unknown parameters. In [17] an almost optimal adaptive control is constructed for a partially known nonlinear stochastic system. To obtain an optimal feedback control in an explicit form, the associated HJB equation must be solved, and for an adaptive control problem, a continuous dependence of the solution of the HJB equation must be verified. Some results for an infinite time horizon discounted cost control problem for a semilinear stochastic system in an infinite dimensional state space are given in [22], where the HJB equation is considered in the mild form, and in [23], where a viscosity solution of the HJB equation is used. Some results for an ergodic cost control problem for a semilinear stochastic system in an infinite dimensional state space are given in [20], where the HJB equation is solved. This is apparently the only result for ergodic cost control for semilinear stochastic systems using the HJB equation. Some other approaches and results for ergodic cost control are given in [16], [18]. For adaptive control, there are results for linear stochastic systems with an ergodic quadratic cost in [13], [14], [15]. Since the results for the discounted cost and the ergodic cost stochastic control problems for known semilinear stochastic systems are relatively recent, it appears that this is the first work on adaptive control for stochastic semilinear systems.

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“…Many of the results of this section are verified similarly to the verifications that are used for a single Hilbert space (cf. [6,8,9]), so some details are omitted. Recall the definitions of (T t , t ≥ 0) in (2.9) and (P α,u t , t ≥ 0) in (2.13).…”

Section: The Mild Kolmogorov Equationmentioning

confidence: 99%

Ergodic Boundary/Point Control of Stochastic Semilinear Systems

Duncan¹,

Maslowski²,

Pasik-Duncan³

1998

SIAM J. Control Optim.

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Abstract.A controlled Markov process in a Hilbert space and an ergodic cost functional are given for a control problem that is solved where the process is a solution of a parameter-dependent semilinear stochastic differential equation and the control can occur only on the boundary or at discrete points in the domain. The linear term of the semilinear differential equation is the infinitesimal generator of an analytic semigroup. The noise for the stochastic differential equation can be distributed, boundary and point. Some ergodic properties of the controlled Markov process are shown to be uniform in the control and the parameter. The existence of an optimal control is verified to solve the ergodic control problem. The optimal cost is shown to depend continuously on the system parameter.

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Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces

Cited by 86 publications

References 14 publications

Uniform Exponential Ergodicity of Stochastic Dissipative Systems

Uniform Exponential Ergodicity of Stochastic Dissipative Systems

Adaptive Control for Semilinear Stochastic Systems

Ergodic Boundary/Point Control of Stochastic Semilinear Systems

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