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We define a state space and a Markov process associated to the stochastic quantisation equation of Yang-Mills-Higgs (YMH) theories. The state space S is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to S and thus define a quotient space of "gauge orbits" O. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on O (up to a potential finite time blow-up) associated to the stochastic YMH flow.

Both human and animals, for their nutritional requirements, mainly rely on the plant-based foods, which provide a wide range of nutrients. Minerals, proteins, vitamins are among the nutrients which are essential and need to be available in adequate amount in edible portion of the staple crops. Increasing nutritional content in staple crops either through agronomic biofortification or through conventional plant-breeding strategies continue to be a huge task for scientists around the globe. Although some success has been achieved in recent past, in most cases, we have fallen short of expected targets. To maximize the nutrient uptake and partitioning to different economic part of plants, scientists have employed and tailored several biofortification strategies. But in present agricultural and environmental concerns, these approaches are not much effective. Henceforth, we are highlighting the recent developments and promising aspects of microbial-assisted and genomic-assisted breeding as candidate biofortification approach, that have contributed significantly in increasing nutritional content in grains of different crops. The methods used to date to accomplish nutrient enrichment with recently emerging strategies that we believe could be the most promising and holistic approach for future biofortification program. Results are encouraging, but for future perspective, the existing knowledge about the strategies needs to be confined. Concerted scientific investment are required to widen up these biofortification strategies, so that it could play an important role in ensuring nutritional security of ever-growing population in growing agricultural and environmental constraints.

We derive a priori bounds for the Φ 4 equation in the full subcritical regime using Hairer's theory of regularity structures. The equation is formally given by (∂t − ∆)φ = −φ 3 + ∞φ + ξ, (⋆) where the term +∞φ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions d < 4 by adjusting the regularity of the noise term ξ, choosing ξ ∈ C −3+δ . Our main result states that if φ satisfies this equation on a space-time cylinder D = (0, 1) × {|x| 1}, then away from the boundary ∂D the solution φ can be bounded in terms of a finite number of explicit polynomial expressions in ξ. The bound holds uniformly over all possible choices of boundary data for φ and thus relies crucially on the super-linear damping effect of the non-linear term −φ 3 .A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of (⋆), which allows to couple the small scale control one obtains from this theory with a suitable large scale argument.Along the way we make several new observations and simplifications: we reduce the number of objects required with respect to Hairer's work. Instead of a model (Πx)x and the family of translation operators (Γx,y)x,y we work with just a single object (Xx,y) which acts on itself for translations, very much in the spirit of Gubinelli's theory of branched rough paths. Furthermore, we show that in the specific context of (⋆) the hierarchy of continuity conditions which constitute Hairer's definition of a modelled distribution can be reduced to the single continuity condition on the "coefficient on the constant level".

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