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Asymptotics of stochastic Burgers equation with jumps. (English) Zbl 1461.60046

Let \(\Delta\) be the Dirichlet Laplacian on \((0,1)\), let \(H=L^2(0,1)\) and \(V\) be the domain of \((-\Delta)^{-1/2}\). Consider the following stochastic differential equation on \(H\) with jump:
\[d X_t= \{\Delta X_t+B(X_t,X_t)\}d t+Qd W_t +\int_U f(X_{t-},u)\tilde N(d t, du),\] where \(B(x,y):= xy'\) for \(x\in H\) and \(y\in V\), \(Q\) is a Hilbert-Schmidt operator on \(H\), \(W_t\) is the cylindrical Brownian motion on \(H\), \(\tilde N\) is the compensated martingale measure induced by \(d t n(d u)\) for a finite measure \(n\) on a measurable space \(U\), and \[f: H\times U\to H\] satisfies \[\sup_{x}\|f(x,\cdot)\|_{L^2(n)}<\infty,\] \[\int_{U}|f(x,\cdot)-f(y,\cdot)|^2d n\le K|x-y|^2\] holds for some constant \(K>0\), and \(f(\cdot,u)\in C_b^1(H)\).
Then the \(\psi\)-exponential ergodicity holds for the solution for \(\psi:=1+\|\cdot\|_H\). Moreover, the moderate deviation principle and large deviation principle are established for the occupation times of the solution.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F10 Large deviations
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References:

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