In this paper, we consider the parabolic Anderson equation that is driven by a Gaussian noise fractional in time and white or fractional in space, and is solved in a mild sense defined by Skorokhod integral. Our objective is the precise moment Lyapunov exponent and high moment asymptotics. As far as the long term asymptotics are concerned, some feature given in our theorems is different from what have been observed in the Stratonovich-regime and in the setting of the white time noise. While the difference disappears when it comes to the high moment asymptotics. To achieve our goal, we introduce a variational inequality and use some newly developed tools such as time-space LDP of Feynman-Kac type, linearization by tangent approximation, together with some techniques developed along the line of probability in Banach spaces.Résumé. Nous considérons l'équation d'Anderson parabolique conduite par un bruit gaussien, fractionnaire en temps, et blanc ou fractionnaire en espace, qu'on résout dans un sens faible défini par une intégrale de Skorokhod. Notre objectif est de donner l'exposant de Lyapounov pour les moments, et les asymptotiques des grands moments. Pour les asymptotiques en temps long, nos résultats mettent en évidence des phénomènes différents de ceux observés pour le régime Stratonovich, et dans le cas d'un bruit blanc en temps. Ces différences s'effacent néanmoins lorsque l'on considère les asymptotiques des grands moments. Nos résultats sont obtenus en introduisant une nouvelle inégalité variationnelle, et à l'aide d'outils nouveaux tels qu'un principe de grandes déviations de type Feynman-Kac, la linéarisation par des approximations tangentes, et des techniques inspirées des probabilités dans les espaces de Banach.
Dickeya zeae is the causal agent of bacterial soft rot disease, with a wide range of hosts all over the world. At present, chemical agents, especially agricultural antibiotics, are commonly used in the prevention and control of bacterial soft rot, causing the emergence of resistant pathogens and therefore increasing the difficulty of disease prevention and control. This study aims to provide a safer and more effective biocontrol method for soft rot disease caused by D. zeae. The spot-on-lawn assay was used to screen antagonistic bacteria, and three strains including SC3, SC11 and 3-10 revealed strong antagonistic effects and were identified as Pseudomonas fluorescens, P. parafulva and Bacillus velezensis, respectively, using multi-locus sequence analysis (MLSA) based on the sequences of 16S rRNA and other housekeeping genes. In vitro antimicrobial activity showed that two Pseudomonas strains SC3 and SC11 were only antagonistic to some pathogenic bacteria, while strain 3-10 had broad-spectrum antimicrobial activity on both pathogenic bacteria and fungi. Evaluation of control efficacy in greenhouse trials showed that they all restrained the occurrence and development of soft rot disease caused by D. zeae MS2 or EC1. Among them, strain SC3 had the most impressive biocontrol efficacy on alleviating the soft rot symptoms on both monocotyledonous and dicotyledonous hosts, and strain 3-10 additionally reduced the occurrence of banana wilt disease caused by Fusarium oxysporum f. sp. cubensis. This is the first report of P. fluorescens, P. parafulva and B. velezensis as potential bio-reagents on controlling soft rot disease caused by D. zeae.
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