# Stochastic Quantization of Laser Propagation Models
111 1 U. S. Air Force Research Laboratory, Wright Patterson Air Force Base,
Ohio 45433, U. S. A. 2 NRC-Senior Research Fellow, National Academies of
Science, Engineering and Medicine, U. S. Air Force Research Laboratory, Wright
Patterson Air Force Base, Ohio 45433, U. S. A. e-mail:
<EMAIL_ADDRESS>∗Corresponding author. e-mail:
<EMAIL_ADDRESS>
Sivaguru S. Sritharan1,2* and Saba Mudaliar1

###### Abstract.

This paper identifies certain interesting mathematical problems of stochastic
quantization type in the modeling of Laser propagation through turbulent
media. In some of the typical physical contexts the problem reduces to
stochastic Schrödinger equation with space-time white noise of Gaussian,
Poisson and Lévy type. We identify their mathematical resolution via
stochastic quantization. Nonlinear phenomena such as Kerr effect can be
modeled by stochastic nonlinear Schrodinger equation in the focusing case with
space-time white noise. A treatment of stochastic transport equation, the
Korteweg-de Vries Equation as well as a number of other nonlinear wave
equations with space-time white noise is also given. Main technique is the
S-transform (we will actually use closely related Hermite transform) which
converts the stochastic partial differential equation with space time white
noise to a deterministic partial differential equation defined on the Hida-
Kondratiev white noise distribution space. We then utlize the inverse
S-transform/Hermite transform known as the characterization theorem combined
with the infinite dimensional implicit function theorem for analytic maps to
establish local existence and uniqueness theorems for path-wise solutions of
these class of problems. The particular focus of this paper on singular white
noise distributions is motivated by practical situations where the refractive
index fluctuations in propagation medium in space and time are intense due to
turbulence, ionospheric plasma turbulence, marine-layer fluctuations, etc.
Since a large class of partial differential equations that arise in nonlinear
wave propagation have polynomial type nonlinearities, white noise distribution
theory is an effective tool in studying these problems subject to different
types of white noises.

Key words: Laser propagation, stochastic nonlinear Schrödinger equation,
stochastic quantization, space-time white noise, Wick product, paraxial
equation, Korteweg-de Vries Equation, white noise calculus, S-transform,
Benjamin-Ono equation, Schrödringer-Hartree equation, Zakharov system, Davey-
Stewartson equation.

Mathematics Subject Classification (2010): 60H15, 81S20, 37N20

###### Contents

  1. 1 Introduction
  2. 2 Derivation of the paraxial equation from the Maxwell equations
  3. 3 Mathematical background on white noise calculus
    1. 3.1 White noise theory: Gaussian, Poisson and Lévy
    2. 3.2 Hida-Kondratiev spaces
    3. 3.3 Wick products and properties
    4. 3.4 S-transform, Hermite transform and characterization theorems
  4. 4 Kato theory of quasilinear abstract evolution equations
    1. 4.1 Analytic Mappings between Banach Spaces and Inverse Mapping Theorem
  5. 5  First-order PDE (transport-type)
    1. 5.1 First order random transport equation with temporal white noise
    2. 5.2 Stochastic transport model with space-time white noise
  6. 6 Stochastic nonlinear wave equations
    1. 6.1 Stochastic Korteweg De Vries equation
    2. 6.2 Stochastic Benjamin-Ono equation
  7. 7 Stochastic reaction diffusion equation and quantization
    1. 7.1 Stochastic heat equation with multiplicative noise and the KPZ Equation
    2. 7.2 Stochastic nonlinear heat equation with white noise Initial Data
  8. 8 Stochastic linear and nonlinear Schrödinger equations with space-time white noise
    1. 8.1 Strichartz estimates
    2. 8.2 Stochastic linear Schrödinger equation with additive space-time white noise
    3. 8.3 Stochastic linear Schrödinger equation with multiplicative space-time white noise
    4. 8.4 Nonlinear Schrödinger equation with multiplicative space-time white noise
  9. 9 Concluding remarks

##  1\. Introduction

Stochastic partial differential equations of the type

$i\frac{\partial}{\partial
t}\psi(x,t,\omega)+[\Delta+\Gamma(x,t,\omega)]\psi(x,t,\omega)+F(\psi(x,t,\omega))=0,$
(1.1)

where $\Gamma(\cdot,\cdot,\cdot)$ is a random field describing the
fluctuations in the medium are often encountered in electromagnetic and
acoustic wave propagation problems in random nonlinear media. They are usually
called either ”paraxial equation” or ”parabolic equation model” in the
engineering literature [60, 64, 32]. As we define later, here
$(\Omega,\mathbb{\mathcal{F}},m)$ is a complete probability space and $\omega$
is an $\mathcal{S}^{\prime}(\mathbb{R}^{d})$-valued random variable. In this
paper we will study Gaussian, Poisson and Levy type white noise models for
$\Gamma(\cdot,\cdot,\cdot)$. In the Gaussian case for example,
$\Gamma(x,t,\omega)$ can also formally represented using generalized
derivative of $\mathbb{R}^{n+1}$-dimensional Brownian sheet $B(x,t,\omega)$:

$\Gamma(x,t,\omega)=\frac{\partial^{n+1}}{\partial x_{1},\cdots\partial
x_{n}\partial t}B(x,t,\omega),\hskip 7.22743ptx=(x_{1},\cdots,x_{n}),$ (1.2)

where the Brownian sheet $B(x,t,\omega)$ has covariance:

$\langle B(x,t,\cdot),B(y,\tau,\cdot)\rangle=\Pi_{k=1}^{n}(x_{k}\wedge
y_{k})(t\wedge\tau),\hskip
7.22743ptx=(x_{1},\cdots,x_{n}),y=(y_{1},\cdots,y_{n}).$ (1.3)

The space-time Gaussian white noise has covariance formally:

$\langle\Gamma(x,t,\cdot),\Gamma(y,\tau,\cdot)\rangle=\delta(x-y)\delta(t-\tau).$
(1.4)

All three type noise structures will be developed in detail in later sections.
In this paper we will provide a systematic treatment of Laser propagation in
random media highlighting some of the well-known physical phenomena such as
deep turbulence. In the simplest yet mathematically nontrivial case, the
problem reduces to stochastic Schrodinger equation with space-time white noise
and we discuss its relationship to stochastic quantization. Nonlinear
phenomena such as Kerr effect can be modeled by stochastic nonlinear
Schrodinger equation [60] in the focusing case again with space-time white
noise. Similar equations also arise in nonlinear fiber optics [1]. We will
also discuss a number of other stochastic partial differential equations
studied in the context of random wave propagation phenomena such as stochastic
transport equation [52, 26] and stochastic Korteweg-de Vries equation [10].
The quantization method we use is in the spirit of Wick expansions used in
quantum field theory (see for example [58]) and the white noise calculus ([35,
49]) method for stochastic partial differential equations is described in [36]
which also gives a comprehensive discussion of the literature in this subject.

##  2\. Derivation of the paraxial equation from the Maxwell equations

In this section we will give a heuristic derivation of the paraxial equation
starting from the Maxwell equation. The paraxial equation we derive is a very
well-known model widely used in the literature on acoustic wave and
electromagnetic wave propagation in random media [65, 53, 64, 60, 32]. Let
$E,H,B,D$ denote respectively the electric field, magnetizing field, magnetic
field and the displacement field.The Maxwell equations are:

$\nabla\times E=\frac{\partial B}{\partial t}$ (2.1) $\nabla\times
H=\frac{\partial D}{\partial t}$ (2.2) $D=\epsilon E\mbox{ and }B=\mu H.$
(2.3)

Here $\epsilon$ is the permitivity and $\mu$ is the permiability of the
medium. Taking curl of the first equation and taking time derivative of the
second equation and substituting in the first we get upon using the vector
identity $\nabla\times\nabla\times E=-\Delta E+\nabla(\nabla\cdot E)$

$\Delta E-\mu\frac{\partial^{2}}{\partial t^{2}}(\epsilon
E)=\nabla(\nabla\cdot E)$ (2.4)

In the absence of free charge $\nabla\cdot D=0$ and hence
$E\cdot\nabla\epsilon+\epsilon\nabla\cdot E=0$ and substituting we get

$\Delta E-\mu\frac{\partial^{2}}{\partial t^{2}}(\epsilon
E)=-\nabla(E\cdot\nabla(log\epsilon))$ (2.5)

We have noting $\epsilon(x,t)=\epsilon_{0}n^{2}(x,t)$ where $n$ is the
refractive index and $c^{2}=1/(\mu\epsilon_{0})$ with $c$ the light speed, we
arrive at

$\Delta E(x,t)-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}(n^{2}(x,t)E(x,t))=-2\nabla(E(x,t)\cdot\nabla log(n(x,t))).$ (2.6)

We assume that the time scale of fluctuations in the medium is much slower
than the light speed and invoke further simplifications based on this
assumption. Thus neglecting the right hand side and also $n^{2}$ term out of
time derivative we arrive at

$\Delta E(x,t)-\frac{n^{2}(x,t)}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}E(x,t)=0.$ (2.7)

Substituting a plane wave solution
$E(x_{1},x_{2},x_{3},t)=\psi(x_{1},x_{2},x_{3},t)\exp(ikx_{3}-i\omega t)$ and
neglecting the back-scatter term
$\frac{\partial^{2}\psi(x_{1},x_{2},x_{3},t)}{\partial x_{3}^{2}}$ (using
simple scaling argument, see for example [53]) we arrive at the paraxial
equation:

$2ik\frac{\partial\psi}{\partial
x_{3}}+[\Delta_{\bot}+k^{2}\\{\frac{n^{2}(x_{1},x_{2},x_{3},t)}{n_{0}^{2}}-1\\}]\psi=0.$
(2.8)

where $\Delta_{\bot}$ is the two dimensional Laplacian in variables
$x_{2},x_{3}$. Renaming the time-like variable $x_{3}$ as $t$ and suppressing
the actual time variable $t$ we end up with the two dimensional linear or
nonlinear stochastic Schrödinger equation:

$i\frac{\partial}{\partial
t}\psi(x,t,\omega)+[\Delta+V(x,t,\omega,\psi)]\psi(x,t,\omega)=0.$ (2.9)

In this paper $V$ will be modeled as a space-time Gaussian white noise[35,
49], Poisson white noise [38, 39, 4] or Levy type white noise[36] and the
paraxial equation will then be framed as a stochastic quantization problem.

