# Guangyu Xi's research while affiliated with Loyola University Maryland and other places

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## Publications (9)

We study the validity of a Smoluchowski-Kramers approximation for a class of wave equations in a bounded domain of $\mathbb{R}^n$ subject to a state-dependent damping and perturbed by a multiplicative noise. We prove that in the small mass limit the solution converges to the solution of a stochastic quasilinear parabolic equation where a noise-indu...

The asymptotic behavior of a class of stochastic reaction-diffusion-advection equations in the plane is studied. We show that as the divergence-free advection term becomes larger and larger, the solutions of such equations converge to the solution of a suitable stochastic PDE defined on the graph associated with the Hamiltonian. Firstly, we deal wi...

In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the c...

In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the c...

In this paper we construct a Markov semi-group with a generator $L=\Delta+b\cdot\nabla$, where $b$ is a divergence-free vector field which belongs to the $L^{p}$-space with $p<n$. A regularity theory of parabolic equations for this case is not available. The research is motivated by the interest in the understanding of blow-up solutions to a certai...

In this paper we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of the parabolic operator $L_{t}=\partial_{t}-\Delta+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the case that $b$ belongs to the Lebesgue space $L^{l}\left(0,T;L^{q}\left(\mathbb{R}^{n}\right)\right)$. The regulari...

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\nu\Delta+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_...

## Citations

... In recent years there has been an intense activity dealing with the Smoluchowski-Kramers approximation of infinite dimensional systems. To this purpose, we refer to [4], [5], [30] and [25] for the case of constant damping term (see also [12] where systems subject to a magnetic field are studied), and to [14] for the case of state-dependent damping. As a matter of fact, these two situations are quite different. ...

... A maximal inequality for processes with controlled increments. The following theorem was found by Liu & Xi [94], without making the connection to the Kolmogorov-Chentsov-Totoki theorem. Let (X t , F t ) t∈[0,T ] be an adapted R d -valued stochastic process satisfying a conditional increment control with ...

Reference: Maximal Inequalities and Some Applications

... zero-divergence b = b(x) with |b| ∈ L p , p > d 2 (essentially, twice more singular than (1)), see [Z1]. See also [QX2]. Although in this case one has to sacrifice much of the regularity theory of (1) and (2), some results can be salvaged. ...

... One class of singular vector fields that is not covered by (3) is b ∈ L ∞ BMO −1 , div b = 0, i.e. the class of divergence-free vector fields whose components are distributional derivatives of functions of bounded mean oscillation (see detailed definition below). Recently, Qian-Xi [QX1] established global in time two-sided Gaussian bound for (2) for vector fields in this class. Their result will follow as a special case of our Theorems 2.2, 2.3. ...