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We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa cite{AL0}. In this article we focus on second order SPDEs of the Allen-Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and a corresponding random PDE of Allen-Cahn type, we prove the existence of semimartingale global attractors for these equations. We also give some results on the finite dimensional asymptotic behavior of the solutions. In particular, we show the finite fractal dimension of this random attractor and give a result on determining modes, both in the forward and the pullback sense.
We prove existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, with drift equal to the sum of a linear maximal monotone operator in variational fo
We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $exp (Phi)_{2}$-quantum field model or H{o}egh-Krohns model. In the present paper, we study the stochastic quantization of this model by
In this paper we investigate the long time behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination
The present paper is a continuation of our previous work on the stochastic quantization of the $exp(Phi)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for si
We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( u Delta X + Delta psi (X) ) dt = sum_{i=1}^N langle b_i, abla X rangle circ dbeta_i$ in $]0,T[ times mathcal{O}$, with $X(0) = x(x