Abstract:
 The BPE (Brownian particle equation) is an SPDE (stochastic partial differential
equation) of the first order including the white noise, at least in its principal
part, that is;
$
\partial_t u+\{a(t,x)+b(t,x)\dot{W}\}\partial_x u=\{A(t,x)+B(t,x)\dot{W}\}u+C(t,x)$,
where $\dot{W}=\frac{d}{dt}W_t$ is the white noise, namely the derivative of
the Brownian motion $W_t(\omega)$.(1)
The equation was first introduced in the article [2] ([3]) as a stochastic model for a transport phenomenon with infinite velocity of propagation. More precisely the model was expected to serve as a bridge between PDEs of two different types, that is the PDE of the first order on one hand and the PDE of parabolic type. At this point it is almost immediate to see that this expectation is realized not with the theory of $It\^{o}$ Calculus. In fact, suppose the multiplication terms $\partial_xu \times \dot{W}$ and $Bu \times \dot{W}$ in [1] are interpreted by $It\^{o}$ integral and take the
expectation on both sides of the equation (1), then we will see that the
average $v(t,x)=E[u(t,x,\omega)]$ comes to satisfy not a parabolic equation
but the following PDE of first order (2) and our expectation is failled;
$\partial_t v+a(t,x)\partial_x v=A(t,x)v+C(t,x)$,(2)
Our aim of constructing the bridge between two PDEs of different categories is realized in the framework of a calculus called {$\it noncausal calculus$} which is based on the
author's {$\it noncausal integral$}
([1],[4]  [9],[10] etc.).
In the two talks, we aim to show some remarkable properties of the BPE by limitting our discussion to the Cauchy problem of the BPE; In the first talk, we establish the fundametal results on the existence and uniqueness of classical solutions of Cauchy problem and on the relation with parabolic equations. We notice that the discussion is also accompanied with a brief introduction to the noncausal calculus. In the second talk, we will show some applications of the BPE theory to the theory of stochastic calculus and to some nonlinear PDEs in mathematical physics ([11]).
We will also refer to the very recent relevant topics in stochastic calculus
([12][15]).
References:
[1] Ogawa,S.: On a Riemann definition of the stochastic integral, (I), Proc.Japan
Acad. 46, pp.153157, 1970
[2] Ogawa,S.: A partial differential equation including the white noise as coefficient,
Z.W.verw.Geb., (1972) Springer Verlag Berlin,
[3] Ogawa,S.: Equation de Schrodinger et equation de particule brownienne, Kyoto
J of Math., 1975
[4] Ogawa,S.: Sur le produit direct du bruit blanc par luimeme, C.R.Acad.Sci.,Serie
A, Paris 288 (1979)
[5] Ogawa,S.: Quelques proprietes de l'integrale stochastique du type noncausal,
Japan J.Appl.Math., (1983)
[6] Ogawa,S. : Une remarque sur l'approximation de l'integrale stochastique du type noncausal par une suite des integrales de Stieltjes, Tohoku Math.J.,vol.36, No.1, pp.4148, 1984, Tohoku Univ.
[7] Ogawa,S. : The stochastic integral of noncausal type as an extension of the
symmetric integrals, JJAM (Japan J.Appl.Math) vol.2, No.1, pp.229240, 1985,
Kinokuniya
[8] Ogawa,S. : Sur la question d'existence des solutions de l'equation differentiale stochastique du type noncausal, Kyoto J.Math., 1986
[9] Ogawa,S. : Stochastic integral equations for the random fields, Seminaire de
Proba. vol.25, 1991. Springer
[10] Ogawa,S.: On the Brownian particle equations and the noncausal stochastic
calculus, Rendi Conti Acad.Nazionale delle Scienze detta dei XL, 119. vol.XXV
2001
[11] Ogawa,S. and KohatsuHiga,A.: A BPE model for Burgers' equation, RIMS
Publication, Kyoto University (2004)
[12] Ogawa,S.: On a stochastic Fourier transformation, Stochastics, Taylor & Fran
cis,iFirst article, 2012, 19, DOI:10.1080/17442508.2011.651621
[13] Ogawa,S. and Uemura,H.: On a stochastic Fourier coefficient, Journ.Theo
Probab, Springer (2013), DOI 10.1007/s109590120464x
[14] Ogawa,S. and Uemura,H.: Identification of noncausal Ito process from its SFCs, Bulletin Sciences Math., Elsevier (2014)
[15] Ogawa,S.: On a direct inversion formula for SFT, (to appear in Sankhya), (2014) DOI: 10.1007/s1317101400561
