arXiv 2107.050282021-07math.PRmath-phmath.MP

具有矩阵漂移的复矩阵布朗运动的奇异值

On the singular values of complex matrix Brownian motion with a matrix drift

Theodoros Assiotis

备注：: Improvements in presentation and some corrections following referee reports. To appear in Bernoulli. Due to the journal's paper length requirement the current version will be split into a main paper and a supplement which will include the proofs of intermediate results, Section 5 and Appendices A and B

摘要：

令$Mat_{\mathbb{C}}(K,N)$为$K\times N$复矩阵空间。设$\mathbf{B}_t$是从零矩阵和$\mathbf{M}\in Mat_{\mathbb{C}}(K,N)$出发的$Mat_{\mathbb{C}}(K,N)$上的布朗运动.我们证明了，与$K\ge N$，$N$特征值的$\left(\mathbf{B}_t+t\mathbf{M}\right)^*\left(\mathbf{B}_t+t\mathbf{M}\right)$形成一个显式的转移核的马尔可夫过程。这推广了Rogers和Pitman关于具有漂移的多维布朗运动的一个经典结果，该结果对应于$N=1$ .然后，我们给出两个更多的描述这个马尔可夫过程。首先，作为独立的平方贝塞尔扩散过程在广义上，介绍了渡边和研究的皮特曼和约，条件是永远不会相交。第二，作为相互作用的平方贝塞尔型扩散在一些相互作用阵列的顶行的分布。最后两个描述也扩展到一般类的一维扩散。

英文摘要：

Let $Mat_{\mathbb{C}}(K,N)$ be the space of $K\times N$ complex matrices. Let $\mathbf{B}_t$ be Brownian motion on $Mat_{\mathbb{C}}(K,N)$ starting from the zero matrix and $\mathbf{M}\in Mat_{\mathbb{C}}(K,N)$. We prove that, with $K\ge N$, the $N$ eigenvalues of $\left(\mathbf{B}_t+t\mathbf{M}\right)^*\left(\mathbf{B}_t+t\mathbf{M}\right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.

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摘要：

英文摘要：

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