##
**Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise.**
*(English)*
Zbl 1075.60072

Let \(x\) be a Markov process on \(\mathbb R^ {d}\) with a transition function \(P_ {t}\). Consider the following two hypotheses: (1) There exist a compact set \(C\subset \mathbb R^ {d}\) with a non-empty interior and a point \(y\in \operatorname {Int}(C)\) such that, for any open neighbourhood \(U\ni y\), there is a time \(\tilde t\) satisfying \(P_ {\tilde t}(x,U)>0\) for all \(x\in C\). Moreover, the function \(P_ {t}\) restricted to \(C\times {\mathcal B}(C)\) admits a density \(p_ {t}\) continuous on \(C\times C\), for all \(t\geq 0\). (Note that this assumption implies the well known minorization condition.) (2) Let \((x_ {n})\) be a Markov chain corresponding to the kernel \(P_ {T}\) for some fixed \(T>0\). Assume there exist a function \(V:\mathbb R^ {d}\to \mathopen [1,\infty \mathclose [\) and constants \(\alpha \in \mathopen ]0,1\mathclose [\), \(\beta \in \mathopen [0,\infty \mathclose [\) such that \(V(x)\to \infty \) as \(| x| \to \infty \) and \(\mathbf E[V(x_ {n+1})\mid {\mathcal F}_ {n}] \leq \alpha V(x_ {n}) + \beta \), \({\mathcal F}_ {n}\) being the \(\sigma \)-algebra generated by \(x_ 1,\ldots ,x_ {n}\). (Note that this assumption is satisfied if \(AV\leq -aV + b\) on \(\mathbb R^ {d}\) for some \(a,b>0\), where \(A\) is the generator of the process \(x\)). Suppose that for some \(T>0\) the embedded chain \((x_ {n})\) obeys Hypotheses (1), (2), with the choice \(C = \{x;\; V(x)\leq 2\beta (\gamma - \alpha )^ {-1} \}\), for some \(\gamma \in \mathopen ]\alpha ^ {1/2},1\mathclose [\). Then there exists a unique invariant measure \(\pi \) for the chain \((x_ {n})\) and the chain is geometrically ergodic, that is, \(| \mathbf E_ {z} f(x_ {n})-\pi (f)| \leq Kr^ {n}V(z)\) for some \(K\geq 0\), \(r\in \mathopen ]0,1\mathclose [\), all \(z\in \mathbb R^ {d}\) and all measurable functions \(f\) such that \(| f| \leq V\) on \(\mathbb R^ {d}\). This theorem follows from more general results which may be found in [S.Meyn and R.Tweedie, “Markov chains and stochastic stability” (1993; Zbl 0925.60001)], but to keep the paper self-contained the authors provide an independent proof. The main aim of the paper is to show that the theorem may be applied to several interesting classes of Markov processes defined by stochastic differential equations whose drift is only locally Lipschitz and diffusion matrix may be degenerate.

First, a second-order stochastic differential equation \(dq = p\,dt\), \(dp = -(\gamma p +\nabla F(q))\,dt + \sigma \,dW\), where \(W\) is a \(d\)-dimensional Wiener process, \(\gamma >0\) and \(\sigma \) is an invertible matrix, is studied. It is proven that the process \((q(t),p(t))\) is geometrically ergodic, provided \(F\) is a nonnegative \(C^ \infty \)-function and \(\langle \nabla F(q),q\rangle /2 \geq \beta F(q) +\gamma ^ 2(8-8\beta )^ {-1} \beta (2-\beta )\| q\| ^ 2 - \alpha \) for some \(\alpha >0\) and \(\beta \in \mathopen ]0,1\mathclose [\). Further, an equation \(dx = Y(x)\,dt + \Sigma \,dW\) is considered, where \(W\) is now an \(m\)-dimensional Wiener process, \(m\leq d\), the columns of the matrix \(\Sigma \) are linearly independent, and the drift is either dissipative, \(\langle Y(x),x\rangle \leq \alpha -\beta \| x\| ^ 2\) for some \(\alpha ,\beta >0\), or of a gradient type, \(Y = \nabla F\). Suitable additional assumptions are found so that geometric ergodicity may be established.

In the second part of the paper, the effect of time discretization on these stochastic differential equations is investigated and ergodicity for several approximation methods is proved. The main difficulty comes from the fact that the Lyapunov condition (2) is sensitive to the choice of a discretization method if the drift is not globally Lipschitz, and is inherited only by specially constructed implicit discretizations.

First, a second-order stochastic differential equation \(dq = p\,dt\), \(dp = -(\gamma p +\nabla F(q))\,dt + \sigma \,dW\), where \(W\) is a \(d\)-dimensional Wiener process, \(\gamma >0\) and \(\sigma \) is an invertible matrix, is studied. It is proven that the process \((q(t),p(t))\) is geometrically ergodic, provided \(F\) is a nonnegative \(C^ \infty \)-function and \(\langle \nabla F(q),q\rangle /2 \geq \beta F(q) +\gamma ^ 2(8-8\beta )^ {-1} \beta (2-\beta )\| q\| ^ 2 - \alpha \) for some \(\alpha >0\) and \(\beta \in \mathopen ]0,1\mathclose [\). Further, an equation \(dx = Y(x)\,dt + \Sigma \,dW\) is considered, where \(W\) is now an \(m\)-dimensional Wiener process, \(m\leq d\), the columns of the matrix \(\Sigma \) are linearly independent, and the drift is either dissipative, \(\langle Y(x),x\rangle \leq \alpha -\beta \| x\| ^ 2\) for some \(\alpha ,\beta >0\), or of a gradient type, \(Y = \nabla F\). Suitable additional assumptions are found so that geometric ergodicity may be established.

In the second part of the paper, the effect of time discretization on these stochastic differential equations is investigated and ergodicity for several approximation methods is proved. The main difficulty comes from the fact that the Lyapunov condition (2) is sensitive to the choice of a discretization method if the drift is not globally Lipschitz, and is inherited only by specially constructed implicit discretizations.

Reviewer: Jan Seidler (Praha)

### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60J35 | Transition functions, generators and resolvents |

65C30 | Numerical solutions to stochastic differential and integral equations |

### Keywords:

stochastic differential equations; invariant measure; geometric ergodicity; time discretization### Citations:

Zbl 0925.60001### Software:

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\textit{J. C. Mattingly} et al., Stochastic Processes Appl. 101, No. 2, 185--232 (2002; Zbl 1075.60072)

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