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In this paper we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogues of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces, and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory, and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations.
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We use a noncommutative generalization of Fourier analysis to define a broad class of pseudo-probability representations, which includes the known bosonic and discrete Wigner functions. We characterize the groups of quantum unitary operations which correspond to phase-space transformations, generalizing Gaussian and Clifford operations. As examples, we find Wigner representations for fermions, hard-core bosons, and angle-number systems.
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In this paper we show the existence of the minimal solution to the multidimensional Lambert-Euler inversion, a multidimensional generalization of $[-e^{-1} ,0)$ branch of Lambert W function $W_0(x)$. Specifically, for a given nonnegative irreducible symmetric matrix $V in mathbb{R}^{k times k}$, we show that for ${bf u}in(0,infty)^k$, if equation $$y_j exp{-{bf e}_j^T V {bf y} } = u_j ~~~~~~forall j=1,...,k,$$ has at least one solution, it must have a minimal solution ${bf y}^*$, where the minimum is achieved in all coordinates $y_j$ simultaneously. Moreover, such ${bf y}^*$ is the unique solution satisfying $rholeft(V D[y^*_j] right) leq 1$, where $D[y^*_j]={sf diag}(y_j^*)$ is the diagonal matrix with entries $y^*_j$ and $rho$ denotes the spectral radius. Our main application is in the vector-multiplicative coalescent process. It is a coalescent process with $k$ types of particles and vector-valued weights that begins with $alpha_1n+...+alpha_k n$ particles partitioned into types of respective sizes, and in which two clusters of weights ${bf x}$ and ${bf y}$ would merge with rate $({bf x}^{sf T} V {bf y})/n$. We use combinatorics to solve the corresponding modified Smoluchowski equations, obtained as a hydrodynamic limit of vector-multiplicative coalescent as $n to infty$, and use multidimensional Lambert-Euler inversion to establish gelation and find a closed form expression for the gelation time. We also find the asymptotic length of the minimal spanning tree for a broad range of graphs equipped with random edge lengths.
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