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Register a new userExponential transform of quadratic functional and multiplicative ergodicity of a Gauss-Markov process
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Authors
Marina Kleptsyna
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The Laplace transform of partial sums of the square of a non-centered Gauss-Markov process, conditioning on its starting point, is explicitly computed. The parameters of multiplicative ergodicity are deduced.
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The mild sufficient conditions for exponential ergodicity of a Markov process, defined as the solution to SDE with a jump noise, are given. These conditions include three principal claims: recurrence condition R, topological irreducibility condition S and non-degeneracy condition N, the latter formulated in the terms of a certain random subspace of Re^m, associated with the initial equation. The examples are given, showing that, in general, none of three principal claims can be removed without losing ergodicity of the process. The key point in the approach, developed in the paper, is that the local Doeblin condition can be derived from N and S via the stratification method and criterium for the convergence in variations of the family of induced measures on Re^m.
By refining a recent result of Xie and Zhang, we prove the exponential ergodicity under a weighted variation norm for singular SDEs with drift containing a local integrable term and a coercive term. This result is then extended to singular reflecting SDEs as well as singular McKean-Vlasov SDEs with or without reflection. We also present a general result deducing the uniform ergodicity of McKean-Vlasov SDEs from that of classical SDEs. As an application, the $L^1$-exponential convergence is derived for a class of non-symmetric singular granular media equations.
The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincar{e} inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by $bullet$ showing that strong functional inequalities are equivalent to Lyapunov type conditions; $bullet$ showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times. We also give some complement concerning the link between Lyapunov conditions and in-tegrability property of the invariant probability measure and as such transportation inequalities , and we show that some unbounded Lyapunov conditions can lead to uniform ergodicity, and coming down from infinity property.
We study the stochastic growth process in discrete time $x_{i+1} = (1 + mu_i) x_i$ with growth rate $mu_i = rho e^{Z_i - frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - gamma Z_t dt + sigma dW_t$ sampled on a grid of uniformly spaced times ${t_i}_{i=0}^n$ with time step $tau$. Using large deviation theory methods we compute the asymptotic growth rate (Lyapunov exponent) $lambda = lim_{nto infty} frac{1}{n} log mathbb{E}[x_n]$. We show that this limit exists, under appropriate scaling of the O-U parameters, and can be expressed as the solution of a variational problem. The asymptotic growth rate is related to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For $Z_t$ a stationary O-U process the lattice gas coincides with a system considered previously by Kac and Helfand. We derive upper and lower bounds on $lambda$. In the large mean-reversion limit $gamma n tau gg 1$ the two bounds converge and the growth rate is given by a lattice version of the van der Waals equation of state. The predictions are tested against numerical simulations of the stochastic growth model.
We consider a general piecewise deterministic Markov process (PDMP) $X={X_t}_{tgeqslant 0}$ with measure-valued generator $mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales is presented as $$M^f_t=frac{f(X_t)}{f(X_0)}left[mathrm{Sexp}left(int_{(0,t]}frac{mathrm{d}L(mathcal{A}f)_s}{f(X_{s-})}right)right]^{-1}.$$ Using this exponential martingale as a likelihood ratio process, we define a new probability measure. It is shown that the original process remains a general PDMP under the new probability measure. And we find the new measure-valued generator and its domain.
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