##  3\. Mathematical background on white noise calculus

###  3.1. White noise theory: Gaussian, Poisson and Lévy

In this section we will recall some of the basic elements of white noise
stochastic calculus [34, 35, 49, 4, 36] needed in this paper. We will start by
defining the Schwartz space $\mathcal{S}=\mathcal{S}(\mathbb{R}^{d})$ of
rapidly decaying real-valued $C^{\infty}$ functions on $\mathbb{R}^{d}$. This
space equipped with the family of seminorms:

$\|f\|_{k,\alpha}:=\sup_{x\in\mathbb{R}^{d}}\\{(1+|x|^{k})|\partial^{\alpha}f(x)|\\}$
(3.1)

is a Frechet space [56]. Here $k$ is a non-negative integer,
$\alpha=\\{\alpha_{1},\cdots,\alpha_{d}\\}$ is a multi-index of non-negative
integers $\alpha_{i},$ $i=1,\cdots,d$ and

$\partial^{\alpha}f(x)=\frac{\partial^{|\alpha|}}{\partial
x_{1}^{\alpha_{1}}\cdots\partial x_{d}^{\alpha_{d}}}f(x),\mbox{ where
}|\alpha|:=\alpha_{1}+\cdots+\alpha_{d}.$ (3.2)

The dual space of $\mathcal{S}(\mathbb{R}^{d})$ denoted
$\mathcal{S}^{\prime}:=\mathcal{S}^{\prime}(\mathbb{R}^{d})$ equipped with the
weak topology is the space of tempered distributions. Let
$(\Omega,\mathbb{\mathcal{F}},m)$ is a complete probability space and $\omega$
is an $\mathcal{S}^{\prime}(\mathbb{R}^{d})$-valued random variable. The law
$\mu$ of this $\mathcal{S}^{\prime}(\mathrm{R}^{d})$-valued random variable
$\omega$ is characterized next. We recall the Bochner-Milnos theorem [56] for
the existence of probability measures on the Borel sets
$\mathcal{B}(\mathcal{S}^{\prime})$:

######  Theorem 3.1.

A necessary and sufficient condition for the existence of a probability
measure $\mu$ on $\mathcal{B}(\mathcal{S}^{\prime})$ and a functional $F$ on
$\mathcal{S}$ such that

$\int_{\mathcal{S}^{\prime}}e^{i<\omega,\phi>}\mu(d\omega)=F(\phi),\hskip
7.22743pt\forall\phi\in\mathcal{S}$ (3.3)

is that $F$ satisfies the following three conditions:

  1. (1)

$F(0)=1$,

  2. (2)

$F$ is positive definite:
$\sum_{j,l=1}^{n}z_{j}\bar{z}_{l}F(\phi_{j}-\phi_{l})\geq 0,\hskip
7.22743pt\forall
z_{k}\in\mathbb{C},\forall\phi_{k}\in\mathcal{S},k=1,\cdots,n,$

  3. (3)

$F$ is continuous in the Frechet topology of $\mathcal{S}$.

We will also utilize three particular cases:
(i) Gaussian measure $\mu_{G}$:

$\int_{\mathcal{S}^{\prime}}e^{i<\omega,\phi>}\mu_{G}(d\omega)=\exp\\{-\frac{1}{2}\|\phi\|_{L^{2}(\mathbb{R}^{d})}^{2}\\},\hskip
7.22743pt\forall\phi\in\mathcal{S},$ (3.4)

(ii) Poisson measure $\mu_{P}$:

$\int_{\mathcal{S}^{\prime}}e^{i<\omega,\phi>}\mu_{P}(d\omega)=\exp\\{\int_{\mathbb{R}^{d}}(e^{i\phi(x)}-1)dx\\},\hskip
7.22743pt\forall\phi\in\mathcal{S},$ (3.5)

(iii) Pure jump Levy measure $\mu_{L}$:

$\int_{\mathcal{S}^{\prime}}e^{i<\omega,\phi>}\mu_{L}(d\omega)=\exp\\{\int_{\mathbb{R}^{d}}\Psi(\phi(x))dx\\},\hskip
7.22743pt\forall\phi\in\mathcal{S}\mbox{ and }$ (3.6)
$\Psi(\phi)=\int_{\mathrm{R}^{d}}(e^{i<\phi,z>}-1-i<\phi,z>)\nu(dz).$ (3.7)

######  Definition 3.1.

  1. (1)

The continuous version of the mapping $\mathbb{R}^{d}\ni
x=(x_{1},\cdots,x_{d})\rightarrow B_{x}(\omega)\in L^{2}(\mu_{G})$ defined by

$B_{x}(\omega)=\langle\omega,\Xi(x_{1})\times\cdots\times\Xi(x_{d})\rangle$
(3.8)

is called the $d$-parameter Brownian motion (Brownian sheet). Here
$\Xi(\cdot)\in L^{2}(\mathbb{R})$ is defined as:

$\Xi(s)=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{ll}\chi_{(0,s]}&\mbox{if
}s\geq 0\\\ -\chi_{(s,0]}&\mbox{if }s<0\end{array}}\right.$ (3.9)

where $\chi$ is the usual indicator function.

  2. (2)

The right continuous interger-valued mapping $\mathrm{R}^{d}\ni
x=(x_{1},\cdots,x_{d})\rightarrow P_{x}(\omega)\in L^{2}(\mu_{P})$ defined by

$P_{x}(\omega)=\langle\omega,\Xi(x_{1})\times\cdots\times\Xi(x_{d})\rangle$
(3.10)

is called the $d$-parameter Poisson process. The mapping $\mathrm{R}^{d}\ni
x=(x_{1},\cdots,x_{d})\rightarrow P_{x}(\omega)-\Pi_{i=1}^{d}x_{i}\in
L^{2}(\mu_{P})$ is called compensated Poisson process.

We will now describe Wiener-Ito expansions:

######  Definition 3.2.

Each $f\in L^{2}(\mu_{G})$ has an expansion in multiple (Brownian) Wiener
integrals:

$f(\omega)=\sum_{n=0}^{\infty}\int_{\mathbb{R}^{nd}}f_{n}(x)dB^{\otimes
n}_{x}(\omega),$ (3.11)

where $f_{n}\in\hat{L^{2}(\mathrm{R}^{nd})}$ are deterministic symmetrized
functions in $nd$ variables and

$\|f\|^{2}_{L^{2}(\mu_{G})}=\sum_{n=0}^{\infty}n!\|f_{n}\|^{2}_{L^{2}(\mathbb{R}^{nd})}.$
(3.12)

Similarly $g\in L^{2}(\mu_{P})$ has an expansion in multiple (Poisson) Wiener
integrals:

$g(\omega)=\sum_{n=0}^{\infty}\int_{\mathbb{R}^{nd}}f_{n}(x)d(P_{x}(\omega)-\Pi_{i=1}^{d}x_{i})^{\otimes
n}_{x}(\omega),$ (3.13)

where $g_{n}\in\hat{L^{2}(\mathbb{R}^{nd})}$ are deterministic symmetrized
functions in $nd$ variables and

$\|g\|^{2}_{L^{2}(\mu_{P})}=\sum_{n=0}^{\infty}n!\|g_{n}\|^{2}_{L^{2}(\mathbb{R}^{nd})}.$
(3.14)

We will now recall equivalent expansions in terms of Hermite and Charlier
polynomials. For $n=1,2,\cdots$ let $\zeta_{n}(x)\in\mathcal{S}(\mathbb{R})$
be the Hermite function of order $n$:

$\zeta_{n}(x):=\pi^{-1/4}((n-1)!)^{-1/2}e^{-x^{2}/2}h_{n-1}(\sqrt{2x}),\hskip
7.22743ptx\in\mathbb{R},$ (3.15)

where $h_{n}(x)$ is the $n$-th Hermite polynomial defined by

$h_{n}(x):=(-1)^{n}e^{x^{2}/2}\frac{d^{n}}{dx^{n}}(e^{-x^{2}/2}),\hskip
7.22743ptx\in\mathbb{R},n=0,1,2,\cdots.$ (3.16)

It is well known that [56] the sequence $\\{\zeta_{n}\\}_{n=1}^{\infty}$ forms
an orthonomal basis for $L^{2}(\mathbb{R})$. Hence the family
$\\{\eta_{\alpha}\\}$ of tensor products

$\eta_{\alpha}=e_{\alpha_{1}\alpha_{2}\cdots\alpha_{d}}:=\zeta_{\alpha_{1}}\otimes\cdots\otimes\zeta_{\alpha_{d}},\hskip
7.22743pt\alpha\in\mathbb{N}^{d}$ (3.17)

forms an orthonomal basis for $L^{2}(\mathbb{R}^{d})$. let us now assume that
the family of all multi-indices $\alpha=(\alpha_{1},\cdots,\alpha_{d})$ is
given a fixed ordering

$(\alpha^{(1)},\alpha^{(2)},\cdots\alpha^{(n)},\cdots),$ (3.18)

where $\alpha^{(k)}=(\alpha^{(k)}_{1},\cdots,\alpha^{(k)}_{d})$ and denote
$\eta_{k}=\eta_{\alpha^{(k)}}$. For a multi-index
$\alpha=(\alpha_{1},\cdots,\alpha_{n})$ and $n\in\mathbb{N}$ define the
Hermite polynomial functionals as

$H_{\alpha}(\omega):=\Pi_{j=1}^{n}h_{\alpha_{j}}(\langle\omega,\eta_{j}\rangle),$
(3.19)

and the Charlier polynomial functionals as

$C_{\alpha}(\omega):=C_{|\alpha|}(\omega;\overbrace{\eta_{1},\cdots,\eta_{1}}^{\alpha_{1}\mbox{
times }},\cdots,\overbrace{\eta_{n},\cdots,\eta_{n}}^{\alpha_{n}\mbox{ times
}}),$ (3.20)

with the convention

$C_{n}(\omega;e_{1},\cdots,e_{n}):=\frac{\partial^{n}}{\partial\theta_{1}\cdots\partial\theta_{n}}e^{\\{\langle\omega,\ln{(1+\sum_{j=1}^{n}\theta_{j}e_{j})}\rangle-\sum_{j=1}^{n}\theta_{j}\int_{\mathbb{R}^{d}}e_{j}(y)dy\\}}|_{\theta_{1}=\cdots=\theta_{n}=0}$
(3.21)

We also have the following equalities:

$H_{\alpha}=\int_{\mathbb{R}^{nd}}e^{\hat{\otimes}|\alpha|}dB_{x}^{\otimes|\alpha|}(\omega)$
(3.22)

and

$C_{\alpha}=\int_{\mathbb{R}^{nd}}e^{\hat{\otimes}|\alpha|}d(P_{x}(\omega)-\Pi_{i=1}^{d}x_{i})_{x}^{\otimes|\alpha|}(\omega)$
(3.23)

Moreover, for any random functional $f(\omega)$ taking values on a separable
Hilbert space $V$, and square integrable in $\mu_{G}$, $f\in L^{2}(\mu_{G};V)$
we have the Wiener Hermite polynomial chaos expansion [13]:

$f(\omega)=\sum_{\alpha}a_{\alpha}H_{\alpha}(\omega),\hskip
7.22743pta_{\alpha}\in V,$ (3.24)

with

$\|f\|^{2}_{L^{2}(\mu_{G};V)}=\sum_{\alpha}\alpha!\|a_{\alpha}\|_{V}^{2},\mbox{
where }\alpha!=\alpha_{1}!\cdots\alpha_{n}!.$ (3.25)

Similarly, for any $g\in L^{2}(\mu_{P};V)$ we have the Wiener Charlier
polynomial chaos expansion

$g(\omega)=\sum_{\alpha}b_{\alpha}C_{\alpha}(\omega),\hskip
7.22743ptb_{\alpha}\in V,$ (3.26)

with

$\|g\|^{2}_{L^{2}(\mu_{P};V)}=\sum_{\alpha}\alpha!\|b_{\alpha}\|_{V}^{2},\mbox{
where }\alpha!=\alpha_{1}!\cdots\alpha_{n}!$ (3.27)

Note also that the correspondence between Gaussian and Poisson spaces
${\mathcal{U}}:L^{2}(\mu_{G};V)\rightarrow L^{2}(\mu_{P};V)$:

${\mathcal{U}}\\{\sum_{\alpha}a_{\alpha}H_{\alpha}(\omega)\\}=\sum_{\alpha}a_{\alpha}C_{\alpha}(\omega),$
(3.28)

is unitary [39, 3].

