##### Department of Mathematics,

University of California San Diego

****************************

### Probability Seminar

## Piotr Grazcyk

#### LAREMA, Universite d'Angers

## SDEs for particle systems and applications in harmonic analysis

##### Abstract:

Consider the following system of SDEs $d\lambda_i = \sigma_i(\lambda_i)dB_i+\left(b_i(\lambda_i)+\sum_{j\neq i}\frac{H_{ij}(\lambda_i,\lambda_j)}{\lambda_i-\lambda_j}\right)dt\/,\quad i=1,\ldots,p\/$, (1)
describing ordered particles $\lambda_1(t)\leq \ldots\leq \lambda_p(t)$, $t\geq0$ on $R$. Here $B_i$ denotes a collection of one-dimensional independent Brownian motions.
Let $Sym(p\times p)$ be the vector space of symmetric real $p\times p$ matrices. The SDEs systems (1) contain the systems describing, for the starting point having no collisions, the eigenvalues of the $Sym(p\times p)$-valued process $X_t$ satisfying the following matrix valued stochastic differential equation $dX_t = g(X_t)dW_th(X_t)+h(X_t)dW_t^Tg(x_t)+b(X_t)dt\/$, where the functions $g,h,b$ act spectrally on $Sym(p\times p)$, and $W_t$ is a Brownian matrix of dimension $p\times p$. Thus the systems (1) contain Dyson Brownian Motions, Squared Bessel particle systems, their $\beta$-versions and other particle systems crucial in mathematical physics and physical statistics.
Note that the functions $\displaystyle{\frac{H_{ij}(lambda_i, \lambda_j)}{\lambda_i-\lambda_j}}$ describe the repulsive forces with which the particle $\lambda_i$ acts on the particle $\lambda_j$. On the other hand the singularities $\displaystyle{\frac{1}{\lambda_i-\lambda_j}}$ make the SDEs system (1) difficult to solve, especially when the starting point $\Lambda(0)$ has a collision $\lambda_i(0)= \lambda_j(0)$. The most degenerate case $\lambda_1(0)= \ldots= \lambda_p(0)$ is of great importance in applications.
In some particular cases (Dyson Brownian Motions, some Squared Bessel particle systems), the existence of strong solutions of (1) has been established by C\'epa and L\'epingle, using the technique of Multivalued SDEs.
We prove the existence of strong and pathwise unique non-colliding solutions of (1), with a degenerate colliding initial point $\Lambda(0)$ in the whole generality, under natural assumptions on the coefficients of the equations in (1). Our approach is based on the classical It\^o calculus, applied to elementary symmetric polynomials in $p$ variables $X=(x_1, \ldots,x_p)$
$e_n(X) = \sum_{i_1<\ldots

Host: Ruth Williams

### June 4, 2015

### 10:00 AM

### AP&M 6402

****************************