Lq‐valued Burkholder–Rosenthal inequalities and sharp estimates for stochastic integrals

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

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Trace operators on fractals, entropy and approximation numbers

Georgian Mathematical Journal ◽

10.1515/gmj.2011.0030 ◽

2011 ◽

Vol 18 (3) ◽

pp. 549-575

Author(s):

Cornelia Schneider

Keyword(s):

Besov Spaces ◽

Trace Operator ◽

Approximation Numbers ◽

Atomic Decompositions ◽

Sharp Estimates ◽

Compact Embeddings ◽

Trace Space

Abstract First we compute the trace space of Besov spaces – characterized via atomic decompositions – on fractals Γ, for parameters 0 < p < ∞, 0 < q ≤ min(1, p) and s = (n – d)/p. New Besov spaces on fractals are defined via traces for 0 < p, q ≤ ∞, s ≥ (n – d)/p and some embedding assertions are established. We conclude by studying the compactness of the trace operator TrΓ by giving sharp estimates for entropy and approximation numbers of compact embeddings between Besov spaces. Our results on Besov spaces remain valid considering the classical spaces defined via differences. The trace results are used to study traces in Triebel–Lizorkin spaces as well.

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Effect of Energy Degeneracy on the Transition Time for a Series of Metastable States

Journal of Statistical Physics ◽

10.1007/s10955-021-02788-0 ◽

2021 ◽

Vol 184 (1) ◽

Author(s):

Gianmarco Bet ◽

Vanessa Jacquier ◽

Francesca R. Nardi

Keyword(s):

Metastable State ◽

Stochastic Dynamics ◽

Transition Probabilities ◽

Mixing Time ◽

Stable State ◽

Temperature Limit ◽

Metastable States ◽

Transition Time ◽

Sharp Estimates ◽

Model Independent

AbstractWe consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in the low temperature limit. We generalize previous model-independent results in several directions. First, we give an estimate of the mixing time of the dynamics in terms of the maximal stability level. Second, assuming the dynamics is reversible, we give an estimate of the associated spectral gap. Third, we give precise asymptotics for the expected transition time from any metastable state to the stable state using potential-theoretic techniques. We do this in a general reversible setting where two or more metastable states are allowed and some of them may even be degenerate. This generalizes previous results that hold for a series of only two metastable states. We then focus on a specific Probabilistic Cellular Automata (PCA) with configuration space $${\mathcal {X}}=\{-1,+1\}^\varLambda $$ X = { - 1 , + 1 } Λ where $$\varLambda \subset {\mathbb {Z}}^2$$ Λ ⊂ Z 2 is a finite box with periodic boundary conditions. We apply our model-independent results to find sharp estimates for the expected transition time from any metastable state in $$\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^e\}$$ { - 1 ̲ , c ̲ o , c ̲ e } to the stable state $$\underline{+1}$$ + 1 ̲ . Here $${\underline{c}}^o,{\underline{c}}^e$$ c ̲ o , c ̲ e denote the odd and the even chessboard respectively. To do this, we identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability.

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On Krylov’s estimates for optional semimartingales

Random Operators and Stochastic Equations ◽

10.1515/rose-2021-2059 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

M. Abdelghani ◽

A. Melnikov ◽

A. Pak

Keyword(s):

Sobolev Space ◽

Measurable Function ◽

Diffusion Processes ◽

Stochastic Integrals ◽

Controlled Diffusion ◽

Convergence Of Solutions ◽

Change Of Variables ◽

General Class Of Functions ◽

Class Of Functions ◽

Controlled Diffusion Processes

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

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