###  3.2. Hida-Kondratiev spaces

We will start with the characterization of the classical embedding of the
Schwartz space in the space of tempered distributions as
$\mathcal{S}(\mathrm{R}^{d})\subset
L^{2}(\mathbb{R}^{d})\subset\mathcal{S}^{\prime}(\mathbb{R}^{d})$ using
Hermite Fourier coefficients [57] and then define the Hida-Kondratiev spaces
in an analogous way using Gaussian, Poisson and Levy measures.

######  Theorem 3.2.

(i) Let $\phi\in L^{2}(\mathbb{R}^{d})$ so that we have the Fourier-Hermite
expansion

$\phi=\sum_{j=1}^{\infty}a_{j}\eta_{j},\hskip 7.22743pt\mbox{ where
}a_{j}=(\phi,\eta_{j}),\hskip 7.22743ptj=1,2,\cdots,$ (3.29)

with

$\eta_{j}:=\zeta_{\delta^{(j)}}=\zeta_{\delta_{1}^{(j)}}\otimes\cdots\zeta_{\delta_{d}^{(j)}},\hskip
7.22743ptj=1,2,\cdots.$ (3.30)

Here $a_{j}$ are the Fourier coefficients of $\phi$ with respect to the tensor
product Hermite functions $\eta_{j}$. Then $\phi\in S(\mathrm{R}^{d})$ if and
only if

$\sum_{j=1}^{\infty}a_{j}^{2}(\delta^{(j)})^{\gamma}<\infty,$ (3.31)

for all $d$-dimensional multi-indices $\gamma=(\gamma_{1},\cdots,\gamma_{d})$.

(ii) A distribution $T\in\mathcal{S}^{\prime}(\mathrm{R}^{d})$ is
characterized by the expansion

$T=\sum_{j=1}^{\infty}b_{j}\eta_{j},\mbox{ with }$ (3.32)
$\sum_{j=1}^{\infty}b_{j}^{2}(\delta^{(j)})^{-\theta}<\infty,$ (3.33)

for some $d$-dimensional multi-index $\theta=(\theta_{1},\cdots,\theta_{d})$.

######  Definition 3.3.

Let $0\leq\rho\leq 1$. We say $f(\omega)=\sum_{\alpha}a_{\alpha}H_{\alpha}\in
L^{2}(\mu_{G};V)$ belongs to Gaussian Hida-Kondratiev stochastic test function
space $(\mathcal{S})_{G}^{\rho}(V)$ if

$\|f\|_{\rho,k}^{2}=\sum_{\alpha}\|a_{\alpha}\|_{V}^{2}(\alpha!)^{1+\rho}(2\mathbb{N})^{\alpha
k}<\infty\hskip 7.22743pt\forall k\in\mathbb{N},$ (3.34)

where

$(2\mathbb{N})^{\alpha}=\Pi_{j=1}^{k}(2j)^{\alpha_{j}},\hskip
7.22743pt\alpha=(\alpha_{1},\cdots,\alpha_{k}).$ (3.35)

We say $f(\omega)=\sum_{\alpha}b_{\alpha}H_{\alpha}(\omega)\in
L^{2}(\mu_{G};V)$ belongs to Gaussian Hida-Kondratiev stochastic distribution
space $(\mathcal{S})_{G}^{-\rho}(V)$ if

$\sum_{\alpha}\|b_{\alpha}\|_{V}^{2}(\alpha!)^{1-\rho}(2\mathbb{N})^{-\alpha
q}<\infty\mbox{ for some }q\in\mathbb{N}.$ (3.36)

This establishes the embedding

$(\mathcal{S})^{1}_{X}(V)\subset(\mathcal{S})^{\rho}_{X}(V)\subset(\mathcal{S})^{+0}_{X}(V)\subset
L^{2}(\mu_{X};V)\subset(\mathcal{S})^{-0}_{X}(V)\subset(\mathcal{S})^{-\rho}_{X}(V)\subset(\mathcal{S})^{-1}_{X}(V),$

with $X=G,P$ or $L$.

Remark: The unitary correspondence ${\mathcal{U}}:L^{2}(\mu_{G};V)\rightarrow
L^{2}(\mu_{P};V)$:

${\mathcal{U}}\\{\sum_{\alpha}a_{\alpha}H_{\alpha}(\omega)\\}=\sum_{\alpha}a_{\alpha}C_{\alpha}(\omega),$
(3.37)

can be extended as a unitary map
${\mathcal{U}}:(\mathcal{S})^{\rho}_{G}(V)\rightarrow(\mathcal{S})^{\rho}_{P}(V),$
$-1\leq\rho\leq 1$ ([39, 4]).

######  Definition 3.4.

We have the following definitions of white noise processes.

  1. (1)

A $d$-parameter Gaussian white noise is defined by

$W_{x}(\omega)=\sum_{k=1}^{\infty}\eta_{k}(x)H_{\varepsilon^{(k)}}(\omega),\hskip
7.22743ptx\in\mathbb{R}^{d},$ (3.38)

where $\varepsilon^{(k)}$ is the multi-index with $1$ in $k$-th entry and zero
otherwise.

  2. (2)

A $d$-parameter compensated Poissonian white noise is defined by

$\dot{P}_{x}(\omega)-1=\sum_{k=1}^{\infty}\eta_{k}(x)C_{\varepsilon^{(k)}}(\omega),\hskip
7.22743ptx\in\mathbb{R}^{d}.$ (3.39)

######  Lemma 3.1.

We have

  1. (1)

$W_{x}(\omega)=\frac{\partial^{d}}{\partial x_{1}\cdots\partial
x_{d}}B_{x}(\omega)\in(\mathcal{S})^{-\rho}_{G}$, $\rho\in[0,1]$,

  2. (2)

$\dot{P}_{x}(\omega)-1=\frac{\partial^{d}}{\partial x_{1}\cdots\partial
x_{d}}(P_{x}(\omega)-\Pi_{i=1}^{d}x_{i})\in(\mathcal{S})^{-\rho}_{P}$,
$\rho\in[0,1]$.

###  3.3. Wick products and properties

The Wick product is defined as below:

######  Definition 3.5.

The Wick product $F\diamond G$ of two elements of
$(\mathcal{S})^{-1}(\mathbb{R}^{n})$ is defined by:

$F=\sum_{\alpha}a_{\alpha}H_{\alpha},G=\sum_{\alpha}b_{\alpha}H_{\alpha}\in(\mathcal{S})^{-1},\mbox{
with }a_{\alpha},b_{\alpha}\in\mathbb{R}^{n},$ (3.40)

$F\diamond G:=\sum_{\alpha,\beta}a_{\alpha}\cdot
b_{\beta}H_{\alpha+\beta}\in(\mathcal{S})^{-1}(\mathbb{R}^{n}).$ (3.41)

The fact that Wick product gives a distribution $F\diamond
G\in(\mathcal{S})^{-1}(\mathbb{R}^{n})$ follows from the estimate below (see
[36]). We have
$F=\sum_{\alpha}a_{\alpha}H_{\alpha},G=\sum_{\beta}b_{\beta}H_{\beta}\in(\mathcal{S})^{-1}(\mathbb{R}^{n})$
means there exists $q_{1}\in\mathbb{N}$ such that

$\sum_{\alpha}|a_{\alpha}|^{2}(2\mathbb{N})^{-q_{1}\alpha}<\infty\hskip
7.22743pt\mbox{ and
}\sum_{\beta}|b_{\beta}|^{2}(2\mathbb{N})^{-q_{1}\beta}<\infty.$

Now rewriting

$F\diamond G:=\sum_{\alpha,\beta}a_{\alpha}\cdot
b_{\beta}H_{\alpha+\beta}=\sum_{\gamma}(\sum_{\alpha+\beta=\gamma}a_{\alpha}\cdot
b_{\beta})H_{\gamma},$ (3.42)

and then setting $C_{\gamma}=\sum_{\alpha+\beta=\gamma}a_{\alpha}\cdot
b_{\beta}$ and $q=q_{1}+k$ we have by Cauchy-Schwartz inequality:

$\sum_{\gamma}(2\mathbb{N})^{-q\gamma}|c_{\gamma}|^{2}\leq\mathopen{}\mathclose{{}\left(\sum_{\gamma}(2\mathbb{N})^{-k\gamma}}\right)\mathopen{}\mathclose{{}\left(\sum_{\alpha}|a_{\alpha}|^{2}(2\mathbb{N})^{-q_{1}\alpha}}\right)\mathopen{}\mathclose{{}\left(\sum_{\beta}|b_{\beta}|^{2}(2\mathbb{N})^{-q_{1}\beta}}\right)<\infty.$

The first term on the right is finite for $k>1$ due to a Lemma by Zhang[71]
and the other two terms are finite by definition of the distributions
$F,G\in(\mathcal{S})^{-1}(\mathbb{R}^{n})$.

###  3.4. S-transform, Hermite transform and characterization theorems

######  Definition 3.6.

Let $F=\sum_{\alpha}b_{\alpha}H_{\alpha}\in(S)_{G}^{-1}(V)$. Then the Hermite
transform of $F$ denoted ${\mathcal{H}}_{G}F$ is defined :

${\mathcal{H}}_{G}F=\sum_{\alpha}b_{\alpha}z^{\alpha}\in V\hskip
7.22743pt\mbox{ (when convergent)},$ (3.43)

where $z=(z_{1},z_{2}\cdots)\in\mathbb{C}^{\mathbb{N}}$,
$z^{\alpha}=z^{\alpha_{1}}\cdots z^{\alpha_{k}}$ for
$\alpha=(\alpha_{1},\cdots,\alpha_{k}).$

Similar statements for the Poisson and Lévy cases. Remark 2.7, [71] clarifies
the convergence of this series.

######  Lemma 3.2.

If $F,G\in(\mathcal{S})_{X}^{-1}(V)$, with $X=G,P,L$ (Gaussian, Poisson and
Levy respectively) then

${\mathcal{H}}_{X}(F\diamond
G)(z)={\mathcal{H}}_{X}F(z)\cdot{\mathcal{H}}_{X}G(z),$ (3.44)

for all $z$ such that ${\mathcal{H}}_{X}F(z)$ and ${\mathcal{H}}_{X}G(z)$
exist (convergent).

