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For continuous time random walks, let (frac {1}{theta _{i}} > 0) be a mean of an exponential holding time of X (i) for each i∈{0,…,k}.
The process L X (·,x =(L X (t,x), t≥0) appearing in relation (25) is called the (right) local time of X at level x.
For example, the leaf shedding time of X. mekongensis started in December and continued until March, i.e., winter and spring seasons, whereas leaf shedding of E. agallocha started from February and continued until May, i.e., spring season.
As a consequence, the formula we obtain contains an infinite-dimensional generalized functional in the sense of Schwartz, defined in terms of Hida's renormalization of the squared derivative of B and in terms of the local time of X at 0. We also compute the divergence on the Wiener space of a class of vector fields not taking values in the Cameron Martin space.
By Berman [30] (see also Lemma 8.1 in Xiao [33]), for any continuous and zero-mean Gaussian process (X=(X t), tin[0,T])) with bounded variance function, the condition int_{[0,T]^{2}}frac{ds,dt}{sqrt{{E}[X t -X s)]^{2}}}< infty is sufficient for the local time of X t -X sst and to be square integrable.
Moreover, we notice that t m e e t is equivalent to the first hitting time of X t to the set begin{array}rcl@ left{(x_{0},ldots,x_{k}) in V^{k+1} ; x_{0} = x_{i} text{for some}, i in Lambdaright}.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com