Measures, Integrals and Martingales

Schilling, René L.

doi:10.1017/cbo9780511810886

Cited by 174 publications

(128 citation statements)

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“…and stays bounded by 1, then by dominated convergence, we can replace at the limit T k (η V i ) in all integrals of (41). Thanks to δ > 0, we then obtain…”

Section: Remark 311mentioning

confidence: 98%

Global existence for reaction–diffusion systems with nonlinear diffusion and control of mass

Laamri

Pierre

2017

Ann. Inst. H. Poincaré Anal. Non Linéaire

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International audienceWe prove here global existence in time of weak solutions for some reaction-diffusion systems with natural structure conditions on the nonlinear reactive terms which provide positivity of the solutions and uniform control of the total mass. The diffusion operators are nonlinear, in particular operators of the porous media type u i → −d i ∆u m i i. Global existence is proved under the assumption that the reactive terms are bounded in L 1. This extends previous similar results obtained in the semilinear case when the diffusion operators are linear of type u i → −d i ∆u i

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“…and stays bounded by 1, then by dominated convergence, we can replace at the limit T k (η V i ) in all integrals of (41). Thanks to δ > 0, we then obtain…”

Section: Remark 311mentioning

confidence: 98%

Global existence for reaction–diffusion systems with nonlinear diffusion and control of mass

Laamri

Pierre

2017

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…[14,Lemma 6.3.5] or [11,Theorem 14.8]: choose χ ∈ C ∞ c (R d ) such that supp χ ⊂ F + B(0, δ 0 ) and…”

Section: Proof Of Proposition 15mentioning

confidence: 99%

On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes

Schilling

Wang

2011

J. Evol. Equ.

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Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein-Uhlenbeck processes. Then we present the explicit coupling property of Ornstein-Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein-Uhlenbeck processes via the symbol. Main resultsLet (X x t ) t≥0 be an n-dimensional Ornstein-Uhlenbeck process, which is defined as the unique strong solution of the following stochastic differential equation(1.1)Here A is a real n × n matrix, B is a real n × d matrix and Z t is a Lévy process in R d ; note that we allow Z t to take values in a proper subspace of R d . It is well known thatThe characteristic exponent or symbol of Z t , defined byenjoys the following Lévy-Khintchine representation:where Q = (q j,k ) d j,k=1 is a positive semi-definite matrix, b ∈ R d is the drift vector and ν is the Lévy measure, i.e. a σ -finite measure on R d \{0} such that the integral z =0 (1 ∧ |z| 2 ) ν(dz) < ∞. For every ε > 0, define ν ε on R d as follows: (2000): 60J25, 60J75 Mathematics Subject Classification

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“…The last assertion is standard (c.f. [22]), and we do not include a proof. Now, fix any R < ∞, and let x ∈ B R (0).…”

Section: Distributed Dynamic Programmingmentioning

confidence: 99%

Distributed dynamic programming for discrete-time stochastic control, and idempotent algorithms

McEneaney

2011

Automatica

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Previously, idempotent methods have been found to be extremely fast for solution of dynamic programming equations associated with deterministic control problems. The original methods exploited the idempotent (e.g., max-plus) linearity of the associated semigroup operator. However, it is now known that the curse-of-dimensionality-free idempotent methods do not require this linearity. Instead, it is sufficient that certain solution forms are retained through application of the associated semigroup operator. Here, we see that idempotent methods may be used to solve some classes of stochastic control problems. The key is the use of the idempotent distributive property. This allows one to apply the curse-of-dimensionality-free idempotent approach. We demonstrate this approach for a class of nonlinear, discrete-time stochastic control problems.

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Measures, Integrals and Martingales

Cited by 174 publications

References 0 publications

Global existence for reaction–diffusion systems with nonlinear diffusion and control of mass

Global existence for reaction–diffusion systems with nonlinear diffusion and control of mass

On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes

Distributed dynamic programming for discrete-time stochastic control, and idempotent algorithms

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