######  Lemma 3.3.

Suppose $g(z_{1},z_{2},\cdots)$ is a bounded analytic function on ${\bf
B}_{q}(\delta)$ for some $\delta>0$, $0<q<\infty$ where

${\bf
B}_{q}(\delta):=\mathopen{}\mathclose{{}\left\\{z=(z_{1},z_{2},\cdots)\in\mathbb{C}_{0}^{N};\hskip
7.22743pt\sum_{\alpha\neq 0}|z^{\alpha}|^{2}(2\mathbb{N})^{\alpha
q}<\delta^{2}.}\right\\}$ (3.45)

then there exists $F\in(\mathcal{S})_{G}^{-1}(V)$ and
$D\in(\mathcal{S})_{P}^{-1}(V)$ such that
${\mathcal{H}}_{G}F=g={\mathcal{H}}_{P}D.$

See also the full statement of characterization theorem for
$(\mathcal{S})^{-1}$ for the Gaussian as well as Poisson and Lévy cases in
[36] (Theorem 2.6.11, Theorem 5.4.19). In this paper we will utilize a vector-
valued version (Hilbert or Banach space valued) of the Hida-Kondratiev spaces,
Hermite transform and inverse Hermite transform (Characterization theorem) and
these extensions are straightforward generalizations of what is stated above.

######  Remark 3.1.

Many authors have introduced ${\mathcal{S}}$-transform of
$(\mathcal{S})^{-1}(V)$ -valued distributions [35, 49] which is closely
related to the Hermite transform:

${\mathcal{S}}\phi(\zeta):=\int_{\mathcal{S}^{{}^{\prime}}}\phi(\omega)\exp^{\diamond}\langle\omega,\zeta\rangle
d\mu(\omega),$ (3.46)

where
$\exp^{\diamond}\langle\omega,\zeta\rangle=\sum_{0}^{\infty}\frac{1}{n!}\langle\omega,\zeta\rangle^{\diamond
n}.$ It can be shown to be related to the Hermite transform as

${\mathcal{H}}\phi(z_{1},z_{2},\cdots,z_{k})={\mathcal{S}}\phi(z_{1}\eta_{1}+\cdots+z_{k}\eta_{k})\hskip
7.22743pt\mbox{ for all }z_{1},z_{2},\cdots,z_{k}\in\mathbb{C}^{k},$ (3.47)

in a suitable neighborhood. We however find it more convenient to work with
the Hermite transform as pointed out in [36].

We also note for later use that the Hermite transform of the above white
noises result in:

  1. (1)

Hermite transform of $d$-parameter Gaussian white noise is given by:

${\mathcal{H}}[W_{x}](z)=\sum_{k=1}^{\infty}\eta_{k}(x)z^{\varepsilon_{k}}.$
(3.48)

  2. (2)

Hermite transform of $d$-parameter compensated Poisson white noise is given by

${\mathcal{H}}[\dot{P}_{x}-1](z)=\sum_{k=1}^{\infty}\eta_{k}(x)z^{\varepsilon_{k}}.$
(3.49)

Similar mathematical theory for pure jump Lévy measures are given in Chapter 5
of Holden [36] which parallel the above developments in polynomial expansions,
distributional spaces as well as Hermite transforms which will be utilized in
our paper as well.

##  4\. Kato theory of quasilinear abstract evolution equations

All the problems considered in this paper, after Wick-quantization followed by
Hermit (or $\mathcal{S}$ )-transform, brought to a general class of
deterministic quasilinear evolutions (complex in general) parameterized by an
infinite sequence of complex numbers $z_{1},z_{2},\cdots,$. We will thus
recall some general results developed by T. Kato [44, 45] on quasilinear
evolutions on a Banach space $X$:

$\frac{du}{dt}+A(t,u)u=0,\hskip 7.22743pt0\leq t\leq T,$ (4.1) $u(0)=u_{0}.$
(4.2)

Here $A(t,u)$ is an unbounded linear operator that nonlinearly depends on $t$
and $u$. Kato also points out in [45] that an inhomogeneous equation with a
right hand side:

$\frac{du}{dt}+A(t,u)u=f(t,u)$ (4.3)

can be recast as a homogeneous problem above by redefining the variables.
Kato’s theory covers parabolic problems where the linearized operator
generates an analytic semigroup as well as hyperbolic problems [42, 43] where
the linearized operator generates a $C_{0}$-semigroup which is not analytic.
The main idea is to construct a mild solution $w\in C([0,T];X)$ for each $u\in
C([0,T];X)$ for the linearized problem:

$\frac{dw}{dt}+A(t,u)w=0,\hskip 7.22743pt0\leq t\leq T,$ (4.4) $w(0)=u_{0}.$
(4.5)

This defines a correspondence $u\rightarrow w=F(u)$ in C([0,T];X). Then use
fixed point theory to construct the solution of the original quasilinear
problem. Kato’s quasilinear theory was extended to the stochastic case using
infinite dimensional Ito calculus in[25, 51]. In this paper we will extend
Kato’s theory to white noise calculus realm to treat problems with singular
noises. We briefly summarize Kato’s theory below [44, 45, 55].

######  Definition 4.1.

Let $B$ be a subset of a Banach space $X$ and for every $0\leq t\leq T$ and
$b\in B$ let $A(t,b)$ be the infinitesimal generator of a $C_{0}$-semigroup
$S_{t,b}(s),$ $s\geq 0$ on $X$. The family of operators
$\\{A(t,b)\\},(t,b)\in[0,T]\times B$, is stable if there are constants $M\geq
1$, $\omega$ such that the resolvent set

$\rho(A(t,b))\supset]\omega,\infty],\hskip 7.22743pt(t,b)\in[0,T]\times B$
(4.6)

and

$\|\Pi_{j=1}^{k}(\lambda-A(t_{j},b_{j}))^{-1}\|\leq
M(\lambda-\omega)^{-k}\hskip 7.22743pt\mbox{ for }\lambda>\omega$ (4.7)

for every finite sequence $0\leq t_{i}\leq t_{2}\cdots\leq T$, $b_{j}\in
B,1\leq j\leq k$.

Stable family of generators $\\{A(t,b)\\},(t,b)\in[0,T]\times B$ has stability
estimate [44]

$\|\Pi_{j=1}^{k}S_{t_{j},b_{j}}(s_{j})\|\leq
Me^{\omega\sum_{j=1}^{k}s_{j}}\hskip 7.22743pt\mbox{ for }s_{j}\geq 0$ (4.8)

for every finite sequence $0\leq t_{i}\leq t_{2}\cdots\leq T$, $b_{j}\in
B,1\leq j\leq k$.

######  Theorem 4.1.

Let $u_{0}\in Y$ and let $B=\\{v\in Y;\|v-u_{0}\|_{Y}\leq r\\},r>0$. Let the
stable family of $C_{0}$-semigroup generators
$\\{A(t,b)\\},(t,b)\in[0,T]\times B$ satisfy the assumptions $H_{1}-H_{4}$ of
[55] (Section 6.4) then there is a $T^{\prime},0<T^{\prime}\leq T$ such that
the initial value problem

$\frac{du}{dt}+A(t,u)u=0,\hskip 7.22743pt0\leq t\leq T,$ (4.9) $u(0)=u_{0},$
(4.10)

has a unique mild solution $u\in C([0,T^{\prime}];X)$ with $u(t)\in B$ for
$t\in[0,T^{\prime}]$.

Details of the hypothesis of this theorem are discussed in [44, 45] and also
[55]. Kato formulated this abstract theory to cover a large class of evolution
system of physics and mechanics including the ones studied in the subsequent
sections.

###  4.1. Analytic Mappings between Banach Spaces and Inverse Mapping Theorem

######  Definition 4.2.

A map $\Phi:Z_{1}\rightarrow Z_{2}$ between Banach spaces $Z_{1},Z_{2}$ is
called analytic in $K_{\delta}=\mathopen{}\mathclose{{}\left\\{u\in
Z_{1};\|u\|_{Z_{1}}<\delta}\right\\}$ if it can be deveoped in to a series of
the form:

$\Phi(u)=\sum_{k=0}^{\infty}\Phi_{k}(u,u,\cdots,u),$ (4.11)

where $\Phi_{k}(\cdot,\cdots,\cdot):Z_{1}^{\otimes k}\rightarrow Z_{2}$ are
symmetric multi-linear operators that are bounded:

$\|\Phi_{k}(\cdot)\|=\sup\mathopen{}\mathclose{{}\left\\{\|\Phi_{k}(u,\cdots,u)\|_{Z_{2}};\|u\|_{Z_{1}}\leq
1}\right\\}<\infty,$

and the series converges in $Z_{2}$-norm for $u\in K_{\delta}$ in the
following sense:

$\sum_{k=0}^{\infty}\|\Phi_{k}\|\rho^{k}<\infty,\hskip 7.22743pt\mbox{ for
}0<\rho\leq\delta.$

We will now state the analytic inverse function theorem [7, 69, 67]:

######  Theorem 4.2.

Suppose that $\Phi:Z_{1}\rightarrow Z_{2}$ is an analytic map in a
neighborhood of the origin $0\in K_{\delta}\subset Z_{1}$ and that its Frechét
derivative at the origin $D\Phi(0)$ is an isomorphism from $Z_{1}\rightarrow
Z_{2}$. Then locally $\Phi$ has a unique inverse operator which is analytic in
the neighborhood of $\Phi(0)\in Z_{2}$.

See Vallent[67] for the analytic version of the implicit function theorem as
well.

##  5\.  First-order PDE (transport-type)

###  5.1. First order random transport equation with temporal white noise

Transport problems arise in many applications including radiative transfer
[16, 17, 18]. In this section we will consider first order random transport
equation with temporal white noise studied by S. Ogawa [52], T. Funaki[26],
and H. Kunita [48] and briefly summarize their results. Let
$(\Omega,\Sigma,m)$ be a complete probability space. Consider

$\frac{\partial}{\partial
t}u(x,t,\omega)+n(x,t,\omega)\frac{\partial}{\partial
x}u(x,t,\omega)=c(x,t,\omega)u(x,t,\omega)+d(x,t,\omega).$ (5.1)

Here the wave speed $c(x,t,\omega)$ and forcing $d(x,t,\omega)$ can be
deterministic or random. We will first discuss the one dimensional Ogawa-
Funaki temporal white noise model with $C,D$ deterministic:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\mathopen{}\mathclose{{}\left\\{\frac{d}{dt}\beta(t,\omega)+b(t,x)}\right\\}\frac{\partial}{\partial
x}u(x,t,\omega)$ $=c(x,t)u(x,t,\omega)+d(x,t).$ (5.2)

Here $\beta(t)$ is the 1-D Brownian motion.

Funaki also considered the n-dimensional scalar random transport model and we
first write with Smooth noise:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\sum_{i=1}^{n}n_{i}(x,t,\omega)\frac{\partial}{\partial
x_{i}}u(x,t,\omega)$ $=c(x,t)u(x,t,\omega)+d(x,t).$ (5.3)

The Funaki’s temporal white noise model takes the form:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left\\{\sum_{j}^{n}a_{ij}(t,x)\frac{d}{dt}\beta_{j}(t,\omega)+b_{i}(x,t)}\right\\}\frac{\partial}{\partial
x_{i}}u(x,t,\omega)$ $=c(x,t)u(x,t,\omega)+d(x,t).$ (5.4)

Here $\beta_{j}(t),j=1,\cdots,n$ are independent 1-D Brownian motions.

Define Stratenovich differential equation

$dX_{t}=a(t,X_{t})\circ dB_{t}+b(t,X_{t})dt,\hskip 7.22743ptt\in(r,T),$ (5.5)

$X_{r}=x.$ (5.6)

This is equivalent to Ito differential equation of the form:

$dX_{t}=a(t,X_{t})dB_{t}+\tilde{b}(t,X_{t})dt,\mbox{ where }$ (5.7)
$\tilde{b}(x,t)=b(t,x)+\frac{1}{2}(a^{\prime}a)(t,x),$ (5.8)
$(a^{\prime}a)(t,x)_{i}=\sum_{k,j}\frac{\partial a_{ij}}{\partial
x_{k}}a_{kj}.$ (5.9)

######  Lemma 5.1.

For each $r,t$ with $0\leq r\leq t\leq T$, $X(r,t,\cdot)$ is a homeomorphism
of $\mathrm{R}^{n}$ on to $\mathrm{R}^{n}$.

Define time reversed process

$Y(r,t,y)=X^{-1}(r,t,\cdot)(y),\hskip 7.22743pt0\leq r\leq t\leq
T,y\in\mathrm{R}^{n}.$ (5.10)

Consider:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left\\{\sum_{j}^{n}a_{ij}(t,x)\frac{d}{dt}\beta_{j}(t,\omega)+b_{i}(x,t)}\right\\}\frac{\partial}{\partial
x_{i}}u(x,t,\omega)$ $=c(x,t)u(x,t,\omega)+d(x,t),$ (5.11)
$u(x,0)=\phi(x),\hskip 7.22743ptx\in G\mbox{ and }u(x,t)=\psi(x,t),\hskip
7.22743pt(x,t)\in\partial G\times[0,T].$ (5.12)

The probabilistic solution to the stochastic transport equation by Funaki is:

$u(x,t,\omega)=\mathopen{}\mathclose{{}\left\\{\phi(Y(0,t,x))+\psi(Y(\sigma(t,x),t,x))}\right\\}\exp\mathopen{}\mathclose{{}\left\\{\int_{0}^{t}c(s,Y(s,t,x))ds}\right\\}$
$+\int_{0}^{t}d(s,Y(s,t,x))\exp\mathopen{}\mathclose{{}\left\\{\int_{s}^{t}c(r,Y(r,t,x))dr}\right\\}ds.$
(5.13)

Here $\sigma(t,x)$ is the exit time for the domain $G$.

###  5.2. Stochastic transport model with space-time white noise

We will begin with the following simple model (similar model with temporal
white noise was considered by Funaki [26]) with space-time white noise
$\Gamma(x,t)$ as characteristic speed:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\Gamma(x,t,\omega)\frac{\partial}{\partial
x}u(x,t,\omega)=0,\hskip 7.22743pt(x,t)\in\mathrm{R}\times[0,T],$ (5.14)

$u(x,0,\omega)=\phi(x),\hskip 7.22743ptx\in\mathrm{R}.$ (5.15)

We quantize this problem as:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\Gamma(x,t,\omega)\diamond\frac{\partial}{\partial
x}u(x,t,\omega)=0,\hskip 7.22743pt(x,t)\in\mathrm{R}\times[0,T],$ (5.16)

$u(x,0,\omega)=\phi(x),\hskip 7.22743ptx\in\mathrm{R}.$ (5.17)

We take Hermite transform to get (denoting $[{\mathcal{H}}u]:=\tilde{u}$ )

$\frac{\partial}{\partial
t}\tilde{u}(x,t,z)+[{\mathcal{H}}\Gamma](x,t,z)\frac{\partial}{\partial
x}\tilde{u}(x,t,z)=0,$ (5.18) $\tilde{u}(x,0,z)=\phi(x).$ (5.19)

We write down the equation of characteristic:

$\frac{d\tilde{u}}{ds}=\frac{\partial\tilde{u}(x,t,z)}{\partial
t}\frac{dt}{ds}+\frac{\partial\tilde{u}(x,t,z)}{\partial x}\frac{dx}{ds}$
(5.20)

and arrive at

$\frac{d\tilde{u}}{ds}=0,\hskip 7.22743pt\frac{dt}{ds}=1,\mbox{ and
}\frac{dx}{ds}=[{\mathcal{H}}\Gamma](x,t,z).$ (5.21)

Solving we get
$x=\zeta_{t}(x_{0})=x_{0}+\int_{0}^{t}[{\mathcal{H}}\Gamma](x,r,z)dr$ as the
equation of characteristics and
$\tilde{u}(x,0,z)=\phi(x_{0})=\phi(\zeta_{t}^{-1}(x))$:

$\tilde{u}(x,t,z)=\phi(\zeta_{t}^{-1}(x))=\phi(x-\int_{0}^{t}[{\mathcal{H}}\Gamma](x,r,z)dr).$
(5.22)

Hence

$u(x,t,\omega)={\mathcal{H}}^{-1}\phi(\zeta_{t}^{-1}(x))={\mathcal{H}}^{-1}\mathopen{}\mathclose{{}\left(\phi(x-\int_{0}^{t}[{\mathcal{H}}\Gamma](x,r,z)dr)}\right).$
(5.23)

We will now turn to the multidimensional stochastic transport model with
space-time white noise:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left\\{\sum_{j}^{n}a_{ij}(t,x)\Gamma_{j}(x,t,\omega)+b_{i}(x,t)}\right\\}\frac{\partial}{\partial
x_{i}}u(x,t,\omega)$ $=c(x,t)u(x,t,\omega)+d(x,t).$ (5.24)

We will also study the quantized random transport model:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left\\{\sum_{j}^{n}a_{ij}(t,x)\Gamma_{j}(x,t,\omega)}\right\\}\diamond\frac{\partial}{\partial
x_{i}}u(x,t,\omega)+\sum_{i}^{n}b_{i}(x,t)\frac{\partial}{\partial
x_{i}}u(x,t,\omega)$ $=c(x,t)u(x,t,\omega)+d(x,t).$ (5.25)

Taking the Hermite transform: For $z\in\mathbb{C}^{\mathbb{N}}$, and denoting
${\mathcal{H}}u$ as $\tilde{u}$, we get

$\frac{\partial}{\partial
t}\tilde{u}(x,t,z)+\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left\\{\sum_{j}^{n}a_{ij}(t,x)[{\mathcal{H}}\Gamma]_{j}(x,t,z)+b_{i}(x,t)}\right\\}\frac{\partial}{\partial
x_{i}}\tilde{u}(x,t,z)$ $=c(x,t)\tilde{u}(x,t,z)+d(x,t),$ (5.26)

This is a deterministic linear hyperbolic equation (for a fixed $z$ ) and we
can solve it similar to the simple model presented above and the apply inverse
Hermite transform to obtain the solution for the original stochastic transport
equation with space-time white noise as characteristic speed coefficient and
spatial white noise initial data. With the help of the deterministic results
in [23] we can also obtain the solvability theorem for stochastic transport
equation with white noise characteristic speeds and spatial white noise
initial data as

######  Proposition 5.1.

Suppose that $a,b\in L^{1}(0,T;L^{1}_{\mbox{loc}}(\mathbb{R}^{n}))$, $c,d\in
L^{1}(0,T;L^{\infty}(\mathbb{R}^{n}))$ and the initial data is a spatial white
noise with $u_{0}\in(\mathcal{S})^{-1}(L^{\infty}(\mathbb{R}^{n}))$, and the
stochastic characteristic speed coefficients
$\Gamma_{j}\in(\mathcal{S})^{-1}(L^{\infty}(\mathbb{R}^{n}\times[0,T]))$,
$j=1,\cdots,n$. Then there exists a unique solution
$u\in(\mathcal{S})^{-1}(L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))$.

We now consider a multidimensional quasilinear stochastic transport model with
space-time white noise:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left\\{\sum_{j}^{n}a_{ij}(u(t,x,\omega))\Gamma_{j}(x,t,\omega)+b_{i}(u(x,t,\omega))}\right\\}\frac{\partial}{\partial
x_{i}}u(x,t,\omega)$ $=c(u(x,t,\omega)),$ (5.27)

where $a_{ij},b_{i},c$ are all polynomials in $u$. We quantize this problem
as:

$\frac{\partial}{\partial
t}u(x,t,\omega)+\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left\\{\sum_{j}^{n}(a_{ij}(u(t,x,\omega))^{\diamond})\diamond\Gamma_{j}(x,t,\omega)+b_{i}(u(x,t,\omega))^{\diamond}}\right\\}\diamond\frac{\partial}{\partial
x_{i}}u(x,t,\omega)$ $=c(u(x,t,\omega))^{\diamond},$ (5.28)

Applying Hermite transform we get

$\frac{\partial}{\partial
t}\tilde{u}(x,t,z)+\sum_{i=1}^{n}\mathopen{}\mathclose{{}\left\\{\sum_{j}^{n}a_{ij}(\tilde{u}(t,x,z))[{\mathcal{H}}\Gamma]_{j}(x,t,z)+b_{i}(\tilde{u}(x,t,z))}\right\\}\frac{\partial}{\partial
x_{i}}\tilde{u}(x,t,z)$ $=c(\tilde{u}(x,t,z)).$ (5.29)

Extending the deterministic result to the Hermite transformed problem may
require some smoothing of the white noise term $\Gamma$ using an operator of
the form $(I-\Delta_{x,t})^{-\gamma}$ for a suitable $\gamma\geq 0$. The local
solvability of this deterministic partial differential equation for a given
fixed $z$ can be obtained from Kato’s theory [44, 45] as $\tilde{u}\in
C([0,T^{\prime}];H^{s}(\mathbb{R}^{n}))$, $s>n/2+1$ and from this we can use
the characterization theorem for Hermite transform to deduce:

######  Proposition 5.2.

For $u_{0}\in(\mathcal{S})^{-1}(H^{s}(\mathbb{R}^{n}))$, $s>n/2+1$ and
$\Gamma_{j}(\cdot,\cdot,\cdot)\in({\mathcal{S}})^{-1}(L^{\infty}_{\mbox{loc}}(\mathbb{R}^{n}\times\mathbb{R}^{+}))$,
$j=1,\cdots,n$, there is a unique local-in-time generalized solution for the
Gaussian white noise forced quasilinear transport equation
$u\in(\mathcal{S})^{-1}(C([0,T^{\prime}];H^{s}(\mathbb{R}^{n})))$. For the
Poisson and Lévy cases the unique correspondence map ${\mathcal{U}}$ gives a
unique solution
$u\in(\mathcal{S})^{-1}(L^{\infty}([0,T^{\prime}];H^{s}(\mathbb{R}^{n})))$.

Global in time generalized solution (that accommodates shock waves) can be
built by applying Kruzkov theory [47] to the Hermite transformed problem with
small bounded variation norm initial data:.

######  Proposition 5.3.

For
$u_{0}\in(\mathcal{S})^{-1}\mathopen{}\mathclose{{}\left(L^{\infty}_{\mbox{loc}}(\mathbb{R}^{n})\cap
BV(\mathbb{R}^{n})}\right)$, with sufficiently small norm, and
$\Gamma(\cdot,\cdot,\cdot)\in({\mathcal{S}})^{-1}(L^{\infty}_{\mbox{loc}}(\mathbb{R}^{n}\times\mathbb{R}^{+}))$
there is a unique global-in-time generalized solution for the
Gaussian/Poisson/Lévy white noise forced quasilinear transport equation
$u\in(\mathcal{S})^{-1}(L^{\infty}([0,T];L^{\infty}_{\mbox{loc}}(\mathbb{R}^{n})))$.

##  6\. Stochastic nonlinear wave equations

###  6.1. Stochastic Korteweg De Vries equation

Let us consider the Korteweg De Vries equation with white noise initial data:

$\frac{\partial}{\partial
t}\varphi(x,t,\omega)+\varphi(x,t,\omega)\frac{\partial}{\partial
x}\varphi(x,t,\omega)+\frac{\partial^{3}}{\partial
x^{3}}\varphi(x,t,\omega)=0,$ (6.1) $\varphi(x,0,\omega)=\Gamma(x,\omega).$
(6.2)

We quantize this problem as

$\frac{\partial}{\partial
t}\varphi(x,t,\omega)+\varphi(x,t,\omega)\diamond\frac{\partial}{\partial
x}\varphi(x,t,\omega)+\frac{\partial^{3}}{\partial
x^{3}}\varphi(x,t,\omega)=0.$ (6.3)

Applying Hermite transform results in:

$\frac{\partial}{\partial
t}\tilde{\varphi}(x,t,z)+\tilde{\varphi}(x,t,z)\frac{\partial}{\partial
x}\tilde{\varphi}(x,t,z)+\frac{\partial^{3}}{\partial
x^{3}}\tilde{\varphi}(x,t,z)=0.$ (6.4)
$\tilde{\varphi}(x,0,z)=[{\mathcal{H}}\Gamma](x,z).$ (6.5)

Note that the Hermite transformed KDV equation will have infinite number of
conservation laws as in P. D. Lax theory for deterministic KDV equation [50]
some of which would be finite depending on the smoothness of noise . However,
for the case of singular white noise, values of these conservation laws will
be all infinity.

Tsutsumi [66] has studied deterministic Korteweg-De Vries equation with
bounded Radon measures as initial data (see also Bourgain [11] for the space
periodic case) and proven existence of a generalized solution and we can use
this theorem the above Hermite transformed problem for a fixed $z$ followed by
inverse Hermite transform to conclude that:

######  Proposition 6.1.

For the Gaussian/Poisson/Lévy white noise initial data
$\varphi(\cdot,0,\cdot)=\Gamma(\cdot,\cdot)\in(\mathcal{S})^{-1}(L^{\infty}_{\mbox{loc}}(\mathbb{R}))$,
there exists a unique solution $u$ to the stochastic KDV equation such that:

$u\in(\mathcal{S})^{-1}\mathopen{}\mathclose{{}\left(L^{\infty}(0,\infty;H^{-1}(\mathbb{R}))\cap
L^{2}((0,T)\times(-R,R))}\right),\hskip 7.22743pt\mbox{ for any }T,R>0.$

###  6.2. Stochastic Benjamin-Ono equation

The stochastic Benjamin-Ono equation is modeled as:

$\frac{\partial}{\partial
t}\varphi(x,t,\omega)+\varphi(x,t,\omega)\frac{\partial}{\partial
x}\varphi(x,t,\omega)+{\bf H}[\frac{\partial^{2}}{\partial
x^{2}}\varphi(x,t,\omega)]=0,$ (6.6)

where ${\bf H}[\cdot]$ is the Hilbert transform defined as

${\bf H}[f](x):=\mbox{ PV
}\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{f(y)}{x-y}dy=\hskip
7.22743pt{\mathcal{F}}^{-1}\mathopen{}\mathclose{{}\left(i\cdot(\mbox{
sign}\zeta){\mathcal{F}}(f)(\zeta)}\right),$ (6.7)

where ${\mathcal{F}}$ and ${\mathcal{F}}^{-1}$ denote Fourier transform and
its inverse respectively. This problem is supplied with random initial data:

$\varphi(x,0)=\Gamma(x,\omega).$ (6.8)

We quantize this equation as

$\frac{\partial}{\partial
t}\varphi(x,t,\omega)+\varphi(x,t,\omega)\diamond\frac{\partial}{\partial
x}\varphi(x,t,\omega)+{\bf H}[\frac{\partial^{2}}{\partial
x^{2}}\varphi(x,t,\omega)]=0,$ (6.9)

Applying Hermite transform we get:

$\frac{\partial}{\partial
t}\tilde{\varphi}(x,t,z)+\tilde{\varphi}(x,t,z)\frac{\partial}{\partial
x}\tilde{\varphi}(x,t,z)+{\bf H}[\frac{\partial^{2}}{\partial
x^{2}}\tilde{\varphi}(x,t,z)]=0,$ (6.10)
$\tilde{\varphi}(x,0,z)={\mathcal{H}}[\Gamma](x,z)].$ (6.11)

We can utilize the sharp results of T. Tao [63] along with a suitable
smoothing of the noise by an operator of the form $(I-\Delta)^{-\gamma}$ to
conclude that there is a unique solution to the above problem for fized $z$ as
$\tilde{\varphi}\in C([0,T];H^{s})$ for $s\geq 1$. Hence using inverse Hermite
transform we obtain:

######  Proposition 6.2.

For the Gaussian white noise initial data
$\varphi(\cdot,0,\cdot)=\Gamma(\cdot,\cdot)\in(\mathcal{S})^{-1}(H^{s}(\mathbb{R}))$,
$s\geq 1$, there exists a unique solution to stochastic Benjamin-Ono equation
as $\varphi\in(\mathcal{S})^{-1}(C([0,T];H^{s})),$ $s\geq 1.$ For Poisson and
Lévy cases the unique correspondence map ${\mathcal{U}}$ gives unique solution
$\varphi\in(\mathcal{S})^{-1}(L^{\infty}([0,T];H^{s})),$ $s\geq 1.$

##  7\. Stochastic reaction diffusion equation and quantization

Parisi and Wu [54] initiated the subject of stochastic quantization [20] which
takes the view that (Euclidean) quantum fields can be constructed by studying
stochastic partial differential equations. Stochastic reaction diffusion
equation with space-time Gaussian noise has been studied by many authors [24]
with results such as construction of invariant measures [5, 27] as well as
pathwise strong solutions[2, 21]. In this section we will treat this class of
problems with white noise theory.

###  7.1. Stochastic heat equation with multiplicative noise and the KPZ
Equation

Kardar, Parisi and Zhang [41] studied the stochastic model (which has come to
be known as the KPZ equation):

$\frac{\partial h}{\partial t}=\nu\Delta h+\frac{\lambda}{2}(\nabla
h)^{2}+\Gamma(x,t).$ (7.1)

As pointed out in this paper, in the case of smooth noise the above stochastic
PDE can be formally transformed in to two other well-known stochastic PDEs
(ignoring some constants). In fact $v=-\nabla h$ gives the stochastic Burgers
equation:

$\frac{\partial v}{\partial t}+\lambda v\cdot\nabla v=\nu\Delta
v+\nabla\Gamma(x,t),$ (7.2)

and the transform $\varphi=\exp((\frac{\lambda}{2\nu})h)$ converts the KPZ
equation to

$\frac{\partial}{\partial
t}\varphi(x,t,\omega)=\nu\Delta\varphi(x,t,\omega)+\frac{\lambda}{2\nu}\varphi(x,t,\omega)\Gamma(x,t,\omega),$
(7.3)

with spatial white noise initial data:

$\varphi(x,0,\omega)=\Gamma_{0}(x,\omega).$

We quantize this problem as (after setting $\nu=\sigma^{2}/2$ and
$\lambda=\sigma^{2}$ for simplicity):

$\frac{\partial}{\partial
t}\varphi(x,t,\omega)=\frac{1}{2}\sigma^{2}\Delta\varphi(x,t,\omega)+\varphi(x,t,\omega)\diamond\Gamma(x,t,\omega).$
(7.4)

Applying the Hermite transform gives

$\frac{\partial}{\partial
t}\tilde{\varphi}(x,t,z)-[\frac{1}{2}\sigma^{2}\Delta+[{\mathcal{H}}\Gamma](x,t,z)]\tilde{\varphi}(x,t,z)=0,$
(7.5) $\tilde{\varphi}(x,0,z)=[{\mathcal{H}}\Gamma_{0}](x,z).$

We write the solution formally using the propagator ${\mathcal{Z}}(t,r)$
generated by [55] the unbounded time dependent operator
$\frac{1}{2}\sigma^{2}\Delta+[{\mathcal{H}}\Gamma](x,t,z)$:

$\tilde{\varphi}(x,t,z)={\mathcal{Z}}(t,0)[{\mathcal{H}}\Gamma_{0}](x,z).$
(7.6)

The solution is also probabilistically expressed by the Feynman-Kac formula
using a Brownian motion $B_{t}$ independent of $\Gamma(x,t,\omega)$ and
$\Gamma_{0}(x,\omega)$:

$\tilde{\varphi}(x,t,z)=E_{x}\mathopen{}\mathclose{{}\left[[{\mathcal{H}}\Gamma_{0}](\sigma
B_{t},z)\exp\mathopen{}\mathclose{{}\left\\{\int_{0}^{t}[{\mathcal{H}}\Gamma](\sigma
B_{s},t-s,z)ds}\right\\}}\right].$ (7.7)

Alternatively, we can consider the solution as a fixed point problem for the
heat semigroup:

$\tilde{\varphi}(x,t,z)=e^{\frac{1}{2}\sigma^{2}t\Delta}[{\mathcal{H}}\Gamma_{0}](x,z)+\int_{0}^{t}e^{\frac{1}{2}\sigma^{2}(t-\tau)\Delta}{\mathcal{H}}[\Gamma](x,\tau,z)\tilde{\varphi}(x,\tau,z)d\tau.$
(7.8)

Heat equation with singular and non-autonomous potentials have been studied in
the literature [30, 31] and for a fixed $z$ as necessary using smoothing
operator of the form $(I-\Delta_{x,t})^{-\gamma}$ for the noise we can
conclude that the above Hermite transformed problem has a unique solution in
$\tilde{\varphi}\in C([0,T];L^{\infty})$ and hence by inverse Hermite
transform we obtain:

######  Proposition 7.1.

For the Gaussian white noise
$\Gamma(\cdot,\cdot,\cdot)\in(\mathcal{S})^{-1}(L^{\infty}_{\mbox{loc}}(\mathbb{R}^{d}\times\mathbb{R}))$,
and initial data $\varphi(\cdot,0)\in
L^{\infty}_{\mbox{loc}}(\mathbb{R}^{d})$, there exists a unique solution to
the stochastic heat equation
$\varphi\in(\mathcal{S})^{-1}(C([0,T];L^{\infty}(\mathbb{R}^{d}))).$ For the
Poisson and Lévy white noises the unique correspondence map ${\mathcal{U}}$
gives a unique solution to the stochastic heat equation
$\varphi\in(\mathcal{S})^{-1}(L^{\infty}([0,T];L^{\infty}(\mathbb{R}^{d}))).$

The solution to the quantized problem is also probabilistically expressed by
the Feynman-Kac formula:

$\varphi(x,t,\omega)=E_{x}\mathopen{}\mathclose{{}\left[\Gamma_{0}(\sigma
B_{t},\omega)\diamond\exp^{\diamond}\mathopen{}\mathclose{{}\left\\{\int_{0}^{t}\Gamma(\sigma
B_{s},t-s,\omega)ds}\right\\}}\right],$ (7.9)

where the Wick exponential of $X\in(\mathcal{S})^{-1}$ is defined by:

$\exp^{\diamond}X=\sum_{0}^{\infty}\frac{1}{n!}X^{\diamond n}.$

We also note here that instead of the multiplicative noise term if we have a
stochastic heat equation with space-time white noise as an additive noise term
or initial data as a spatial white noise then after Hermite transform we end
up with heat equation with singular initial data or forcing term and the
problem can be resolved using the fundamental solution of the heat equation
followed by inverse Hermite transform.

###  7.2. Stochastic nonlinear heat equation with white noise Initial Data

We will now consider:

$\frac{\partial}{\partial
t}\varphi(x,t,\omega)+\varphi(x,t,\omega)^{p}=\Delta\varphi(x,t,\omega),\hskip
7.22743ptp=2,3,x\in\mathbb{R}^{n},$ (7.10)
$\varphi(x,0,\omega)=\Gamma(x,\omega),x\in\mathbb{R}^{n}.$

We will quantize this equation as:

$\frac{\partial}{\partial t}\varphi(x,t,\omega)+\varphi(x,t,\omega)^{\diamond
p}=\Delta\varphi(x,t,\omega),$ (7.11)

Applying the Hermite transform gives

$\frac{\partial}{\partial
t}\tilde{\varphi}(x,t,z)-[\Delta-\tilde{\varphi}(x,t,z)^{p-1}]\tilde{\varphi}(x,t,z)=0,$
(7.12) $\tilde{\varphi}(x,0,z)=[{\mathcal{H}}\Gamma](x,z).$ (7.13)

Nonlinear heat equations with measure initial data has been studied in the
literature with positive as well as negative results. When the initial data is
a Dirac measure it has been shown in [8] that this problem with the
nonlinearity selected above $(p=2$ and $3)$ has no solution in any space
dimension $n\geq 1$. This means that we need to smooth the measure by an
operator of the form $(I-\Delta)^{-\gamma}$ so that we can use results for
slightly less singular but still measure data such as that presented in [9]
with initial data in $L^{q},1\leq q<\infty$. This provides a unique short time
solution to the Hermite transformed problem as $\tilde{\varphi}\in
C([0,T^{\prime}];L^{q})$ and we then inverse Hermite transform to obtain:

######  Proposition 7.2.

There exists a unique solution
$\varphi\in(\mathcal{S})^{-1}(C([0,T^{\prime}];L^{q}))$ to the stochastic
nonlinear heat equation with spatial white noise initial data
$\Gamma(\cdot,\cdot)\in(\mathcal{S})^{-1}(L^{q})$.

##  8\. Stochastic linear and nonlinear Schrödinger equations with space-time
white noise

In this section we will address the stochastic linear and nonlinear
Schrödinger equations with space-time white noise that arise in the Laser
propagation problems as discussed earlier. Once again we will utilize the
Hermite transform to convert the problem to deterministic linear and nonlinear
Schrodinger equations parameterized by an infinite sequence of complex
variables $z$. There is a wealth of literature on linear Schrödinger semigroup
with a range of potentials [59, 6] and nonlinear Schrödinger equations [12,
62] which will enable the solvability of the deterministic problems obtained
by Hermite transforms as we discuss below.

###  8.1. Strichartz estimates

We recall the following estimates for the Schrödinger free propagator [61,
46]:

######  Lemma 8.1.

For $(q,r)$ and $(\tilde{q},\tilde{r})$ such that for $d\geq 1$ both exponent
pair satisfying:

$2\leq q,r\leq\infty,\frac{1}{q}=\frac{d}{2}(\frac{1}{2}-\frac{1}{r})\hskip
7.22743pt(q,r,d)\neq(2,\infty,2),$

we have:

$\|e^{it\Delta}u_{0}\|_{L^{q}_{t}L^{r}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\lesssim\|u_{0}\|_{L^{2}_{x}(\mathbb{R}^{d})},$
(8.1)

and

$\|\int_{0}^{t}e^{i(t-s)\Delta}F(s,\cdot)ds\|_{L^{q}_{t}L^{r}_{x}(\mathbb{R}\times\mathbb{R}^{d})}\lesssim\|F\|_{L^{\tilde{q}^{\prime}}_{t}L^{\tilde{r}^{\prime}}_{x}(\mathbb{R}\times\mathbb{R}^{d})}.$
(8.2)

###  8.2. Stochastic linear Schrödinger equation with additive space-time
white noise

$i\frac{\partial}{\partial
t}\psi(x,t,\omega)+\Delta\psi(x,t,\omega)=\Gamma(x,t,\omega),\hskip
7.22743pt(x,t)\in\mathbb{R}^{d}\times\mathbb{R}^{+},$ (8.3)

with spatial white noise initial data:

$\psi(x,0,\omega)=\Gamma_{0}(x,\omega),\hskip 7.22743ptx\in\mathbb{R}^{d}.$
(8.4)

Applying Hermite transform we get the Schrödinger equation in free space:

$i\frac{\partial}{\partial
t}\tilde{\psi}(x,t,z)+\Delta\tilde{\psi}(x,t,z)={\mathcal{H}}[\Gamma](x,t,z),\hskip
7.22743ptx\in\mathbb{R}^{d},z\in\mathbb{C}^{\mathbb{N}},$ (8.5)
$\tilde{\psi}(x,0,z)={\mathcal{H}}[\Gamma_{0}](x,z),\hskip
7.22743ptx\in\mathbb{R}^{d},z\in\mathbb{C}^{\mathbb{N}}.$ (8.6)

The solution is written formally using the free Schrödinger propagator
$e^{it\Delta}$ as

$\tilde{\psi}(x,t,z)=e^{it\Delta}{\mathcal{H}}[\Gamma_{0}](x,z)-i\int_{0}^{t}e^{i(t-\tau)\Delta}{\mathcal{H}}[\Gamma](x,\tau,z)d\tau.$
(8.7)

The d-dimensional Schrödinger kernel is:

$K_{t}(x)=\frac{1}{(4\pi it)^{d/2}}e^{i\frac{|x|^{2}}{4t}}.$ (8.8)

Hence we also have:

$\tilde{\psi}(x,t)=\frac{1}{(4\pi
it)^{d/2}}\mathopen{}\mathclose{{}\left[\int_{\mathrm{R}^{d}}e^{i\frac{|x-y|^{2}}{4t}}{\mathcal{H}}[\Gamma_{0}](y,z)dy-i\int_{0}^{t}\int_{\mathrm{R}^{d}}e^{i\frac{|x-y|^{2}}{4(t-\tau)}}{\mathcal{H}}[\Gamma](x,\tau,z)dyd\tau}\right].$
(8.9)

This equation makes sense for a fixed $z\in\mathbb{C}^{\mathbb{N}}$ if
${\mathcal{H}}[\Gamma_{0}](x,z)\in L^{2}(\mathbb{R}^{d})$ and
${\mathcal{H}}[\Gamma](\cdot,\cdot,z)\in
L^{q}_{t}L^{r}_{x}(\mathbb{R}\times\mathbb{R}^{d})$ due to the Strichartz
estimates recalled above. Hence inverse Hermite transform gives:

######  Proposition 8.1.

For initial data noise distribution
$\Gamma_{0}\in(\mathcal{S})^{-1}(L^{2}(\mathbb{R}^{d}))$ and noise forcing
$\Gamma\in(\mathcal{S})^{-1}(L^{q}_{t}L^{r}_{x}(\mathbb{R}\times\mathbb{R}^{d}))$
there exists a unique solution
$\psi\in(\mathcal{S})^{-1}(L^{q}_{t}L^{r}_{x}(\mathbb{R}\times\mathbb{R}^{d}))$
to the stochastic Schrödinger equation.

###  8.3. Stochastic linear Schrödinger equation with multiplicative space-
time white noise

Consider the linear stochastic Schrödinger equation with multiplicative space-
time noise: For $\omega\in\Omega,x\in\mathrm{R}^{d},d\geq 1,t>0$

$i\frac{\partial}{\partial
t}\psi(x,t,\omega)+[\Delta+\Gamma(x,t,\omega)]\psi(x,t,\omega)=0,$ (8.10)

and spatial white noise initial data:

$\psi(x,0,\omega)=\Gamma_{0}(x,\omega),\hskip 7.22743ptx\in\mathbb{R}^{d}.$
(8.11)

Here the time-like coordinate $t$ is the propagation direction, $\varphi$ is
the (complex) electric field and the potential $V$ in general depends on the
refractive index of the medium.

We consider the quantized linear Schrödinger equation with multiplicative
white noise:

$i\frac{\partial}{\partial
t}\psi(x,t,\omega)+\Delta\psi(x,t,\omega)+\psi(x,t,\omega)\diamond\Gamma(x,t,\omega)=0.$
(8.12)

Applying Hermite transform we get the Schrödinger equation with a potential
$V(x,t,z)={\mathcal{H}}[\Gamma](x,t,z)$:

$i\frac{\partial}{\partial
t}\tilde{\psi}(x,t,z)+\mathopen{}\mathclose{{}\left(\Delta+{\mathcal{H}}[\Gamma](x,t,z)}\right)\tilde{\psi}(x,t,z)=0.$
(8.13) $\tilde{\psi}(x,0,z)={\mathcal{H}}[\Gamma_{0}](x,z),\hskip
7.22743ptx\in\mathbb{R}^{d},z\in\mathbb{C}^{\mathbb{N}}.$ (8.14)

We write the solution formally using the propagator ${\mathcal{Z}}(t,r)$
generated by [59, 55] the unbounded Hamiltonian time dependent
$(\Delta+V(x,t,z))$:

$\tilde{\psi}(x,t,z)={\mathcal{Z}}(t,0)\tilde{\psi}(x,0,z).$ (8.15)

Alternatively, we can consider the solution as a fixed point problem for the
free propagator:

$\tilde{\psi}(x,t,z)=e^{it\Delta}\tilde{\psi}(x,0,z)-i\int_{0}^{t}e^{i(t-\tau)\Delta}{\mathcal{H}}[\Gamma](x,\tau,z)\tilde{\psi}(x,\tau,z)d\tau.$
(8.16)

Schrödinger equations with time dependent unbounded singular potential have
been studied in the literature [70] where it is shown that the two parameter
propagator ${\mathcal{Z}}(t,r)$ is unitary in $L^{2}(\mathbb{R}^{2})$
(actually K. Yajima’s results holds in any dimension.) With an introduction to
a suitable smoothing for the noise in the form of $(I-\Delta_{x,t})^{-\gamma}$
we can fit the Hermite transformed problem above in Yajima’s framework and
deduce a unique solution $\tilde{\psi}\in C([0,T];L^{2}(\mathbb{R}^{2}))$ and
hence inverse Hermite transform gives:

######  Proposition 8.2.

For Gaussian initial data noise distribution
$\Gamma_{0}\in(\mathcal{S})^{-1}(L^{2}(\mathbb{R}^{d}))$ and Gaussian white
noise forcing
$\Gamma\in(\mathcal{S})^{-1}(L^{\infty}_{\mbox{loc}}(\mathbb{R}\times\mathbb{R}^{d}))$,
there exists a unique solution to the stochastic Schrödinger equation with
multiplicative space-time white noise as
$\psi\in(\mathcal{S})^{-1}(C([0,T];L^{2}(\mathbb{R}^{2})))$. For Gaussian and
Lévy cases unique correspondence map ${\mathcal{U}}$ gives a unique solution
$\psi\in(\mathcal{S})^{-1}(L^{\infty}([0,T];L^{2}(\mathbb{R}^{2})))$

###  8.4. Nonlinear Schrödinger equation with multiplicative space-time white
noise

Nonlinear defocusing cubic Stochastic Schrödinger equation with multiplicative
space-time white noise:

$i\frac{\partial}{\partial
t}\psi(x,t,\omega)+\Delta\psi(x,t,\omega)+\mathopen{}\mathclose{{}\left(\Gamma(x,t,\omega)-|\psi|^{2}}\right)\psi(x,t,\omega)=0,\hskip
7.22743ptx\in\mathbb{R}^{3},t>0$ (8.17)

and spatial white noise initial data:

$\psi(x,0,\omega)=\Gamma_{0}(x,\omega),\hskip 7.22743ptx\in\mathbb{R}^{3}.$
(8.18)

We consider the quantized nonlinear Schrodinger equation with multiplicative
white noise:

$i\frac{\partial}{\partial
t}\psi(x,t,\omega)+\Delta\psi(x,t,\omega)+\mathopen{}\mathclose{{}\left(\Gamma(x,t,\omega)-\psi(x,t,\omega)\diamond\psi^{*}(x,t,\omega)}\right)\diamond\psi(x,t,\omega)=0.$
(8.19)

Here (in the Gaussian white noise case) we take

$\psi(x,t,\omega)=\sum_{\alpha}\Phi_{\alpha}(x,t)H_{\alpha}(\omega)\mbox{ and
its complex conjugate
}\psi^{*}(x,t,\omega)=\sum_{\alpha}\Phi^{*}_{\alpha}(x,t)H_{\alpha}(\omega),$

Applying Hermite transform we get a deterministic nonlinear Schrödinger
equation:

$i\frac{\partial}{\partial
t}\tilde{\psi}(x,t,z)+[(\Delta+{\mathcal{H}}[\Gamma](x,t,z)-\tilde{\psi}(x,t,z)\tilde{(\psi^{*})}(x,t,z)]\tilde{\psi}(x,t,z)=0,$
(8.20)

where

$\tilde{\psi}(x,t,\omega)=\sum_{\alpha}\Phi_{\alpha}(x,t)z^{\alpha}\mbox{ and
}\tilde{(\psi^{*})}(x,t,z)=\sum_{\alpha}\Phi^{*}_{\alpha}(x,t)z^{\alpha}.$
$\tilde{\psi}(x,0,z)={\mathcal{H}}[\Gamma_{0}](x,z),\hskip
7.22743ptx\in\mathbb{R}^{3},z\in\mathbb{C}^{\mathbb{N}}.$ (8.21)

Nonlinear Schrödinger equation with time dependent potential of the above type
(with fixed $z$ ) is studied in [15]. With an introduction to a suitable
smoothing for the noise in the form of $(I-\Delta_{x,t})^{-\gamma}$ the
Hermite transformed problem above in the framework of [15] (in particular
Assumption 1.3 regarding the potential in that paper) and deduce a unique
solution $\tilde{\psi}\in C([0,T];L^{2}(\mathbb{R}^{3}))\cap
L^{8/3}([0,T];L^{4}(\mathbb{R}^{3}))$ and hence inverse Hermite transform
gives:

######  Proposition 8.3.

For Gaussian initial data noise distribution
$\Gamma_{0}\in(\mathcal{S})^{-1}(L^{2}(\mathbb{R}^{d}))$ and Gaussian noise
forcing
$\Gamma\in(\mathcal{S})^{-1}(L^{\infty}_{\mbox{loc}}(\mathbb{R}\times\mathbb{R}^{d}))$,
there exists a unique solution to the stochastic nonlinear Schrödinger
equation with multiplicative space-time white noise as
$\psi\in(\mathcal{S})^{-1}(C([0,T];L^{2}(\mathbb{R}^{3}))\cap
L^{8/3}([0,T];L^{4}(\mathbb{R}^{3})))$. For Poisson and Lévy cases unique
correspondence map ${\mathcal{U}}$ gives a unique solution
$\psi\in(\mathcal{S})^{-1}(L^{\infty}([0,T];L^{2}(\mathbb{R}^{3}))\cap
L^{8/3}([0,T];L^{4}(\mathbb{R}^{3})))$

##  9\. Concluding remarks

In this paper we have casted a number of Laser propagation problems in random
media in the white noise distribution theory framework to stimulate further
research in their mathematical structure. Some of the most prominent problems
are selected for our discussion and a number of other nonlinear wave equations
that arise in Laser interaction with plasma such as the Schrodinger-Hartree
equation [33], Zakharov system [72] and Davey-Stewartson equation [22, 14] can
also be treated by the method initiated here when dealing with stochastic
medium effects. We briefly indicate below how one may proceed with Wick
quantization in these well-known nonlinear wave problems and for simplicity we
formulate them with spatially random initial data.

(1) Random (quantized) Schrödinger-Hartree equation:

$i\frac{\partial}{\partial
t}\varphi+\Delta\varphi=\pm[|x|^{n}\star(\varphi^{*}\diamond\varphi)]\diamond\varphi,\hskip
7.22743ptx\in\mathbb{R}^{d},\hskip 7.22743pt0<n<d,$ (9.1)
$\varphi(x,\omega,0)=\Gamma(x,\omega).$ (9.2)

Upon Hermite transform this system (for a fixed $z$) will result in the usual
deterministic Schrödinger-Hartree system measure initial data. We can apply a
smoothing to the noise in the form $(I-\Delta)^{-\gamma}$ and then use a
solvability theorem such as in [37] and utilize the analytic implicit function
theorem [7, 67] and inverse Hermite transform to deduce the solvability:

######  Proposition 9.1.

For $0<\gamma<\mbox{ min }(2,n)$, and Gaussian/Poisson/Lévy white noise
initial data
$\Gamma(\cdot,\cdot)\in(\mathcal{S})^{-1}(L^{2}(\mathbb{R}^{n}))$, there is a
unique solution to the (quantized) Schrödinger-Hartree equation
$\varphi\in(\mathcal{S})^{-1}\mathopen{}\mathclose{{}\left(C(\mathbb{R};L^{2}(\mathbb{R}^{n}))\cap
L^{8/\gamma}_{\mbox{loc}}(\mathbb{R};L^{\frac{4n}{2n-\gamma}}(\mathbb{R}^{n}))}\right)$.

(2) Random (quantized) Zakharov system in $\mathbb{R}^{d+1}$, $d=2,3$:

$i\frac{\partial}{\partial t}\varphi+\Delta\varphi=\varphi\diamond n,$ (9.3)
$[\frac{\partial^{2}}{\partial
t^{2}}-\Delta]n=-\Delta(\varphi^{*}\diamond\varphi),$ (9.4)
$\varphi(x,\omega,0)=\Gamma_{1}(x,\omega),n(x,\omega,0)=\Gamma_{2}(x,\omega),\mbox{
and }\frac{\partial}{\partial t}n(x,\omega,0)=\Gamma_{3}(x,\omega).$ (9.5)

Upon Hermite transform this system (for a fixed $z$) will result in the usual
deterministic Zakharov system measure initial data. We may have to apply a
smoothing to the noise in the form $(I-\Delta)^{-\gamma}$ before we can start
with a suitable solvability theorem such as in [29, 19] and utilize the
analytic implicit function theorem [7, 67] and inverse Hermite transform [36]
to deduce the solvability of the stochastic Zakharov model.

######  Proposition 9.2.

Suppose the initial data Gaussian white noise distribution satisfy
$(\Gamma_{1},\Gamma_{2},\Gamma_{3})\in(\mathcal{S})^{-1}\mathopen{}\mathclose{{}\left(H^{1/2}(\mathbb{R}^{d}))\times
L^{2}(\mathbb{R}^{d})\times H^{-1}(\mathbb{R}^{d})}\right)$ then there exists
a unique local-in-time solution to stochastic Zakharov equation with white
noise initial data such that

$(\varphi,n,\partial_{t}\varphi)\in(\mathcal{S})^{-1}\mathopen{}\mathclose{{}\left(C([0,T];H^{1/2}(\mathbb{R}^{d})\times
L^{2}(\mathbb{R}^{d})\times H^{-1}(\mathbb{R}^{d}))}\right).$ (9.6)

(3) Random (quantized) Davey-Stewartson system in $\mathbb{R}^{2+1}$: The
system was derived by Davey and Stewartson [22] and see [28] for a rigorous
study. We consider the Wick-quantized problem:

$i\frac{\partial}{\partial t}u+\delta\frac{\partial^{2}}{\partial
x^{2}}u+\frac{\partial^{2}}{\partial y^{2}}u=\chi(u^{*}\diamond
u)\diamond\varphi+bu\diamond\frac{\partial}{\partial x}\varphi,$ (9.7)
$\frac{\partial^{2}}{\partial x^{2}}\varphi+m\frac{\partial^{2}}{\partial
y^{2}}\varphi=\frac{\partial}{\partial x}(u^{*}\diamond u),$ (9.8)
$u(x,y,\omega,0)=\Gamma(x,y,\omega).$ (9.9)

The four parameters $\delta,\chi,b,m$ are real, $|\delta|=|\chi|=1$. The
system is classified as elliptic-elliptic, elliptic-hyperbolic, hyperbolic-
elliptic and hyperbolic-hyperbolic according to the respective signs of
$(\delta,m):(+.+),(+,-),(-,+),(-,-)$. Upon Hermite transform this system (for
a fixed $z$) will result in the usual deterministic Davey-Stewartson system
measure initial data and as discussed in the paper we can start with a
suitable solvability theorem such as in [28, 68] and utilize the analytic
implicit function theorem [7, 67] and inverse Hermite transform [36] to deduce
the solvability of the stochastic model:

######  Proposition 9.3.

For the Gaussian/Poisson/Lévy initial data white noise distribution
$\Gamma\in(\mathcal{S})^{-1}(L^{2}(\mathbb{R}^{2}))$, there exists a unique
local in time solution in the elliptic-elliptic and hyperbolic-elliptic cases
($m>0)$:

$u\in(\mathcal{S})^{-1}\mathopen{}\mathclose{{}\left(C([0,T^{*}[;L^{2}(\mathbb{R}^{2}))\cap
L^{4}((0,T^{*})\times\mathbb{R}^{2})}\right),$ (9.10)
$\nabla\varphi\in(\mathcal{S})^{-1}\mathopen{}\mathclose{{}\left(L^{2}((0,T^{*})\times\mathbb{R}^{2})}\right).$
(9.11)

Acknowledgment: The first author’s research has been supported by the U. S.
Air Force Research Laboratory through the National Research Council Senior
Research Fellowship of the National Academies of Science, Engineering and
Medicine.

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